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Stream: learning: questions

Topic: Resolutions and Descent

John Onstead (May 15 2024 at 14:13):

Continuing off of this discussion, I wanted to learn more about resolutions and hopefully eventually learn the full story that connects them with descent, local-global conditions, and potentially other concepts (such as cohomology, which also seems to be closely related to these).
To start off, my first question is about the relationship between simplicial resolutions and bar constructions/resolutions- I think I understand some of this but I just want to clarify so I'm completely clear on the connection. On nlab, they offer a way, given a comonad, to determine a simplicial resolution from it known as a comonadic resolution. Based on the article, it seems to generate something very similar to the bar construction, where the objects of the augmented simplex category are mapped to continuous applications of the monad to some object in the category the comonad is defined over. The article even also states that this comonadic resolution should be acyclic, just like we saw with bar constructions. First, is the comonadic resolution the same as the augmented simplicial object you get from the fact that Del_a^op is the walking comonoid (and so a functor will select a comonad in C's endofunctor category, which can then be followed by composition with an evaluation functor back to C), like with bar constructions? Are bar constructions a direct special case of a simplicial comonadic resolution (and if so, in what way- for instance, is it the special case when the comonoid corresponds to some monad T and the category C is EM(T))? If so, then what "nice" properties do bar constructions have that general comonadic resolutions do not, such that they "stand out" among all the comonadic resolutions?

John Baez (May 16 2024 at 13:51):

By the way, you can link to the nLab just using square brackets, making it easier for people to look at what you're talking about. Now I'll look to see if there's an article called [[comonadic resolution]].

John Baez (May 16 2024 at 13:59):

Close enough: the link works even though the article is called 'simplicial resolution'. Okay, now I get what you're talking about and can respond soon.

John Onstead (May 17 2024 at 06:46):

Sorry about that, I'll make sure to include links from now on! Look forward to hearing your thoughts.

John Baez (May 17 2024 at 10:12):

Okay. I don't know what a 'comonadic resolution' is, and the nLab article [[simplicial resolution]] doesn't say. So I'd be tempted to say it's just another name for the bar resolution. If anyone here knows any other 'comonadic resolutions' that aren't special cases of the bar resolution, they should say.

John Baez (May 17 2024 at 10:13):

I would say that there's a general concept of 'simplicial resolution' that has the bar resolution as an important special case - but also has many other interesting special cases.

John Baez (May 17 2024 at 10:15):

So, until someone tells us something interesting about 'comonadic resolutions', I recommend forgetting about comonadic resolutions and focusing on the concept of [[simplicial resolution]] and understanding 1) how the bar resolution is a special case, and 2) why the bar resolution is special.

John Baez (May 17 2024 at 10:17):

I already explained 2), to some extent: I pointed to @Todd Trimble's paper Bar construction, which proves

Theorem: B(M, M, X) is initial in the category of X-acyclic M-algebras.

John Baez (May 17 2024 at 10:18):

(Todd uses the term 'bar construction' instead of 'bar resolution', and I don't think it's realistic to hope that people draw a clear distinction between the two concepts.)

John Onstead (May 17 2024 at 10:41):

John Baez said:

So, until someone tells us something interesting about 'comonadic resolutions', I recommend forgetting about comonadic resolutions and focusing on the concept of [[simplicial resolution]] and understanding 1) how the bar resolution is a special case

"Comonadic resolution" appears 4 paragraphs above here and is mentioned in the "idea" section ("Simplicial resolutions can be constructed in various ways, for instance, by a comonad"), but in any case I think it's discussed in the context of how the bar resolution/construction is a special case of the simplicial resolution. This was to be my next question anyways: how does the notion of simplicial resolution given at the top of the nlab page ("a simplicial resolution of an object c in C is a simplicial object in C such that c is its colimit") have anything to do with bar constructions, monads, comonads, and so on? For instance, what if the monad/comonad is defined over a category that does not have colimits- then shouldn't it be even harder to connect these two notions?

John Baez said:

(Todd uses the term 'bar construction' instead of 'bar resolution', and I don't think it's realistic to hope that people draw a clear distinction between the two concepts.)

This is one reason I like the two-sided bar construction notation like B(M, M, X) since that seems to be unambiguous!

Todd Trimble (May 17 2024 at 13:29):

If someone says to me "comonadic resolution", I would understand them to be referring to what I have sometimes called the "bar construction" or the "bar resolution".

Todd Trimble (May 17 2024 at 13:29):

John B. referred to some very old notes of mine. The nLab article bar construction, which was largely written by me, can be read as a more polished version of at least part of those old notes. I also wrote an n-Category Cafe blog post on this back in the early days of the Cafe. Anyway, there should be more cross-linking between the various nLab entries.

Todd Trimble (May 17 2024 at 13:29):

To discuss these things cleanly, it's preferable to work with the "algebraist's simplex category", consisting of finite ordinals (including the empty ordinal) and order-preserving maps between them. I'll denote this as $\Delta_a$. It has one more object than the topologist's $\Delta$, which omits the empty ordinal. Under ordinal sum, $\Delta_a$ is a strict monoidal category whose monoidal unit is the empty ordinal $\mathbf{0}$. The terminal object $\mathbf{1}$, the one-element ordinal, is a monoid object, as terminal objects always are. $\Delta_a$ considered together with the monoid $\mathbf{1}$ is sometimes called "the walking monoid": given a (strict) monoidal category $\mathbf{M}$ together with a monoid object $M$, there is a unique strict monoidal functor $\Delta_a \to \mathbf{M}$ that carries the monoid structure on $\mathbf{1}$ to the monoid structure on $M$. Similarly, $\Delta_a^{op}$ is the "walking comonoid".

Todd Trimble (May 17 2024 at 13:30):

If $\mathbf{A}$ is a category and $C$ is a comonad on $\mathbf{A}$, then we may regard $C$ as a comonoid object in the category of endofunctors $[\mathbf{A}, \mathbf{A}]$, regarding the latter as a strict monoidal category under endofunctor composition. Since $\Delta_a^{op}$ is the "walking comonoid", we get a unique monoidal functor

$\Delta^{op} \to [\mathbf{A}, \mathbf{A}]$

that carries the comonoid $\mathbf{1}$ to the comonoid $C$. If, further, we have an object $A$ of $\mathbf{A}$, then we have a functor $\mathrm{ev}_A: [\mathbf{A}, \mathbf{A}] \to \mathbf{A}$ which evaluates an endofunctor $F$ at $A$, i.e., sends $F$ to $F(A)$. The evident composite

$\Delta_a^{op} \to [\mathbf{A}, \mathbf{A}] \overset{\mathrm{ev}_A}{\to} \mathbf{A}$

then defines a simplicial object in $\mathbf{A}$, and this is the basic construction we are discussing.

Todd Trimble (May 17 2024 at 13:30):

In simplicial language, it's actually what one calls an "augmented" simplicial object. It's augmented because we have this extra object $\mathbf{0}$ in $\Delta_a$, where the functor $\Delta_a^{op} \to \mathbf{A}$ takes $\mathbf{0}$ to $A$, and in general takes the $n$-element ordinal $\mathbf{n}$ to $C^n(A)$. We also have this extra map $\mathbf{0} \to \mathbf{1}$, and the functor $\Delta_a^{op} \to \mathbf{A}$ takes this map to an extra face map of the form $C(A) \to A$, and this is referred to as the augmentation. Sometimes the experts refer to the receiving object $A$ of the augmentation as being in (simplicial) degree $-1$, so that the face maps in low degree end with the shape

$\ldots C^2(A) \rightrightarrows C(A) \to A.$

Todd Trimble (May 17 2024 at 13:30):

Now suppose we have a monad $M$ on a category $\mathbf{E}$, and an $M$-algebra $A$. Let $\mathbf{A} = \mathbf{E}^M$ be the category of $M$-algebras. There's a forgetful functor $U: \mathbf{E}^M \to \mathbf{E}$, with a free functor $F: \mathbf{E} \to \mathbf{E}^M$ that is left adjoint to it. The free-forgetful adjunction induces a comonad $C = FU$ on $\mathbf{E}^M$. So then we can apply the basic construction above to this situation, resulting in an augmented simplicial object

$\Delta_a^{op} \to [\mathbf{E}^M, \mathbf{E}^M] \overset{\mathrm{ev}_A}{\to} \mathbf{E}^M$

and this is what I (nowadays) would call the bar construction on $A$.

Todd Trimble (May 17 2024 at 13:31):

In my idiolect, I make a distinction between this and what I would (nowadays) call the bar resolution on $A$. The bar resolution is the augmented simplicial object in $\mathbf{E}$ that we get by appending to this construction the forgetful functor $U$:

$\Delta_a^{op} \to [\mathbf{E}^M, \mathbf{E}^M] \overset{\mathrm{ev}_A}{\to} \mathbf{E}^M \overset{U}{\to} \mathbf{E}$

and this is what I would denote by $B(M, M, A)$. The object in simplicial degree $n-1$ is $U C^n A = UFU\ldots FUA$, but this is the same as $M^n UA$.

Todd Trimble (May 17 2024 at 13:31):

The reason for further composing with $U$ is that it allows us to pick up some extra degeneracy maps that play the role of a contracting homotopy, one that witnesses a homotopy equivalence between this augmented simplicial object, and the constant augmented simplicial object (that is constantly $A$ in every simplicial degree). It's this contracting homotopy that I think of as putting the "r" in "resolution". Namely, the contracting homotopy, given by maps

$B(M, M, A)_{n-1} \to B(M, M, A)_n,$

is given by components of the unit of the monad $M$:

$u_{M^n UA}: M^n UA \to M^{n+1} UA.$

Notice that these unit maps live in $\mathbf{E}$, not in $\mathbf{E}^M$ (unit maps are not algebra maps!). That's why we had to map down to $\mathbf{E}$.

Todd Trimble (May 17 2024 at 13:32):

By the way, also notice that in this situation, the augmentation map for the $M$-algebra $A$ is a coequalizer of the algebra face maps coming before it,

$FUFU A \rightrightarrows FU A \to A$

and in fact if we descend down to $\mathbf{E}$, the portion of the contracting homotopy gives a splitting of the fork

$UFUF A \rightrightarrows UFU A \to UA$

which makes this a split coequalizer, the basic sort of thing appearing in all those monadicity theorems. I thought I'd mention that, to tie this up with the nLab article on simplicial resolutions.

Todd Trimble (May 17 2024 at 13:33):

There is much more to say, but I'll stop here for now.

Todd Trimble (May 17 2024 at 15:32):

(And now I see [I've been away from this Zulip for a while] that I may have been repeating some stuff over in the effective descent thread. Oh well.)

John Onstead (May 17 2024 at 19:36):

Todd Trimble said:

(And now I see [I've been away from this Zulip for a while] that I may have been repeating some stuff over in the effective descent thread. Oh well.)

That's ok, it was really nice to see everything written out and come together and to have everything clarified. This seems to answer basically all of my questions and confusions up until now!

Todd Trimble said:

To discuss these things cleanly, it's preferable to work with the "algebraist's simplex category", consisting of finite ordinals (including the empty ordinal) and order-preserving maps between them. I'll denote this as Δa

That's interesting, I guess I assumed the "a" subscript stood for "augmented" and not "algebraist" since functors out of this category are augmented (co)simplicial objects!

Todd Trimble said:

The free-forgetful adjunction induces a comonad C=FU on EM. So then we can apply the basic construction above to this situation, resulting in an augmented simplicial object

This answers my question of the relation between comonadic resolution and bar construction- indeed the latter is a case of the former. I suspected it to be the case (I guess my thinking was on the right track) but it's nice to see it written out here!

Todd Trimble said:

In my idiolect, I make a distinction between this and what I would (nowadays) call the bar resolution on A

I admit the terminologies did give me some confusion when I was going through the nlab, but I think just being explicit about the bar construction notation IE B(M, M, A) can solve this.

Todd Trimble said:

the portion of the contracting homotopy gives a splitting of the fork

UFUFA⇉UFUA→UA

which makes this a split coequalizer, the basic sort of thing appearing in all those monadicity theorems. I thought I'd mention that, to tie this up with the nLab article on simplicial resolutions.

This must be how the bar construction/comonadic resolution induces a simplicial resolution- it creates a scenario where we can view the object UA (the object in E underlying the algebra A) as a colimit of a simplicial object (in this case the split coequalizer of the fork diagram). But it still makes me wonder about what happens in a category E that does not have colimits or coequalizers. Surely we could still do a bar construction since monads would still be definable, but would it just be that these constructions are not "simplicial resolutions" given in the nlab sense?

Todd Trimble said:

There is much more to say, but I'll stop here for now.

Thanks for your help! It's very much appreciated!

Todd Trimble (May 17 2024 at 21:05):

That's interesting, I guess I assumed the "a" subscript stood for "augmented" and not "algebraist" since functors out of this category are augmented (co)simplicial objects!

I picked up this subscript $a$ and this "explanation" from Street (I think it might be in his Myhill Lectures). I'm not sure I've seen anyone else use it (not counting myself of course) except for Street.

Todd Trimble (May 17 2024 at 21:05):

I admit the terminologies did give me some confusion when I was going through the nlab, but I think just being explicit about the bar construction notation IE B(M, M, A) can solve this.

Yes, this $B(M, M, A)$ is a special case of a two-sided bar construction $B(Y, M, X)$, where $M: \mathbf{C} \to \mathbf{C}$ has a monad structure, where $X: \mathbf{B} \to \mathbf{C}$ has a left $M$-module structure $\alpha: MX \to X$, and $Y: \mathbf{C} \to \mathbf{D}$ has a right $M$-module structure $\gamma: YM \to Y$. The 2-sided bar construction is a simplicial object of the form

$\Delta_a^{op} \to \hom(\mathbf{B}, \mathbf{D})$

where the component in simplicial degree $n$ is $Y M^{n+1} X$. This can be carried out in any 2-category or bicategory in which such suitable 1-cells $M, X, Y$ live; the hom here being a local hom-category.

Todd Trimble (May 17 2024 at 21:05):

I first learned of this from J.P. May's The Geometry of Iterated Loop Spaces. Like most busy mathematicians who use these things, he defines the items by giving a quick nuts-and-bolts definition of the face and degeneracy maps, but there is also a pleasant "walking two-sided structure" way to set up the construction that deserves to be better known, because it ties a lot of stuff together, including this stuff about decalage. Some day...

Todd Trimble (May 17 2024 at 21:06):

This must be how the bar construction/comonadic resolution induces a simplicial resolution- it creates a scenario where we can view the object UA (the object in E underlying the algebra A) as a colimit of a simplicial object (in this case the split coequalizer of the fork diagram).

Yes, that's what I was suggesting. But I think this simplicial resolution also plays nicely with higher-dimensional notions of (reflexive) coequalizer, like codescent object, by truncating the simplicial resolution further back, from simplicial degree 1 to simplicial degree 2. (I'll just treat this as a bookmark for now, because it would be good to make this idea more explicit -- but I don't plan on doing this now.)

Todd Trimble (May 17 2024 at 21:06):

But it still makes me wonder about what happens in a category E that does not have colimits or coequalizers. Surely we could still do a bar construction since monads would still be definable, but would it just be that these constructions are not "simplicial resolutions" given in the nlab sense?

If you have an algebra $\alpha: MA \to A$ of a monad, then independent of however many colimits there are in the surrounding category of algebras, you can always count on the fact that

$MMA \overset{M\alpha}{\underset{\mu A}{\rightrightarrows}} MA \overset{\alpha}{\to} A$

is a (reflexive) coequalizer in the category of algebras. So you'll get a simplicial resolution regardless.

John Baez (May 17 2024 at 21:49):

Thanks for joining in, Todd!

Mike Shulman (May 17 2024 at 22:02):

Todd Trimble said:

I think this simplicial resolution also plays nicely with higher-dimensional notions of (reflexive) coequalizer, like codescent object, by truncating the simplicial resolution further back, from simplicial degree 1 to simplicial degree 2. (I'll just treat this as a bookmark for now, because it would be good to make this idea more explicit -- but I don't plan on doing this now.)

The classical-homotopy-theoretic way to say this, in the untruncated case, is that the augmented simplicial object $B_\bullet(M,M,A)$ has an "extra degeneracy" and therefore is "simplicially contractible". The latter means that the unaugmented simplicial object $B_\bullet(M,M,A)$ is simplicially-homotopy-equivalent to the constant simplicial object $A$, and therefore they have equivalent geometric realizations. (I believe this is also in GoILS.) There's a more modern-looking categorical gloss on pp36-38 of https://arxiv.org/abs/1904.07004.

Todd Trimble (May 17 2024 at 22:04):

Thank you, Mike! I'll have a look at those pages.

Todd Trimble (May 17 2024 at 22:09):

The "walking two-sided structure" I alluded to here is similar in flavor to $\mathcal{Adj}$, but it has instead three objects, not two. But I'm sure you can guess what I have in mind.

If you can: do you know where it might be written down?

Mike Shulman (May 17 2024 at 22:12):

Hmm, not offhand, sorry.

Todd Trimble (May 17 2024 at 22:19):

Well, for anyone reading this, I don't have to be so coy about the nature of this walking two-sided structure. If the three objects are $0, 1, 2$, then the hom-categories go something like this (going by memory):

• $\hom(0, 1)$ is the category of nonempty finite ordinals and bottom-preserving monotone maps.
• $\hom(1, 1)$ is the category $\Delta_a$
• $\hom(1, 2)$ is the category of nonempty finite ordinals and top-preserving maps
• $\hom(0, 2)$ is the category of nonempty finite ordinals and maps that preserve both top and bottom
• There are no maps $i \to j$ if $i > j$

(Not a complete description of the 2-category, just a rough idea.)

John Onstead (May 18 2024 at 03:05):

Todd Trimble said:

Well, for anyone reading this, I don't have to be so coy about the nature of this walking two-sided structure. If the three objects are 0,1,2, then the hom-categories go something like this (going by memory):

I don't know if there's any papers with this written down, but I did find a decent writeup on the nlab here. I found it quite informative for understanding how these constructions work in a generic 2-category (IE, outside Cat).

Todd Trimble said:

If you have an algebra α:MA→A of a monad, then independent of however many colimits there are in the surrounding category of algebras, you can always count on the fact that

MMAμA⇉​Mα​MA→αA

is a (reflexive) coequalizer in the category of algebras. So you'll get a simplicial resolution regardless.

This is interesting to know and really adds to how fascinating bar constructions really are. If I interpret this correctly, so long as you can define some monad, any monad at all, you automatically know that EM(T) has at least some coequalizers (even if it doesn't have all coequalizers)! If I didn't know any better, I'd even say that bar constructions are "magical" in this way. If you give any monad at all on your category, it doesn't have to be special or anything, it could just be any old monad: you automatically now have a way of expressing any algebra of that monad via a colimit in EM(T)! That is an amazing result, especially because so many algebraic constructions (groups, monoids, etc.) can be given as monad algebras.
But if I'm understanding things correctly (please correct me if I'm not), these equalizers are split equalizers which means they are preserved by all functors, including U. Thus, descending back down to the original category from EM(T), this equalizer is preserved. This seems even more magical for two reasons: first, if you are able to define any monad at all on your category, in any way, you automatically know your category has at least some equalizers. But second, it means that any object in any category with a monad on it can be expressed via colimits, and in fact if your category has n monads on it, you now have up to potentially n ways of expressing any old object in your category via colimits! This just makes my head spin!

So long as I'm understanding things correctly, I think my first batch of questions are settled. But this is just the tip of the iceberg as I eventually want to get to descent so I'll be sure to return with more questions about the next concept I want to uncover soon!

Todd Trimble (May 18 2024 at 03:07):

I don't know if there's any papers with this written down, but I did find a decent writeup on the nlab here. I found it quite informative for understanding how these constructions work in a generic 2-category (IE, outside Cat).

You know who wrote that, don't you? ;-)

Todd Trimble (May 18 2024 at 03:11):

But if I'm understanding things correctly (please correct me if I'm not), these equalizers are split equalizers

No, they're not split coequalizers in the category of algebras. But they are $U$-split coequalizers (the application of $U$ to them gives you a fork that admits a splitting, making it a split coequalizer). One uses the units to obtain the splittings, as I was saying in the long series of posts.

John Onstead (May 18 2024 at 03:13):

Ah, my bad, I'd forgotten what a split coequalizer was! There's so many concepts and definitions in category theory sometimes it's hard to keep track of them all!

Mike Shulman (May 18 2024 at 05:05):

Note that all this business with $U$-split coequalizers is also the starting point of the [[monadicity theorem]].

John Onstead (May 18 2024 at 07:13):

So here's my next question which is a natural extension of the previous discussion. I wanted to extend this discussion, which has currently tied together bar constructions/resolutions and simplicial resolutions, with the concept of a [[canonical resolution]]. At first I thought this was the same thing as a bar construction because it is another instance of resolution associated to a monad just like a bar construction, but the top of the page clarifies that canonical resolutions need not be acyclic or contractible, which contrasts with bar constructions which are always acyclic. That said, the concept of [[monadic cohomology]] appears on both the nlab page for simplicial resolutions (where it is mentioned under the comonadic resolutions) and on the canonical resolution page. This leaves me wondering: is there a relationship between these notions (bar vs canonical) and if so what is it? Second, what are some "use cases" of both structures that coincide and some that diverge?

Nathanael Arkor (May 18 2024 at 08:01):

Todd Trimble said:

Well, for anyone reading this, I don't have to be so coy about the nature of this walking two-sided structure. If the three objects are $0, 1, 2$, then the hom-categories go something like this (going by memory):

Is there a slick construction of this 2-category as some kind of classifier for lax functors equipped with lax (co)cones, analogously to how the free-standing monad can be obtained by considering monads as lax functors from $1$, and applying the lax functor classifier?

Nathanael Arkor (May 18 2024 at 08:05):

John Onstead said:

This is interesting to know and really adds to how fascinating bar constructions really are. If I interpret this correctly, so long as you can define some monad, any monad at all, you automatically know that EM(T) has at least some coequalizers (even if it doesn't have all coequalizers)! If I didn't know any better, I'd even say that bar constructions are "magical" in this way. If you give any monad at all on your category, it doesn't have to be special or anything, it could just be any old monad: you automatically now have a way of expressing any algebra of that monad via a colimit in EM(T)! That is an amazing result, especially because so many algebraic constructions (groups, monoids, etc.) can be given as monad algebras.

Relatedly, there's a coherent way to see the algebras of a monad as being presented by colimits of free algebras, via the pullback theorem, which expresses the entire category of algebras as a certain subcategory of the presheaf category on the category of free algebras.

John Baez (May 18 2024 at 08:34):

John Onstead said:

Todd Trimble said:

Well, for anyone reading this, I don't have to be so coy about the nature of this walking two-sided structure. If the three objects are 0,1,2, then the hom-categories go something like this (going by memory):

I don't know if there's any papers with this written down, but I did find a decent writeup on the nlab here. I found it quite informative for understanding how these constructions work in a generic 2-category (IE, outside Cat).

Todd Trimble said:

If you have an algebra $\alpha: MA \to A$ of a monad, then independent of however many colimits there are in the surrounding category of algebras, you can always count on the fact that

$MMA \overset{M\alpha}{\underset{\mu A}{\rightrightarrows}} MA \overset{\alpha}{\to} A$

is a (reflexive) coequalizer in the category of algebras. So you'll get a simplicial resolution regardless.

This is interesting to know and really adds to how fascinating bar constructions really are. If I interpret this correctly, so long as you can define some monad, any monad at all, you automatically know that EM(T) has at least some coequalizers (even if it doesn't have all coequalizers)! If I didn't know any better, I'd even say that bar constructions are "magical" in this way. If you give any monad at all on your category, it doesn't have to be special or anything, it could just be any old monad: you automatically now have a way of expressing any algebra of that monad via a colimit in EM(T)! That is an amazing result, especially because so many algebraic constructions (groups, monoids, etc.) can be given as monad algebras.

I wouldn't say this is "magical" and I wouldn't say it's a property of the bar construction per se. I think what it's saying is that monads describe gadgets that admit presentations in terms of generators and relations, so any algebra of a monad is, more or less by definition, obtainable from a free algebra by taking a coequalizer. This is easiest to talk about for monads on the category of sets, but the theory of monads distills that idea so it applies more generally.

For example suppose $M$ is the "free group" monad on the category of sets. Then if $A$ is any group, there's a free group $MA$ with all the elements of $A$ as generators, and $\alpha MA \to A$ expresses $A$ as a quotient of $MA$. Moreover it's not just that we know $\alpha$ is an epimorphism with some kernel giving us the relations in our presentation. We know

$MMA \overset{M\alpha}{\underset{\mu A}{\rightrightarrows}} MA \overset{\alpha}{\to} A$

is a coequalizer, and this tells us what the relations are. Elements $x \in MMA$ are formal products of elements in our free group $MA$, and the relations are all the equations of the form

$(M\alpha)(x) = (\mu A)(x)$

This is a very "wasteful" presentation of our group $A$, since we've taken all possible group elements as generators and all possible relations between these as relations. But it's a "natural" presentation: finding it didn't require making arbitrary choices, so if we have a group homomorphism $f: A \to B$ we get a naturality diagram that looks like

$MMA \overset{M\alpha}{\underset{\mu A}{\rightrightarrows}} MA \overset{\alpha}{\to} A$

on top and

$MMB \overset{M\alpha}{\underset{\mu B}{\rightrightarrows}} MB \overset{\alpha}{\to} B$

with vertical arrows giving two naturality squares.

John Baez (May 18 2024 at 08:41):

Well, maybe this is magical: it's great stuff. But I would say this is the magic of monads, not the bar construction - since we haven't yet invoked the simplicial object that emerges when we go further and start looking at $MMMA,$ $MMMMA,$ etc. Indeed, from this perspective it seems completely unnecessary to bring in that extra stuff. $MA$ is describing generators, $MMA$ is describing relations, but $MMMA$ is describing relations between relations, and $MMMMA$ is describing relations between relations between relations - and who gives a damn about those?

Well, it turns out they're important! Relations between relations are called "syzygies", and Hilbert's syzgy theorem showed how important these are back in 1890!

This ultimately led to a more "homological" or "homotopical" approach to monads and their algebras, in which the bar construction becomes important. First people looked at monads on abelian categories, where simplicial objects are the same as chain complexes - that's homological algebra. Historically part of what happened is that Hilbert's student Noether invented chain complexes around 1926-1928 while listening to some lectures on topology by Alexandroff and Hopf. Later Eilenberg and Mac Lane invented the bar construction. Only later did people realize that chain complexes are secretly the same as simplicial abelian groups! In1958 Godement invented the bar construction for a general monad.

At some point people noticed the bar construction was also interesting in Set, where simplicial objects are a way of describing homotopy types - that's what Quillen called homotopical algebra.

John Onstead (May 18 2024 at 20:56):

Nathanael Arkor said:

Relatedly, there's a coherent way to see the algebras of a monad as being presented by colimits of free algebras, via the pullback theorem, which expresses the entire category of algebras as a certain subcategory of the presheaf category on the category of free algebras.

That's a very interesting connection, I'll have to look more into it!

John Baez said:

Well, maybe this is magical: it's great stuff. But I would say this is the magic of monads, not the bar construction - since we haven't yet invoked the simplicial object that emerges when we go further and start looking at MMMA, MMMMA, etc. Indeed, from this perspective it seems completely unnecessary to bring in that extra stuff. MA is describing generators, MMA is describing relations, but MMMA is describing relations between relations, and MMMMA is describing relations between relations between relations - and who gives a damn about those?

I am always amazed at some of the results you can get from category theory, especially the unexpected ones since any theorem about categories in turn can apply across all mathematics.

I think I'll come back to continue this post in a week or so, I have questions about other things in the meantime! Thanks for the help so far!

John Onstead (May 22 2024 at 06:37):

On second thought I wanted to clarify a few more things before I moved on. First, I wanted to bring "density" to all this colimit talk. "Density" is when you have some objects in a category (a subcategory of your category) that you can use to express any other object in the category in terms of via colimits, and this notion is intrinsically related to that of Kan extensions. The algebras of a monad being presented by free algebras means that for any EM(T), EM(T) is dense in the subcategory defined by Kleisli(T), IE, the image of the free functor into EM(T).
Interestingly, there's a "density theorem" known as the "Co-Yoneda lemma" that states how any presheaf category is dense in the Yoneda embedding. If it really is true that by the pullback theorem mentioned above, the functor given in the pullback EM(T) -> Presheaf(Kleisli(T)) is actually an embedding, then we can actually apply the Co-Yoneda lemma to get yet another way of expressing algebras in EM(T) in terms of free algebras. In this case, the Co-Yoneda gives a way to express any object in Presheaf(Kleisli(T)) via colimits of objects in Kleisli(T), and so if EM(T) is a subcategory of Presheaf(Kleisli(T)), that would include all objects in it too! I'm not sure how this colimit expression relates to the presentation one, but I just thought I'd share it because it's a cool fact!

John Onstead (May 22 2024 at 06:39):

But then, descending back down to C from EM(T), we get the split coequalizers. Does this mean that any arbitrary category C is dense in the image of any arbitrary monad T defined on it? I'm a little iffy on this one since not all objects in a category for some arbitrary monad T might be underlying objects for some algebra of T. For instance, a morphism TA -> A might not always exist? What do you think?
Lastly, I wanted to ask for clarification on something. I might have already gotten the answer but I may have forgotten it, but here it is anyways. If you get the bar construction, its full simplicial object, of an algebra, you can truncate it to get the fork diagram. We know that the target algebra can be expressed as both a colimit of the full simplicial object and the truncated one (in the latter case, the coequalizer). What about arbitrary truncations (IE, the ones including the "syzygies" as discussed above)?

Nathanael Arkor (May 22 2024 at 07:42):

John Onstead said:

The algebras of a monad being presented by free algebras means that for any EM(T), EM(T) is dense in the subcategory defined by Kleisli(T), IE, the image of the free functor into EM(T).

That algebras are presented by free algebras expresses that the comparison functor $i_T : \text{Kl}(T) \to \text{Alg}(T)$ is dense, which people might abbreviate to "the category of free algebras is dense in the category of algebras", i.e. the subcategory is what's dense, not the supercategory.

Nathanael Arkor (May 22 2024 at 07:44):

But density of the comparison functor is exactly the notion of "every algebra is presented as a colimit of free algebras" expressed by the pullback theorem, because density of the comparison is equivalent to full faithfulness of its nerve $\text{Alg}(T) \to \widehat{\text{Kl}(T)}$, which is (up to isomorphism) the functor appearing in the pullback diagram.

John Onstead (May 23 2024 at 06:55):

Thanks! I admit I was a little confused at first by the usage of the term "nerve" but it seems you are using it to talk about a restricted Yoneda Embedding. But this did bring up something else I meant to ask when a discussion above was happening about realizations (since nerves and realizations are adjoint). Both nerve realizations and resolutions give simplicial things for some category: the realization gives some simplicial set for its objects while the resolution gives a simplicial object that "resolves" via colimits to the objects in your category. Are realizations and resolutions related at all (besides both having to do with simplicial things and sounding similar in name), and if so, what are some scenarios in which a realization doubles as a resolution (IE, would geometric realizations count)?
Indeed it seems a fully faithful restricted Yoneda embedding and density are interchangeable, which is fascinating to me because it means that every instance of density can be recontextualized as part of the canonical case of density- that is, the presheaf category being dense in the Yoneda embedding. This interchangeability in one direction is easy to see due to the co-Yoneda lemma: if you have a fully faithful restricted Yoneda embedding, you can apply the co-Yoneda lemma and get the density property, but what is much more interesting to me is that it also somehow applies in the other direction- given any functor with density property, it has a fully faithful restricted Yoneda embedding!
I also have a related question. Every monad gives rise to a comonad named the density comonad by taking the left Kan extension of the monad along itself. For which kinds of monad are the density comonad isomorphic to the identity comonad?

Nathan Corbyn (May 23 2024 at 07:34):

For the last part: the left Kan extension of a functor along itself is the identity iff the functor is dense. So the density comonad of a monad is the identity iff the monad itself is dense as a functor. (See here.)

John Baez (May 23 2024 at 14:42):

John Onstead said:

If you get the bar construction, its full simplicial object, of an algebra, you can truncate it to get the fork diagram. We know that the target algebra can be expressed as both a colimit of the full simplicial object and the truncated one (in the latter case, the coequalizer). What about arbitrary truncations?

I think the colimit of any one of the truncations you're talking about gives the algebra, at least if the truncation is big enough to include the fork.

Mike Shulman (May 23 2024 at 15:49):

At least, if by "colimit" you mean an ordinary 1-categorical colimit.

Mike Shulman (May 23 2024 at 15:50):

If you want an $n$-categorical colimit, you probably have to truncate at the (n+1)st stage or above.

John Onstead (May 24 2024 at 07:45):

Thanks for the help on these questions so far! I appreciate it!
I had a question about geometric realization and density. So a geometric realization is a process that takes in a simplicial set del^op -> Set, does some sort of coend machinery on it, and then spits out some topological space. That is, there's a functor GR: [del^op, Set] -> Top. I'm not sure if every topological space is geometrically realizable, so let's restrict our attention to the image subcategory of GR and call it Real, the subcategory of Top on all geometrically realizable topological spaces.
But notice that every simplicial set can be transformed into a simplicial topological space: that is, you can compose the functors del^op -> Set -> Real via the inclusion i: Set -> Real, that sends a set to a discrete topological space. You can then presumably do the same sort of coend things and get any other object in Real from these simplicial topological spaces. But since a set-enriched coend can be written as a colimit, doesn't this imply that Set is a dense subcategory of Real?

John Baez (May 24 2024 at 09:02):

John Onstead said:

But since a set-enriched coend can be written as a colimit, doesn't this imply that Set is a dense subcategory of Real?

Are you asking if every topological space that's a geometric realization of a simplicial set can be expressed as a colimit of discrete topological spaces? That seems really wrong to me, because at least in Top I think a colimit of discrete topological spaces is discrete again - right? I don't see how you could get something interesting like a circle. So it seems the only way out would be if colimits in your category Real behave much differently than I expect from Top.

John Baez (May 24 2024 at 09:11):

Mike Shulman said:

At least, if by "colimit" you mean an ordinary 1-categorical colimit.

If you want an $n$-categorical colimit, you probably have to truncate at the (n+1)st stage or above.

I'd been trying to avoid such $n$-categorical considerations to avoid scaring @John Onstead, but this touches on why the bar construction is really important. The bar construction, applied to an algebra of a monad, magically gives an infinite-dimensional 'puffed up' version of that algebra where all equations holding in the algebra have been replaced by edges in a simplicial object, and all equations between equations have been replaced by triangles, and so on. So it hurls us from the world of 1-category theory into the world of infinite-dimensional geometry.

If we are working purely 1-categorically with monads and their algebras, a 1-dimensional truncation of the bar construction is enough for most purposes: the fork $TTA \rightrightarrows TA \to A$. It's when we start working $n$-categorically, or start doing homological algebra, or homotopy theory - which are all just different ways of bringing in higher dimensions - that we need to go further, and use higher truncations of the bar construction, and ultimately to the whole bar construction. So we can't really understand the point of the bar construction until we get into some of this stuff.

Todd Trimble (May 24 2024 at 11:45):

There are a number of things to say here.

One is a somewhat technical aside, that it's best in practice to replace the category $\mathsf{Top}$ of ordinary topological spaces with a "convenient category" of spaces, such as some variation of compactly generated spaces (compactly generated weakly Hausdorff is usually a good choice). The basic idea is that for various reasons, one really wants geometric realization to preserve finite products, but this will not hold in the generality one wants, for example if one wants to take geometric realizations of simplicial spaces, unless one works with a convenient category from the get-go. But I think we can more or less put a pin through this for now, and maybe come back to it later. Whatever the desirable category of spaces, let's just call it $\mathsf{Top}$.

Todd Trimble (May 24 2024 at 11:45):

Notice that the geometric realization of a simplicial set is a CW-complex, and most topological spaces are not so nice as that. CW-complexes make up only a very tiny fraction of all spaces.

Todd Trimble (May 24 2024 at 11:45):

You should think of this "mysterious" coend though as a recipe or specification for gluing together affine simplices

$\{(x_0, x_1, \ldots, x_n)|\; 0 = x_0 \leq x_1 \leq \ldots \leq x_n = 1\}$

in their ordinary Euclidean topology. Not the discrete topology! In your question, it sounds like you're pulling a fast one and trying to consider the coend as a colimit of discrete spaces, but it's certainly not that. Each time you apply the geometric realization, you'll be dealing with these honest-to-god affine simplices, and you just can't get those as colimits of discrete spaces. John is right: the colimit of a diagram of discrete spaces will again be discrete, because the discrete space functor from sets to spaces preserves colimits, because it has a right adjoint, namely the forgetful functor from spaces back to sets.

Todd Trimble (May 24 2024 at 11:46):

In case it helps, another way to describe geometric realization is that it is the left Kan extension of the affine simplex functor

$\Delta \to \mathsf{Top}$
along the Yoneda embedding $\Delta \to \mathsf{Set}^{\Delta^{op}}$. It takes a presheaf, which is a colimit of representables a la the "co-Yoneda" embedding you were talking about a moment ago, to the corresponding colimit of affine simplexes in $\mathsf{Top}$. This is all connected with the way that presheaf categories are "free cocompletions" of their (let us say small) generating categories, in this case $\Delta$.

Todd Trimble (May 24 2024 at 12:00):

(By the way, yes I put "mysterious" in somewhat snarky quotes. The snark is not seriously intended, but it does seem to be the case that coends and ends are stumbling blocks for so many people, and I personally think it's a really good idea for a student of category theory to demystify them at their earliest convenience, especially by getting on close terms with some of the important examples. The most important examples of coends for me are what I would call (generalized) "tensor products". For now I can point to an MO discussion where the opening example I give is actually geometric realization.)

John Onstead (May 24 2024 at 22:12):

Todd Trimble said:

Notice that the geometric realization of a simplicial set is a CW-complex, and most topological spaces are not so nice as that. CW-complexes make up only a very tiny fraction of all spaces.

It seems this means the category "Real" I described above is best given as the category of CW complexes. Good to know!

Todd Trimble said:

Each time you apply the geometric realization, you'll be dealing with these honest-to-god affine simplices, and you just can't get those as colimits of discrete spaces. John is right: the colimit of a diagram of discrete spaces will again be discrete

This seems to clear up my confusion. I figured something was up with my conclusion since I suspected something like "colimit of a discrete space will again be discrete"! It seems what I was doing was completely ignoring the influence of the affine simplices, when in reality they are arguably the more important ingredients.

So in this regard, would you say that geometric realizations aren't actually special cases of simplicial resolutions, and that they are two distinct concepts that just so happen to both deal with simplicial-y things and colimit-y things? Sorry if it seems I'm asking the same question over and over but I really want to make sure I have the precise relation between these two ideas made extremely explicit and concrete!

John Onstead (May 24 2024 at 22:21):

John Baez said:

The bar construction, applied to an algebra of a monad, magically gives an infinite-dimensional 'puffed up' version of that algebra where all equations holding in the algebra have been replaced by edges in a simplicial object, and all equations between equations have been replaced by triangles, and so on. So it hurls us from the world of 1-category theory into the world of infinite-dimensional geometry.

If we are working purely 1-categorically with monads and their algebras, a 1-dimensional truncation of the bar construction is enough for most purposes: the fork TTA⇉TA→A. It's when we start working n-categorically, or start doing homological algebra, or homotopy theory - which are all just different ways of bringing in higher dimensions - that we need to go further, and use higher truncations of the bar construction, and ultimately to the whole bar construction. So we can't really understand the point of the bar construction until we get into some of this stuff.

This certainly clears things up for me, and indeed I feel like somewhere I'm missing a "big picture" of why simplicial objects are relevant and of what exactly they are doing. It seems to tie this discussion of simplicial resolutions and objects into the even broader discussion of categorical homotopy theory (that branch of category theory concerned with model categories, homotopy limits, weak equivalences, fibrant objects, and all the other fun stuff that comes with those things!) The basic gist I get is that categorical homotopy theory tells us to reinterpret the objects of some categories as acting like spaces that we can do homotopy-like things with. Since there's a close relationship between simplicial things and at least some topological spaces, it makes sense that this connection would "generalize" into this broader setting. I hope I'm understanding this right. But even if so I don't know if I'm ready to get into all this just yet!

Todd Trimble (May 25 2024 at 01:41):

John Onstead said:

Todd Trimble said:

Notice that the geometric realization of a simplicial set is a CW-complex, and most topological spaces are not so nice as that. CW-complexes make up only a very tiny fraction of all spaces.

It seems this means the category "Real" I described above is best given as the category of CW complexes. Good to know!

But I didn't say that! All I meant is one direction: that the realization of a simplicial set is a CW-complex. Not that every CW-complex is homeomorphic to the geometric realization of a simplicial set. (The MO discussion here discusses this; it's not a trivial question.)

Todd Trimble (May 25 2024 at 01:41):

However, geometric realization $R$ is left adjoint to "singularization" $S$, and the unit and counit of this adjunction are equivalences in a homotopical sense. What this means on the topological side is that the counit induces an isomorphism on connected components, and on each connected component, an isomorphism on all homotopy groups. If you further restrict to CW-complexes, this amounts to an actual homotopy equivalence. And this is what people mostly care about, much more than homeomorphism. The homotopical statements are part of the lore of (Quillen) model categories.

Todd Trimble (May 25 2024 at 01:52):

So in this regard, would you say that geometric realizations aren't actually special cases of simplicial resolutions, and that they are two distinct concepts that just so happen to both deal with simplicial-y things and colimit-y things? Sorry if it seems I'm asking the same question over and over but I really want to make sure I have the precise relation between these two ideas made extremely explicit and concrete!

They aren't comparable concepts, because the output of a simplicial resolution is, well, a simplicial object of some form, whereas geometric realization takes a simplicial object as input, and outputs a topological space. But of course they both have something to do with colimits and simplicial objects.

It is nice that the [[bar construction]] can be regarded as a left adjoint of some sort, as described on that nLab page. As can [[geometric realization]]. But that's just an aspect of the Mac Lane motto "adjoints are everywhere". :-)

John Onstead (May 25 2024 at 04:00):

Todd Trimble said:

But I didn't say that! All I meant is one direction: that the realization of a simplicial set is a CW-complex. Not that every CW-complex is homeomorphic to the geometric realization of a simplicial set.

That's my bad! So "Simplicial-izable" spaces must be an even more restricted notion than CW complexes.
Thanks for the clarifications!

John Onstead (May 29 2024 at 06:25):

The next topic in this web of resolution-y concepts I wanted to cover was canonical resolutions. These also appear to be associated to a monad like the bar construction, and indeed the nlab page gives a seeming connection here. However I'm not exactly sure what this is trying to say. Are bar constructions directly a special case of canonical resolutions? Or are they related in some other way? If they are a special case, is it in a different sense to how bar constructions are a special case of simplicial resolutions as we covered above?

John Baez (May 29 2024 at 08:15):

There are lots of canonical resolutions used in homological algebra, and this page doesn't talk about any of them except the bar resolution. If you grab an introduction to homological algebra like Rotman's Homological Algebra you'll meet a bunch of them. In many cases you can get the same results - that is, calculate the same homology and cohomology groups - as you do with the bar resolution (which is the mother of all canonical resolutions, since it applies in many categories). The point of these other resolutions is that they're more "efficient": they produce simplicial objects that are "smaller" and more easy to calculate with.

John Baez (May 29 2024 at 08:16):

The only canonical resolution discussed on this page is the bar resolution, but confusingly it's introduced 3 times, under the names "monadic cohomology", "resolution of a T-algebra" and "the bar resolution for abelian groups". So it's bar, bar, bar all the day long here.

John Baez (May 29 2024 at 08:32):

To see some other resolutions, go to Section 9.3 of the new edition of Rotman's book, called "bar resolutions", and compare the bar resolution to the "homogeneous resolution" of $\mathbb{Z}$ as a trivial $G$-module for a group $G$. The bar resolution gets the job done, but the homogeneous resolution is more efficient.

Kevin Carlson (May 29 2024 at 17:25):

John Baez said:

So it's bar, bar, bar all the day long here.

How barbaric.

John Onstead (May 30 2024 at 06:56):

Thanks for the clarifications!
Before getting into monadic cohomology, I wanted to make sure I fully understand how to properly dualize the bar construction. On the nlab page for canonical resolution, it gives the canonical resolution of a monad as an augmented cosimplicial object as opposed to an augmented simplicial object (that we would get for a comonad). This is because, just as del_a^op is the walking comonoid, del_a is the walking monoid. But then, my question is: wouldn't the monadic resolution express an object in the category as a limit (not a colimit) of a cosimplicial object relative to the monad, dually to how the comonadic resolution expresses an object in a category as a colimit of a simplicial object relative to a comonad? Wouldn't a better term be a "coresolution" if so?

John Baez (May 30 2024 at 07:32):

Before "getting into monadic cohomology", or discussing how to dualize the concept of comonadic resolution, I would want to look at examples of what people actually use it for. Unlike the bar construction, which is ubiquitous, I don't have a good sense of this.

Does anyone here know a good concrete application of this?

If not, I could read some of the references at [[canonical resolution]].

Tim Hosgood (May 30 2024 at 10:07):

John Onstead said:

Wouldn't a better term be a "coresolution" if so?

not sure if this is completely made up and ahistorical, but I would guess that this is related to how algebraists/algebraic geometers call(ed) limits and colimits (of certain forms) "inverse limits" and "direct limits", without using the "co" prefix. for example, both projective resolutions and injective resolutions are called "resolutions", but the latter looks more like what you might want to call a coresolution

Tim Hosgood (May 30 2024 at 10:08):

a question for the category theorists here: are projective/injective resolutions in any form explained by some sort of monadic story? or are these really resolutions in a different sense? (my naive guess would be that to bring monadic resolutions into the story you do dold–kan somewhere)

Todd Trimble (May 30 2024 at 11:23):

John Onstead said:

Thanks for the clarifications!
Before getting into monadic cohomology, I wanted to make sure I fully understand how to properly dualize the bar construction. On the nlab page for canonical resolution, it gives the canonical resolution of a monad as an augmented cosimplicial object as opposed to an augmented simplicial object (that we would get for a comonad). This is because, just as del_a^op is the walking comonoid, del_a is the walking monoid. But then, my question is: wouldn't the monadic resolution express an object in the category as a limit (not a colimit) of a cosimplicial object relative to the monad, dually to how the comonadic resolution expresses an object in a category as a colimit of a simplicial object relative to a comonad? Wouldn't a better term be a "coresolution" if so?

I think of it less in terms of a dualization procedure, and more as looking at the other end of the adjunction between a category and an Eilenberg-Moore category $U: \mathbf{E}^M \leftrightarrows \mathbf{E}: F$. It really would be a dual construction if we knew that $\mathbf{E}$ is the category of coalgebras of the comonad $FU$, but we don't always know that. If we did know that in the case of some example we were looking at, then indeed we would get the object as the limit of that cosimplicial object.

Todd Trimble (May 30 2024 at 11:24):

I see all this in very simple terms. The comonad $FU: \mathbf{E}^M \to \mathbf{E}^M$ gives you a comonoid in the endofunctor category $[\mathbf{E}^M, \mathbf{E}^M]$, and we get an induced functor

$\Delta^{op} \to [\mathbf{E}^M, \mathbf{E}^M]$

and the ordinary bar construction is gotten by following this by evaluation $\mathrm{ev}_A: [\mathbf{E}^M, \mathbf{E}^M] \to \mathbf{E}^M$ at some object $A$ of $\mathbf{E}^M$. The monad $UF: \mathbf{E} \to \mathbf{E}$ gives you a monoid in the endofunctor category $[\mathbf{E}, \mathbf{E}]$, and we get an induced functor

$\Delta \to [\mathbf{E}, \mathbf{E}]$

and the "monadic resolution" as defined on the nLab page is gotten by following this by evaluation $\mathrm{ev}_X: [\mathbf{E}, \mathbf{E}] \to \mathbf{E}$ at some object $X$ of $\mathbf{E}$.

John Baez (May 30 2024 at 11:46):

I know lots of concrete applications of the bar construction applied to some object of $\mathbf{E}^M$. What are some concrete applications of the monadic resolution of some object of $\mathbf{E}$?

John Baez (May 30 2024 at 11:50):

Tim Hosgood said:

a question for the category theorists here: are projective/injective resolutions in any form explained by some sort of monadic story?

Many of the most famous projective resolutions are instances of the bar construction applied to an object of some Eilenberg-Moore category $\mathbf{E}^M$. I could regale you with examples if desired.

John Baez (May 30 2024 at 11:51):

Hence my question above!

Todd Trimble (May 30 2024 at 11:55):

John Baez said:

I know lots of concrete applications of the bar construction applied to some object of $\mathbf{E}^M$. What are some concrete applications of the monadic resolution of some object of $\mathbf{E}$?

Shamefully, I don't know myself. I was hoping someone (you!) would tell us. I was just adding a little categorical commentary on the construction itself, and why it's not quite the dual.

Tim Hosgood (May 30 2024 at 12:21):

John Baez said:

Tim Hosgood said:

a question for the category theorists here: are projective/injective resolutions in any form explained by some sort of monadic story?

Many of the most famous projective resolutions are instances of the bar construction applied to an object of some Eilenberg-Moore category $\mathbf{E}^M$. I could regale you with examples if desired.

for me, the "most famous" projective resolution is given by taking free modules generated by generators and then looking at the kernel and iterating the procedure; is this a bar construction?

Tim Hosgood (May 30 2024 at 12:22):

the two "most famous" specific projective resolutions I can think of are $0\to\mathbb{Z}\xrightarrow{\cdot n}\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ and $0\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}$; are these bar constructions?

Todd Trimble (May 30 2024 at 12:32):

Tim Hosgood said:

the two "most famous" specific projective resolutions I can think of are $0\to\mathbb{Z}\xrightarrow{\cdot n}\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ and $0\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}$; are these bar constructions?

They don't look like bar constructions, on initial appearance, but they might be relatable to those in some disguised way. I'd have to think harder about it. (But that second one -- I think you meant injective resolution for that one, yes?)

The resolutions of more "bar" type that we're discussing are simplicial or cosimplicial objects, but of course you can get chain or cochain complexes out of them, and you did after all mention Dold-Kan a little while ago. The specific sequences you mention are short exact sequences, but as we know a long exact sequence can be factored into itty-bitty short exact sequences. So maybe someone can cook up something along these lines.

Tim Hosgood (May 30 2024 at 12:49):

woops, yes, I meant injective for the latter! but yeah, this was the distinction that I wanted to point out maybe: I understand in what sense "most" resolutions look like the bar resolution, but if you take this motto seriously then I think it'll be very confusing to read anything "classical" about resolutions

Tim Hosgood (May 30 2024 at 12:51):

when I hear "resolution" I think of chain complexes and resolutions in that sense; when I think about e.g. the Čech nerve I think more about (co)fibrant replacements. I know that you can sort of go between these two through the lens of model structures, but as somebody who lives mostly on the dg-category side of things, I do get a bit confused about stuff like where the bar resolution is secretly hiding

John Baez (May 30 2024 at 14:19):

Tim Hosgood said:

John Baez said:

Many of the most famous projective resolutions are instances of the bar construction applied to an object of some Eilenberg-Moore category $\mathbf{E}^M$. I could regale you with examples if desired.

for me, the "most famous" projective resolution is given by taking free modules generated by generators and then looking at the kernel and iterating the procedure; is this a bar construction?

Yes, if I understand you correctly. For some ring R we are taking an R-module M, then the free R-module on the underlying abelian group of M, say FUM, and then FUFUM, etc. These and the obvious maps between them give us an augmented simplicial R-module (the bar construction) which is 'the same as' an augmented chain complex of R-modules... and this augmented chain complex is a free resolution of M called the bar resolution of M.

John Baez (May 30 2024 at 14:22):

All this relies on the fact that FU is a comonad on the category of R-modules, which in turn is the Eilenberg-Moore category of the monad UF on the category of abelian groups.

John Baez (May 30 2024 at 14:27):

Tim Hosgood said:

the two "most famous" specific projective resolutions I can think of are $0\to\mathbb{Z}\xrightarrow{\cdot n}\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ and $0\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}$; are these bar constructions?

No. The second isn't a projective resolution and the first isn't a bar resolution because it's not the thing I just described: it doesn't go on forever. It's a hand-crafted projective resolution that's 'more efficient', taking advantage of special features of $\mathbb{Z}/n$.

John Baez (May 30 2024 at 14:42):

The bar construction (and bar resolution) is a brutal big machine that always works. Its main advantage is that it's functorial and requires no special cleverness. Its main disadvantage is that it goes on forever and the individual terms get huge. Whenever you see a cute little resolution like $0\to\mathbb{Z}\stackrel{n}{\to}\mathbb{Z}\to\mathbb{Z}/n$, or indeed any one of finite length, you know that's not the bar resolution!

An object in an abelian category that has a finite-length projective resolution is said to have finite projective dimension, and you just showed the projective dimension of the $\mathbb{Z}$-module $\mathbb{Z}/n$ is at most $1$. (It's also at least $1$. The numbering convention is a wee bit confusing but I think the way it works is that free objects have projective dimension 0, and $\mathbb{Z}/n$ is only slightly worse since it's the quotient of a free object by the image of some free object.)

Tim Hosgood (May 30 2024 at 14:47):

so what does the bar construction give as a resolution of $\mathbb{Z}/2\mathbb{Z}$? following what you said above, the first term should be the free abelian group on $\mathbb{Z}/2\mathbb{Z}$, which sounds like $\mathbb{Z}$, and then the next term should be free on this, but I don't really know what it means to take the "free $R$-module on an abelian group"

John Baez (May 30 2024 at 14:51):

The forgetful functor from $R$-modules to abelian groups has a left adjoint and that's called "the free $R$-module on an abelian group".

If you were a dyed-in-the-wool category theorist that would count as a sufficient answer. But it sounds like you want to know how to actually compute it!

It's not hard: you take your abelian group $A$ and 'freely' make it into an $R$-module by creating symbols $r a$ where $r \in R, a \in A$, and then formal sums of those, obeying the relations they need to obey.

When you think about this a while, you realize you've reinvented $R \otimes_\mathbb{Z} A$. So that's the answer I would have given if I were in a "don't teach a man to fish, just throw the poor sod a fish" mood: the free $R$-module on an abelian group $A$ is $R \otimes_\mathbb{Z} A$. But it's much better to learn how to guess what the free gizmo on a gadget is.

Tim Hosgood (May 30 2024 at 14:56):

hm, so if $R=\mathbb{Z}$ then this says that you just get back $A$ again, since $\mathbb{Z}\otimes_{\mathbb{Z}}-$ is the identity, no?

John Baez (May 30 2024 at 14:58):

Is that bad? Are you trying to scare me?

Tim Hosgood (May 30 2024 at 15:00):

well then it sounds like the bar construction applied to $\mathbb{Z}/2\mathbb{Z}$ gives me lots and lots of copies of $\mathbb{Z}/2\mathbb{Z}$, but this is very much not a projective resolution!

John Baez (May 30 2024 at 15:01):

Oh. Okay, maybe I shouldn't be using the adjunction between $R$-modules and abelian groups to get what you're after here, because when $R = \mathbb{Z}$ that adjunction consists of identity functors going both ways: $R$-modules just are abelian groups.

John Baez (May 30 2024 at 15:02):

Maybe we want the adjunction between $R$-modules and sets.

John Baez (May 30 2024 at 15:03):

The free $R$-module on a set $S$ is probably $R \times S$.

John Baez (May 30 2024 at 15:04):

So then if you do $R = \mathbb{Z}$ and start with the $R$-module $\mathbb{Z}/2$ and do that FU, FUFU, FUFUFU stuff you first get the free abelian group on the underlying set of $\mathbb{Z}/2$, which is $\mathbb{Z}^2$, and so on.

John Baez (May 30 2024 at 15:06):

But if that seems ungainly you could probably get equivalent results doing the adjunction between $R$-modules and pointed sets, where the zero element of your $R$-module gives the basepoint in your pointed set. The free $R$-module on the underlying pointed set of $\mathbb{Z}/2$ is just $\mathbb{Z}$, which is what you were hoping for:

the first term should be the free abelian group on $\mathbb{Z}/2\mathbb{Z}$, which sounds like $\mathbb{Z}$.

John Baez (May 30 2024 at 15:24):

Sorry for screwing up at first: I couldn't believe that our forgetful functor had to go from R-modules all the way down to sets, since the category of abelian groups beckons so temptingly as an intermediate way-station, and algebraists are so loath to sink to the level of mere sets. But when you're talking about a free resolution of an abelian group, what can 'free' mean but free on a set (or pointed set)?

John Baez (May 30 2024 at 18:17):

John Baez said:

The forgetful functor from $R$-modules to abelian groups has a left adjoint and that's called "the free $R$-module on an abelian group".

If we use this adjunction to set up a bar resolution, we get from any ring $R$ and any $R$-module $M$ a chain complex of $R$-modules

$M \leftarrow R \otimes_{\mathbb{Z}} M \leftarrow R \otimes_{\mathbb{Z}} R \otimes_{\mathbb{Z}} M \leftarrow \cdots$

which I believe is a free resolution of $M$ as an $R$-module. It's not what @Tim Hosgood was wanting to look at! But it's interesting. It seems closely connected to the Hochschild complex. But I'm confused about the precise relation.

John Onstead (May 30 2024 at 19:13):

Todd Trimble said:

It really would be a dual construction if we knew that E is the category of coalgebras of the comonad FU, but we don't always know that.

It's not true that in a monadic adjunction for a monad T, the category C is the co-EM category of EM(T) for the corresponding comonad on EM(T)? I've always assumed that to be the case. Wouldn't this make monadicity non-symmetric such that a monadic adjunction doesn't double as a comonadic adjunction for the corresponding comonad? But aside from that, the rest here (composing evaluation with the augmented cosimplicial object in [E, E]) makes sense and is how I envisioned the monadic resolution to work.
In light of this (potential) revelation I have to check something I wrote down in my notes again. I wrote that bar constructions and comonadic resolutions are the same thing because any time we have a comonad on a category C, we can view C as being the EM category for some monad on another category. I showed this by arguing that all comonads have coEM categories with which there's a comonadic adjunction, and every comonad induces a monad on this coEM category by flipping this adjunction. I then used this assumption of mine to argue that the original C is just the EM category of this monad. Even if this assumption is not true, I'm still absolutely convinced that any category with a comonad can be thought of as an EM category for some monad, but now I have no way to prove this. It would involve taking a comonad, finding all possible ways to produce it via adjunction composition, flipping those adjunctions to get all possible corresponding monads, and then showing that at least one of these flipped adjunctions is monadic. I'm not sure where to start on the proof this can be done, so any hints would be helpful!

Todd Trimble (May 30 2024 at 19:35):

It's not true that in a monadic adjunction for a monad T, the category C is the co-EM category of EM(T) for the corresponding comonad on EM(T)?

No, and here's an easy example: take a monad $c$ on a poset $P$, which is the same as a closure operator $c: P \to P$. Then the EM-category for $c$ is the poset of fixed points of $c$. Note that the forgetful functor from the category of algebras is the inclusion $\mathrm{Fix}(P) \hookrightarrow P$! The free functor left adjoint to it is $p \mapsto c(p)$. The comonad given by the composite is the identity comonad! And the coEM-category of fixed points of the identity comonad gives you nothing new.

The same thing happens with any idempotent monad, for example the one where sheafifying a presheaf is left adjoint to the inclusion of presheaves into sheaves. Any situation where the right adjoint part is fully faithful. For example, the inclusion of abelian groups into all groups.

Todd Trimble (May 30 2024 at 19:41):

That being said, there are some positive results. If $T$ is a monad on $\mathsf{Set}$ such that the unique algebra map $T(0) \to T(1)$ is a regular monomorphism but not an isomorphism, then the free functor $F: \mathsf{Set} \to \mathsf{Set}^T$ is comonadic. This happens for example for any Lawvere theory with at least one constant, and a model with at least two elements, which cuts a pretty wide swath.

John Onstead (May 30 2024 at 23:21):

Todd Trimble said:

And the coEM-category of fixed points of the identity comonad gives you nothing new.

The same thing happens with any idempotent monad, for example the one where sheafifying a presheaf is left adjoint to the inclusion of presheaves into sheaves. Any situation where the right adjoint part is fully faithful. For example, the inclusion of abelian groups into all groups.

I see, thanks for letting me know. This is one of the main difficulties I face in learning math: I unknowingly make assumptions along the way and then carry on as if I understand something fully. Inevitably upon review down the line, I might suddenly realize that not only did I end up making an assumption, but that the assumption I made is wrong (it almost always is) and then have to go back to the drawing board. But this is only if I get lucky and just so happen to read something that contradicts what I wrote in my notes: who knows how many untrue assumptions lurk in all my hundreds of pages of notes so far?
As for the more specific matter at hand, this means that I do not understand the relation between comonadic resolutions and bar constructions, the initial point of this thread, as well as I thought. My assumption made me think these were equivalent notions, but now I know that bar constructions are strictly a special case of comonadic resolutions with this being untrue in the opposite direction. That means we will have to revisit these notions before moving on. I'll get my thoughts together first, but I think what I want to do next here is to give myself an exercise: prove that the comonadic resolution is actually a simplicial resolution. That is, given any comonadic resolution for some object in a category with a comonad on it, prove once and for all that the colimit of the corresponding simplicial object returns that original object. Since the bar construction is a special case, this will automatically prove the resolution abilities of the bar construction. I'm not sure if I'll be able to prove this in the end as I'm still very much a beginner, but I am inspired to try it out and see how far I can get!

John Baez (May 31 2024 at 05:55):

That is, given any comonadic resolution for some object in a category with a comonad on it, prove once and for all that the colimit of the corresponding simplicial object returns that original object.

What makes you think this is true, btw?

I'm not sure if I'll be able to prove this in the end....

Especially if it's not true. Whenever you're trying to prove something that might not be true, it's important to spend an equal amount of time trying to disprove it, i.e. find a counterexample. Looking for counterexamples is also good because it gets your mind away from generalities and makes you focus on specifics: to disprove a generality $\forall x P(x)$ the best way to find a specific $x$ with $\neg P(x)$.

dusko (May 31 2024 at 06:13):

John Onstead said:

Todd Trimble said:

It really would be a dual construction if we knew that E is the category of coalgebras of the comonad FU, but we don't always know that.

It's not true that in a monadic adjunction for a monad T, the category C is the co-EM category of EM(T) for the corresponding comonad on EM(T)? I've always assumed that to be the case. Wouldn't this make monadicity non-symmetric such that a monadic adjunction doesn't double as a comonadic adjunction for the corresponding comonad? But aside from that, the rest here (composing evaluation with the augmented cosimplicial object in [E, E]) makes sense and is how I envisioned the monadic resolution to work.
[...]
It would involve taking a comonad, finding all possible ways to produce it via adjunction composition, flipping those adjunctions to get all possible corresponding monads, and then showing that at least one of these flipped adjunctions is monadic. I'm not sure where to start on the proof this can be done, so any hints would be helpful!

with an apology if this is besides the point since i only just logged in and looked up a small part of the conversation, but just in case, the general state of affairs with the EM-resolution of a monad induced by an adjunction and the coEM resolution of the comonad induced by the same adjunction looks like this:
nuc.jpeg
there is an adjunction between the EM-category of algebras and the coEM-category of coalgebras, and in that adjunction the right adjoint is monadic and the left adjoint is comonadic.

dusko (May 31 2024 at 06:19):

and incidentally, an algebra that carries a coalgebra-over-algebra obviously has to be a projective algebra. and a coalgebra that carries an algebra-over-coalgebra has to be injective. so the projective and the injective resolutions very much play a role.

Paolo Perrone (May 31 2024 at 08:55):

In case this helps, a few years ago, together with the people from our ACT school group, we got some insight on the bar construction (mostly coming from probability!) as encoding "partial evaluation" of the operations induced by a monad.
See here and here.
For the monad induced by a group action, this recovers the classical "shape of the action" picture in group cohomology.
(For probability monads, this is related to martingales.)

John Onstead (May 31 2024 at 09:05):

John Baez said:

What makes you think this is true, btw?

My belief comes from the nlab page on simplicial resolutions, which lists comonadic resolution as a direct special case of simplicial resolution. It writes: "Of course this construction did not depend on the fact that we were handling groups, so we could apply it to any comonad (within reason!)". If it turns out that only the special case of "comonadic resolutions" known as bar constructions possess the property of being a simplicial resolution, that's ok, but then the nlab page should correct all mentions of "comonadic resolutions" to "bar resolutions". The page does say "within reason" so maybe the "reasonable" comonads they are talking about are precisely the ones on EM categories of some monad, in which case they are specifically talking about bar constructions (then, why not just come out and state you are talking about bar constructions instead of being so vague about it?)

dusko said:

with an apology if this is besides the point since i only just logged in and looked up a small part of the conversation, but just in case, the general state of affairs with the EM-resolution of a monad induced by an adjunction and the coEM resolution of the comonad induced by the same adjunction looks like this:

there is an adjunction between the EM-category of algebras and the coEM-category of coalgebras, and in that adjunction the right adjoint is monadic and the left adjoint is comonadic.

This is really cool for many reasons, not the least of which is that it is a good example of an adjunction that is both monadic and comonadic. What I am interested in is how this corresponds to this MO post by Todd Trimble since it says there that there's no reason for such an adjunction between EM and coEM categories to exist in general? (It might not be related, in which case I might have gotten confused somewhere!)

John Onstead (May 31 2024 at 09:10):

John Baez said:

Especially if it's not true. Whenever you're trying to prove something that might not be true, it's important to spend an equal amount of time trying to disprove it, i.e. find a counterexample.

That's a good idea! But to do this I'd have to find an example of a comonad on a category that cannot be thought of as the EM category for any possible monad. Not sure how that could be verified!
Edit: I looked at the "examples" section of the nlab article on comonads and found the jet bundle comonad, which takes a bundle over a smooth manifold to its jet bundle. Since topology-esque things generally are not algebraic, I doubt any category of bundles is an EM category for any monad. So maybe I can use this to find a counterexample, but unfortunately I don't know enough about jet bundles to attempt this for too long.

dusko (May 31 2024 at 10:33):

John Onstead said:

John Baez said:

What makes you think this is true, btw?

My belief comes from the nlab page on simplicial resolutions, which lists comonadic resolution as a direct special case of simplicial resolution. It writes: "Of course this construction did not depend on the fact that we were handling groups, so we could apply it to any comonad (within reason!)". If it turns out that only the special case of "comonadic resolutions" known as bar constructions possess the property of being a simplicial resolution, that's ok, but then the nlab page should correct all mentions of "comonadic resolutions" to "bar resolutions". The page does say "within reason" so maybe the "reasonable" comonads they are talking about are precisely the ones on EM categories of some monad, in which case they are specifically talking about bar constructions (then, why not just come out and state you are talking about bar constructions instead of being so vague about it?)

dusko said:

with an apology if this is besides the point since i only just logged in and looked up a small part of the conversation, but just in case, the general state of affairs with the EM-resolution of a monad induced by an adjunction and the coEM resolution of the comonad induced by the same adjunction looks like this:

there is an adjunction between the EM-category of algebras and the coEM-category of coalgebras, and in that adjunction the right adjoint is monadic and the left adjoint is comonadic.

This is really cool for many reasons, not the least of which is that it is a good example of an adjunction that is both monadic and comonadic. What I am interested in is how this corresponds to this MO post by Todd Trimble since it says there that there's no reason for such an adjunction between EM and coEM categories to exist in general? (It might not be related, in which case I might have gotten confused somewhere!)

well, todd says that the category of algebras and the category of coalgebras induced by the monad and the comonad of an adjunction are generally not equivalent, and that is true. (they are equivalent when the categories are preorders, in which case the monad is a closure and the comonad is an interior operator. this is the classical galois connection. the original question seemed to be concerned with that special case.) but todd then goes beyond the claim that they are not equivalent, and says that he doesn't even see an adjunction between them. the composites of each of the forgetful functors and each of the comparison functors are adjoint. and they are monadic and comonadic, respectively. the proof is in my nucleus paper.

if I understand your conjecture, I think it is a corollary of a part of the claim above: that the category of algebras for a monad is equivalent to the category of algebras over the category of coalgebras for the corresponding comonad. just take the first layer of the comonadic resolution. it is an algebra over coalgebras. every algebra is the coequalizer of the U-split pair that it induces.

Todd Trimble (May 31 2024 at 11:40):

Dusko wrote:

but todd then goes beyond the claim that they are not equivalent, and says that he doesn't even see an adjunction between them. the composites of each of the forgetful functors and each of the comparison functors are adjoint. and they are monadic and comonadic, respectively. the proof is in my nucleus paper.

Dusko seems to be right. I wrote that MO post quite a while back, and so while I can't be absolutely sure of what was in my mind then, I can guess what was probably in my mind, and I can also forgive my blindness at that time a little because the situation is actually somewhat "twisty", as I'll explain in a moment.

Todd Trimble (May 31 2024 at 11:40):

Suppose we have an adjunction $F: \mathbf{D} \leftrightarrows \mathbf{C}: U$, with $F \dashv U$. Let $\varepsilon: FU \to 1_\mathbf{C}$ be the counit and $\eta: 1_\mathbf{D} \to UF$ the unit. We get an induced monad $T = UF$ on $\mathbf{D}$ and a comonad $K = FU$ on $\mathbf{C}$ then let $\mathbf{D}^T$ denote the category of $T$-algebras and $\mathbf{C}_K$ the category of $K$-coalgebras. There is a comparison functor

$\hat{U}: \mathbf{C}_K \to \mathbf{D}^T$

which takes a coalgebra $(c, \gamma: c \to Kc)$ to $Uc$ with its canonical algebra structure $U\varepsilon c: UFU c \to Uc$. Dually, we get a comparison functor going the other way,

$\hat{F}: \mathbf{D}^T \to \mathbf{C}_K.$

Todd Trimble (May 31 2024 at 11:41):

Remember that $F \dashv U$. What I was probably telling myself at the time is that contrary to what one might hastily expect, we do not however have an adjunction $\hat{F} \dashv \hat{U}$ between the lifts -- and that observation is correct. Because, given an algebra structure $\alpha: UFd \to d$, the only candidate for a putative unit $(d, \alpha) \to \hat{U} \hat{F}(d, \alpha)$ would be, at the level of the underlying category $\mathbf{D}$, the unit $\eta d: d \to UFd$, but this map is not an algebra map.

Todd Trimble (May 31 2024 at 11:41):

However, what I failed to observe then is that actually the adjunction is the other way around, $\hat{U} \dashv \hat{F}$! That's the twistiness: that $F \dashv U$ implies $\hat{U} \dashv \hat{F}$. It's easy to verify once you see it: at each algebra $(d, \alpha)$, the counit component $\hat{U}\hat{F} (d, \alpha) \to (d, \alpha)$ is given by $\alpha: UFd \to d$, which is of course a map of algebras. Dually for over on the coalgebra side, and the triangle identities for this twisted lifted adjunction come down to the unit and counit conditions for algebras and coalgebras. (This stuff may be more of a participatory sport than a spectator sport, so you may be better off writing it out for yourself.)

I took a look at that "nucleus paper" by Dusko and Dominic Hughes, here, where the twisty lift makes a first appearance at the bottom of page 16.

John Onstead (May 31 2024 at 18:42):

dusko said:

the composites of each of the forgetful functors and each of the comparison functors are adjoint. and they are monadic and comonadic, respectively. the proof is in my nucleus paper.

I will certainly look through the nucleus paper when I have time, at first glance it seems relevant to some of the things we are discussing here.

Todd Trimble said:

Dusko seems to be right. I wrote that MO post quite a while back, and so while I can't be absolutely sure of what was in my mind then, I can guess what was probably in my mind, and I can also forgive my blindness at that time a little because the situation is actually somewhat "twisty", as I'll explain in a moment.

Thanks for the explanation, it clears things up!

John Onstead (May 31 2024 at 19:15):

For now, I am going to take a break for a few days to recover and get my thoughts together. But first I wanted to write a summary of all I have learned on this thread:

• You can construct a simplicial object (a "comonadic simplicial object") from any comonad due to the amazing properties of the opposite augmented simplex category being the "walking comonoid" (and dually for monads and cosimplicial objects)
• In the very special case that the comonad just so happens to be on an EM category for a monad T, AND that monad T is the "flip" of the comonad when it comes to their composite adjunctions, then the above construction reduces to a "bar construction". This is a very special case because not every comonad arises in this way!
• A simplicial resolution is when you have an object you can interpret as being the colimit of a simplicial object; together they form an augmented simplicial object
• A bar construction is a special case of BOTH the comonadic simplicial object AND a simplicial resolution. However, comonadic simplicial objects and simplicial resolutions are NOT special cases of one another (thus warranting a correction to the nlab page for "simplicial resolution" under the Examples section)!
• A canonical resolution is usually just the same as the bar construction, but in some cases one can find a "more efficient" way to present an object via colimits than an entire simplicial object. While bar constructions are always acyclic and contractible, a canonical resolution does not necessarily have to be, thus making bar constructions a special case of yet another concept. These canonical resolutions are meant to generalize "presentations in terms of generators and relations"
• Through the above observation, a bar construction is related to the idea of a "canonical presentation in terms of generators and relations" since you can always get such a presentation when you truncate the bar construction down to a coequalizer fork. Higher categories might require higher truncations, with these truncations tying into "syzygies"
• A simplicial realization and simplicial resolution might seem related since they both deal with simplicial objects and have similar names, but they generally do different things or are meant for different purposes

Next Steps:

• What are some examples of simplicial resolutions that are not bar constructions? What would be a general form of a simplicial resolution (IE, in terms of what it means for a simplicial object in a category to be "homotopy equivalent" to some standard object in that category and what this has to do with colimits)?
• Monadic cohomology and understanding cochain complexes

Let me know if there's any mistakes above as I'd like to correct them before moving on. As always thanks for all your help! Your patience and guidance is very much appreciated, especially when I make silly mistakes as I am want to do.

John Onstead (Jun 05 2024 at 06:11):

I think I'm up to the task again. I want to start by connecting a general definition of resolution to simplicial resolutions. A resolution is defined to be a choice of homotopy / weak equivalence between two objects in some place where these notions can be defined (IE, a model category, since weak equivalences are part of the definition for that). A simplicial resolution is then a weak equivalence between a simplicial object and an object in some category, but the obvious problem here is a typing problem: a simplicial object and a typical object are incommensurable and so cannot strictly have a weak equivalence between them. I have a guess as to how to resolve this, but I'm probably wrong. Let's start in a category of form [del^op, C]. First, recall a diagonal functor dia: C -> [del^op, C] will send any object in C to the constant functor from del^op to that object. We can use this to treat constant functors into an object of C as that object itself (a similar game to what we do when defining limits in terms of cones). My hypothesis is that a simplicial resolution of an object A in C is simply a weak equivalence between A, viewed as a constant functor in [del^op, C], and some other object in [del^op, C], which we then call the "simplicial resolution of A".

So three questions emerge: First, is my hypothesis anywhere close to correct? Second, if so, then do ALL categories of form [del^op, C] have some model structure on them? Thirdly, what does this have anything to do with colimits as defined under the "simplicial resolution" nlab page? Colimits and weak equivalences are completely different concepts, so it doesn't seem obvious how they connect in any way at all here!

John Baez (Jun 05 2024 at 10:10):

I don't think that for an arbitrary category C you can make the category of simplicial objects in C into a model category in some standard way.

John Baez (Jun 05 2024 at 10:13):

If C is already a model category, you can give the category of simplicial objects in C three model structures, called the [[Reedy model structure]], the [[projective model structure]] and the [[injective model structure]]. But none of these help us when C = Set, C = Grp, etc.

John Baez (Jun 05 2024 at 10:16):

So, if you want to understand simplicial resolutions in such cases, you can either talk about the particular model structures that happen to be present for simplicial sets, simplicial groups, etc., or avoid the machinery of model categories and try to work with simplicial homotopies, which make sense for simplicial objects in any category.

John Baez (Jun 05 2024 at 10:17):

For the bar resolution, the latter approach works admirably and was laid out by @Todd Trimble in the nLab article [[bar construction]].

John Baez (Jun 05 2024 at 10:18):

For other simplicial resolutions... I don't know.

John Onstead (Jun 07 2024 at 06:28):

John Baez said:

or avoid the machinery of model categories and try to work with simplicial homotopies, which make sense for simplicial objects in any category.

Ah this might be where my confusion is. I haven't learned too much about model category theory so I assumed that you need a model category structure to be able to do anything at all related to homotopy. My reasoning was that if you had a notion of "homotopy" in some category, you then also have a notion of homotopy equivalence, and then you could just define a model structure on that category by taking the class of weak equivalences (that you need to define the model structure) to be those homotopy equivalences. Maybe it's a little more involved than that?

Also for a bit of context, my previous questions and hypothesis are based on this comment from Mike Shulman above:
Mike Shulman said:

The latter means that the unaugmented simplicial object B∙​(M,M,A) is simplicially-homotopy-equivalent to the constant simplicial object A, and therefore they have equivalent geometric realizations.

It really does kind of seem from this comment that my hypothesis of a simplicial resolution consisting of finding a homotopy equivalence in [del^op, C] between a simplicial object in C and a constant functor on some object in C is true. But I was worried I might be misinterpreting something!

Kevin Carlson (Jun 07 2024 at 07:06):

Yeah, it's more involved; model categories include a notion of homotopy and homotopy equivalence, but these notions are central only in their relation to a given class of weak equivalences, inspired by maps of topological spaces which are isomorphisms on homotopy groups (similarly, maps of chain complexes which are isomorphisms on homology groups.) Weak equivalences are sometimes equivalent to homotopy equivalences, but only, in most model categories, for special classes of objects: cofibrant and fibrant ones, concepts which lead to the need for the cofibrations and fibrations that are also central to the definition of a model category. All of this can be dispiritingly complicated at first but it seems to be the way homotopy theory wants to work (though some aspects of model category theory can be avoided in numerous ways, such as focusing on the $\infty$-categories model categories are now conceptualized as presenting.)

John Baez (Jun 07 2024 at 07:12):

John Onstead said:

I haven't learned too much about model category theory so I assumed that you need a model category structure to be able to do anything at all related to homotopy. My reasoning was that if you had a notion of "homotopy" in some category, you then also have a notion of homotopy equivalence, and then you could just define a model structure on that category by taking the class of weak equivalences (that you need to define the model structure) to be those homotopy equivalences. Maybe it's a little more involved than that?

Yes, it's often quite hard to define fibrations, cofibrations and weak equivalences and show the trio obey the conditions of a [[model category]], and there are many situations where all we have is some lesser amount of structure. The hard part is often the fibrations and cofibrations, and there are often different ways to choose those given a notion of weak equivalence... or no way at all. Thus, it can be useful to content ourselves with a lesser amount of structure, like a [[category with weak equivalences]].

(Btw, have you seen how hard it is to construct the usual Quillen model structure on the category of simplicial sets? For a long time the only way people could do it was bringing in the category Top. I thought I heard that Joyal and Tierney were going to write up a construction that only uses simplicial sets, but I don't know if they - or anyone else - has done it.)

John Baez (Jun 07 2024 at 07:13):

But as Kevin just pointed out, there are many situations where weak equivalences are weaker than homotopy equivalences - e.g. in Top with its usual model structure.

John Baez (Jun 07 2024 at 07:15):

Anyway, it's very nice that we don't need to get into any of this stuff to understand the sense in which the bar resolution of any algebra of any monad on any category is a 'resolution'. I keep recommending @Todd Trimble's work on the bar construction and I'll do it again. The key idea is here:

Theorem: B(M, M, X) is initial in the category of X-acyclic M-algebras.

Once you've learned the jargon and notation, this is a terse statement of the universal property of the bar construction: roughly, the bar resolution is an initial object among all resolutions. I think I should add some extra verbiage to the nLab page [[bar construction]], where this key thought is buried amid notation here:

Theorem: The functor $\mathrm{hom}_{\mathbf{E}^T}(A, G-): \mathrm{AlgRes}_T \to \mathrm{Set}$ is represented by $\mathrm{Bar}_T(A)$; i.e., $\mathrm{Bar}_T(-): \mathbf{E}^T \to \mathrm{AlgRes}_T$ is left adjoint to $G$.

Kevin Carlson (Jun 07 2024 at 07:23):

John Baez said:

(Btw, have you seen how hard it is to construct the usual Quillen model structure on the category of simplicial sets? For a long time the only way people could do it was bringing in the category Top. I thought I heard that Joyal and Tierney were going to write up a construction that only uses simplicial sets, but I don't know if they - or anyone else - has done it.)

I know this is off-topic but this is apparently a widely believed myth—at least the nLab claims Quillen’s original proof is purely combinatorial, and I think I’ve heard Cisinski make the same claim, though I don’t remember whether I’ve read Quillen’s book myself. Cisinski model structures definitely give a more general purely combinatorial approach, and Christian Sattler has papers on various aspects of defining this model structure better, lots of citations here: https://ncatlab.org/nlab/show/classical+model+structure+on+simplicial+sets

John Baez (Jun 07 2024 at 08:00):

Hmm, I wonder how I got indoctrinated in that myth. I spent a bunch of time trying to read Quillen's Homotopical Algebra but I was not at all concerned with his construction of the model structure on simplicial sets! Then later I looked at some other constructions of this model structure and saw that weak equivalences in SimpSet were defined to be those that mapped to weak equivalences in Top under geometric realization. This is great if you're trying to quickly prove that SimpSet and Top are Quillen equivalent model categories, but horrible if you're trying to argue that SimpSet can serve as a self-standing framework for homotopy theory.

So I'm very glad this isn't the only way to do it!

John Baez (Jun 07 2024 at 08:23):

The nLab still starts by describing the weak equivalences in SimpSet in terms of Top:

https://ncatlab.org/nlab/show/classical+model+structure+on+simplicial+sets#TheClassicalModelStructure

so you have to be a bit more persistent to find another way. But it clearly states that Quillen used another way! So that myth should die out.

Mike Shulman (Jun 07 2024 at 08:49):

It looks like Quillen defined weak equivalences of simplicial sets to be the maps that factor as a trivial cofibration followed by a trivial fibration, but to prove that this satisfies 2-out-of-3 he showed that they are also the maps that induce bijections on homotopy classes of maps into any Kan complex.

Mike Shulman (Jun 07 2024 at 08:50):

His proof also cites some technology from other places, including minimal fibrations.

John Baez (Jun 07 2024 at 08:51):

Thanks! So he's sort of using Kan complexes rather than Top here.

John Baez (Jun 07 2024 at 08:53):

While all this is a bit of a digression from "resolution and descent", it at least teaches us how bringing model categories into the definition of "resolution" is a nontrivial commitment, which at least in the case of the bar resolution can be avoided.

John Onstead (Jun 07 2024 at 09:22):

Kevin Carlson said:

Weak equivalences are sometimes equivalent to homotopy equivalences, but only, in most model categories, for special classes of objects: cofibrant and fibrant ones, concepts which lead to the need for the cofibrations and fibrations that are also central to the definition of a model category.

John Baez said:

The hard part is often the fibrations and cofibrations, and there are often different ways to choose those given a notion of weak equivalence... or no way at all. Thus, it can be useful to content ourselves with a lesser amount of structure, like a category with weak equivalences.

Ah I see, I was glossing over the importance of the fibrations and cofibrations. Thanks for the help! I'll look more into this part of category theory (when I've got more time) since it seems very interesting, but also quite complex!

John Onstead (Jun 07 2024 at 09:42):

John Baez said:

Anyway, it's very nice that we don't need to get into any of this stuff to understand the sense in which the bar resolution of any algebra of any monad on any category is a 'resolution'. I keep recommending @Todd Trimble's work on the bar construction and I'll do it again. The key idea is here:

Theorem: B(M, M, X) is initial in the category of X-acyclic M-algebras.

I've gone through this again and I think I see how this converges with my idea. On the article it defines a comonad P on the category [del_a^op, C] that is the composite of the evaluation at 1 with the diagonal functor (which is what I was mentioning earlier). IE, if X is the bar resolution of an object A in C (if C is some category of algebras for some monad), the functor P will send X to the constant functor into A. An "acyclic structure" is then a coalgebra of this comonad given by X -> PX. Thus, any homotopy equivalences between a simplicial object in C and an object in C viewed as a constant functor into it will take on the form X -> PX and will be a coalgebra of P. So all simplicial resolutions will be coalgebras of this comonad!

I think I realize what was confounding me the first few times around trying to understand this: all simplicial resolutions were coalgebras of P, but how many coalgebras of P were simplicial resolutions? The writeup on the nlab states that a coalgebra X -> PX can be viewed as a homotopy of sorts. But it doesn't state if this is a homotopy equivalence or not. In my mind, such a map has to be a homotopy equivalence, not a mere homotopy, for it to count as a "resolution". But if I remember correctly, a key property of being acyclic is contractibility, and since any coalgebra X -> PX is an acyclic structure (proven on the nlab page), it seems every such coalgebra is a "contraction" of X into PX. If a space contracting to a point means it is homotopy equivalent to the point, then it seems a contraction of X into PX means should be a homotopy equivalence. If this reasoning is true, then indeed all coalgebras X -> PX are simplicial homotopy equivalences/simplicial resolutions and thus solves my problem.

There's still a lot more details for me to cover in the proof specifics so I might still be missing things but this is what I have so far to say about it.

John Onstead (Jun 07 2024 at 21:51):

After taking a break to clear the mind I wanted to correct some things I stated above. First, I was wrong that a key property of acyclicity is contractibility- it's the other way around. Fortunately, this isn't too much of a problem here because the acyclicity of PX, I believe, is proven through contractibility: As nlab says, "Each space of based paths is contractible and therefore PX is acyclic."

But I think a bigger point I was missing was that I had somehow already forgotten what Todd Trimble wrote at the top of this page:
Todd Trimble said:

The reason for further composing with U is that it allows us to pick up some extra degeneracy maps that play the role of a contracting homotopy, one that witnesses a homotopy equivalence between this augmented simplicial object, and the constant augmented simplicial object (that is constantly A in every simplicial degree). It's this contracting homotopy that I think of as putting the "r" in "resolution".

My apologies to him if he is reading this! In any case, what is interesting here is that one first proves this homotopy equivalence exists for a bar construction not within the category EM(T), but in the category C on which T is defined, through following U: EM(T) -> C. But if I am understanding remark 3.2 correctly, one can use remark 3.2 to show that this also implies a homotopy equivalence for the bar construction in EM(T) as well, which proves that both notions of bar construction give a simplicial resolution!

I think my biggest takeaway here is that you can view any simplicial resolution itself as a coalgebra for a comonad you can define for any [del^op, C] and thus actually get categories of simplicial resolutions. They will only work "nice" in the case of the bar construction scenarios, but as long as I'm understanding everything well this time around this would answer my question of how to define a general simplicial resolution.

John Baez (Jun 07 2024 at 22:07):

But if I am understanding remark 3.2 correctly, one can use remark 3.2 to show that this also implies a homotopy equivalence for the bar construction in EM(T) as well!

How are you getting that out of remark 3.2?

John Baez (Jun 07 2024 at 22:10):

(By the way, is there a typo in Remark 3.2? I think $Gh:FX \to FXD$ should be $Gh: GX \to GXD$.)

John Onstead (Jun 08 2024 at 01:05):

This comes from the paragraph right above "4. Properties" on the nlab page. It says "By Remark 3.2, it follows that Bar(T, A), obtained by applying evaluation at a T-algebra A, carries an acyclic structure as well..." To be honest I'm not exactly sure how this follows, but if we let G from Remark 3.2 be the free functor F from C to EM(T), then if UBar(T, A) has an acyclic structure as was proven, then by remark 3.2 F compose UBar(T, A) would also have an acyclic structure (but again this is an incomplete thought since you would then have to prove that F compose UBar(T, A) being acyclic implies Bar(T, A) is acyclic).

Todd Trimble (Jun 08 2024 at 01:31):

John Baez said:

(By the way, is there a typo in Remark 3.2? I think $Gh:FX \to FXD$ should be $Gh: GX \to GXD$.)

Fixed.

Todd Trimble (Jun 08 2024 at 01:34):

John Onstead said:

This comes from the paragraph right above "4. Properties" on the nlab page. It says "By Remark 3.2, it follows that Bar(T, A), obtained by applying evaluation at a T-algebra A, carries an acyclic structure as well..." To be honest I'm not exactly sure how this follows, but if we let G from Remark 3.2 be the free functor F from C to EM(T), then if UBar(T, A) has an acyclic structure as was proven, then by remark 3.2 F compose UBar(T, A) would also have an acyclic structure (but again this is an incomplete thought since you would then have to prove that F compose UBar(T, A) being acyclic implies Bar(T, A) is acyclic).

Yeah, I'm not sure where you're going here. Yes, FUBar(T, A) has an acyclic structure. That doesn't imply Bar(T, A) has an acyclic structure.

John Baez (Jun 08 2024 at 07:36):

I bet we can see in examples that it doesn't. I think @John Onstead should give it a try. Examples are very good for disproving things, and I think looking at examples would do this discussion a world of good. There's a kind of understanding you get only from calculating with examples.

Todd Trimble (Jun 08 2024 at 08:24):

Sent from my Verizon, Samsung Galaxy smartphone

John Onstead (Jun 09 2024 at 00:50):

I'll try to find some counterexamples, but I'm not sure why there would be any. As discussed above, the bar construction in EM(T) allows one to express any T-algebra as a colimit of a simplicial object, its bar construction, in EM(T). Is this not enough to give a simplicial homotopy equivalence between the T-algebra and the simplicial object? In other words, is the concept of a "simplicial resolution as a simplicial homotopy equivalence" different from the concept of a "simplicial resolution as a colimit of a simplicial object"?
It doesn't make sense that they'd be two different concepts. A simplicial homotopy equivalence is a two way morphism in the category of simplicial objects; as a functor category, this is thus a two way natural transformation. One of these is a natural transformation into a constant functor in this functor category (representing the object being resolved) from its simplicial resolution. This is precisely just a cocone in the category. If the object is a colimit of the simplicial object, then this also defines a cocone- in this case, the colimiting cocone. Wouldn't this colimiting cocone satisfy all the properties necessary to also be a simplicial homotopy equivalence?

John Baez (Jun 09 2024 at 11:35):

John Onstead said:

I'll try to find some counterexamples, but I'm not sure why there would be any. As discussed above, the bar construction in EM(T) allows one to express any T-algebra as a colimit of a simplicial object, its bar construction, in EM(T). Is this not enough to give a simplicial homotopy equivalence between the T-algebra and the simplicial object?

How does expressing a T-algebra A as a colimit of a simplicial object X in T-algebras give a simplicial homotopy equivalence between X and A (or more precisely, the discrete simplicial T-algebra corresponding to A)?

John Baez (Jun 09 2024 at 11:40):

The bar construction gives, for any monad T on any category C, a very particular simplicial object X in T-algebras whose colimit is A. Todd has shown that in this particular case, there's a simplicial homotopy equivalence between UX and UA, where U: EM(T) $\to$ C is the functor sending any T-algebra to its underlying object in C.

John Baez (Jun 09 2024 at 11:43):

But note this is a simplicial homotopy equivalence between simplicial objects in C, not in EM(T).

Obviously if we had a simplicial homotopy equivalence between simplicial objects X, A in EM(T), it would give a simplicial homotopy equivalence between UX and UA in C. That's going downhill.

But in your earlier comments you were trying to go uphill. Starting from Todd's simplicial homotopy equivalence between UX and UA, you were trying to get a simplicial homotopy equivalence between X and A. I think you suggested applying F: C $\to$ EM(T). This gives a simplicial homotopy equivalence between FUX and FUA. But it doesn't give one between X and A.

John Baez (Jun 09 2024 at 11:47):

Now you seem to be giving some other argument, but I don't understand it, because your argument doesn't seem to be proving any of the properties required of a [[simplicial homotopy equivalence]] between X and A:

A morphism $f \colon A\to B$ of simplicial objects is a simplicial homotopy equivalence if there is a morphism $g \colon B\to A$ and morphisms $p\colon \Delta^1\times A\to A$ from $1_A$ to $g \circ f$ and $q\colon \Delta^1\times B\to B$ from $f \circ g$ to $1_B$.

John Baez (Jun 09 2024 at 11:49):

I don't see you providing these morphisms $p$ and $q$.

Todd Trimble (Jun 09 2024 at 16:29):

I was in airports all day yesterday and didn't have proper internet access. So while I had tried to type out a reply and send it, it didn't catch. Now that I'm back home, I can explain.

Todd Trimble (Jun 09 2024 at 16:29):

Anyway, @John Onstead seems to be misunderstanding that Remark 3.2, which is referring to the notion of "acyclic structure" in the sense of that article. Let $[n]$ denote the $n$-element ordinal. Then $[1]$, being terminal in $\Delta$ (which here will be the "algebraist's $\Delta$" which includes $[0]$) is a monoid, and so $[1]$ is a comonoid in $\Delta^{op}$. Letting $+$ denote the ordinal sum on $\Delta$, where $[m]+[n] = [m+n]$, it follows that we have a comonad $D$ on $\Delta^{op}$, defined by $(-) + [1]$, called "decalage" (sp.? not sure about accents). This in turn induces a comonad on (augmented) simplicial objects in any category $\mathbf{E}$, taking a simplicial object $X: \Delta^{op} \to \mathbf{E}$ to $XD: \Delta^{op} \to \mathbf{E}$. An "acyclic structure" is defined to be a coalgebra $X: \Delta^{op} \to \mathbf{E}$ over this comonad. It is given by a morphism $h: X \to XD$, the coalgebra structure, satisfying the coalgebra axioms. (There might be some other term instead of "acyclic structure" in the literature, but I don't know what it is.)

Todd Trimble (Jun 09 2024 at 16:30):

You need to unravel this terse definition of acyclic structure in order to fully appreciate what is going on in Remark 3.2, but briefly, the coalgebra structure provides a purely algebraic and absolute notion of acyclicity of simplicial $\mathbf{E}$-objects, one that is preserved by any functor $G: \mathbf{E} \to \mathbf{D}$. That is to say, if $h: X \to XD$ is a coalgebra structure, then $Gh: GX \to GXD$ provides a coalgebra structure on

$GX = (\Delta^{op} \overset{X}{\to} \mathbf{E} \overset{G}{\to} \mathbf{D}).$

It's an easy exercise to see this, although you should verify it for yourself and not just take my word for it. But morally, if we're thinking of a simplicial resolution as giving the augmented component as a colimit of the diagram of face and degeneracy maps coming before it, then the presence of an acyclic structure guarantees that this this colimit is an absolute colimit, one that is preserved by any functor $G$.

Todd Trimble (Jun 09 2024 at 16:30):

Let's dig into this by making it more concrete. According to the definition of $D$, we have $XD([n]) = X[n+1]$. Thus an acyclic structure $h: X \to XD$ is given by a sequence of maps $h[n]: X[n] \to X[n+1]$ satisfying some equations involving the face and degeneracy maps on $X$. One set of equations is given by the counit axiom, which would say that the composite

$X[n] \overset{h[n]}{\to} X[n+1] \overset{X[n+!]}{\to} X[n+0] = X[n]$

is the identity map for all $n$. So $h[n]$ would be a section of $X[n+!]$. Here $!$ denotes the unique map $0 \to 1$ in $\Delta$. Another set of equations is imposed by the naturality condition on $h$. There is also a coassociativity axiom which provides a third set of equations, but we won't need this for the current discussion.

Todd Trimble (Jun 09 2024 at 16:30):

Now let us focus in on the particular case of the bar construction $\mathrm{Bar}(T, A)$, where $T$ is a monad given by an adjunction $F \dashv U$ where $U$ is a forgetful functor mapping from the category of $T$-algebras. You can pretty much pick any $T$ you want, but I'll take it as the one where $U$ is the forgetful functor from abelian groups to sets. In this example, I'll take $A$ to be the abelian group $\mathbb{Z}_2$, with two elements. Part of the simplicial resolution $\mathrm{Bar}(T, A)$ will have face maps that look like this:

$FUFU(\mathbb{Z}_2) \overset{\varepsilon FU\mathbb{Z}_2}{\underset{FU\varepsilon \mathbb{Z}_2}{\rightrightarrows}} FU(\mathbb{Z}_2) \overset{\varepsilon \mathbb{Z}_2}{\longrightarrow} \mathbb{Z}_2$

which is the canonical generators-and-relations presentation of $\mathbb{Z}_2$ as an abelian group. Here $FU(\mathbb{Z}_2)$ is the free abelian group on two generators $0, 1$, isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$. All this was probably explained by John B. somewhere up in this thread, or a related thread. Anyway, if this simplicial object carried an acyclic structure, we'd have to have an abelian group map

$h[0]: \mathbb{Z}_2 \to \mathbb{Z} \oplus \mathbb{Z}$

that is a section of the canonical map $\varepsilon \mathbb{Z}_2: \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}_2$. But there is no map having the same domain and codomain as this putative $h[0]$, except the zero map. So this bar construction cannot possibly carry an acyclic structure.

Todd Trimble (Jun 09 2024 at 16:31):

That's enough of a counterexample, I suppose, but my larger point would have been that the data $h[0], h[1]$ that gives maps going from right to left, in addition to the face maps that form a fork

$X[2] \rightrightarrows X[1] \to X[0],$

give you a splitting of that fork, so that the face map $X[1] \to X[0]$ is in fact a split coequalizer, and that this happens essentially never at the level of the Eilenberg-Moore category if we're talking about the bar construction itself. You have to descend back down to the base category $\mathbf{E}$ on which $T$ acts, by applying the forgetful functor $U$, in order to get the splitting (this splitting is given by unit maps for the adjunction). The result that you get after applying $U$ is what I call the bar resolution, as distinct from the bar construction, which takes place at the level of algebras. So in other words, the coequalizer at the level of algebras is essentially never a split coequalizer, which is what the acyclic structure would give you.

John Onstead (Jun 09 2024 at 19:50):

I really appreciate the help, it's really helping me to disentangle concepts and put them in their proper place. My mind has the weird property that it entangles concepts when they should be distinct, usually based on similar terminology. For instance, the nlab page on "simplicial resolution" defines it both in terms of colimits (an object that is the colimit of some simplicial object) and as a special case of resolutions (so in terms of homotopy equivalences). I just assumed these were the same thing (admittedly, the nlab page didn't do much to dissuade this assumption, so maybe this mixup isn't entirely my fault!) but at least I should have checked for myself that this would even make sense!

John Onstead (Jun 09 2024 at 19:51):

Todd Trimble said:

You need to unravel this terse definition of acyclic structure in order to fully appreciate what is going on in Remark 3.2

Todd Trimble said:

Let's dig into this by making it more concrete.

I think a big lesson here is to get more concrete as John Baez often suggests. Maybe I don't do it often enough due to my preconceived notion of category theory as being an entirely abstract kind of math, but the counterexample with the groups was extremely helpful in illustrating why there couldn't be an acyclic structure on the bar construction. But since I don't have much experience yet with "getting concrete" due to this avoidance I think it'll take quite a bit of practice and getting used to! For instance, I understood well enough what an "acyclic structure" was from the comonad point of view, where it comes from, etc. (the stuff in the first paragraph by Todd Trimble), but not when it came time to "unravel" this into sequences of maps.

John Onstead (Jun 09 2024 at 19:54):

Todd Trimble said:

The result that you get after applying U is what I call the bar resolution, as distinct from the bar construction, which takes place at the level of algebras. So in other words, the coequalizer at the level of algebras is essentially never a split coequalizer, which is what the acyclic structure would give you.

Now there's good cause for a distinction between these! This has certainly come full circle from the bar construction vs resolution discussion that started off this thread.

John Onstead (Jun 09 2024 at 19:55):

Todd Trimble said:

It's an easy exercise to see this, although you should verify it for yourself and not just take my word for it.

Maybe to get some extra practice I'll give this a try in a little bit!

Todd Trimble (Jun 09 2024 at 21:41):

I think a big lesson here is to get more concrete as John Baez often suggests.

Fully agree. Abstraction can be quite wonderful because of its power of unification, its clarity, its cleanliness -- but sometimes that sense can even be enhanced by seeing what it means in a few different examples. Working through some telling examples is where the rubber meets the road, and it makes you a stronger category theorist! :-)

Todd Trimble (Jun 09 2024 at 21:48):

the nlab page on "simplicial resolution" defines it both in terms of colimits (an object that is the colimit of some simplicial object) and as a special case of resolutions (so in terms of homotopy equivalences)

That should probably be fixed. When people speak of projective resolutions in e.g. homological algebra, the sense of "resolution" is exactness of the chain complex, and this is the acyclicity aspect. You were right to find the terminology confusing.

John Baez (Jun 10 2024 at 08:17):

Now I want to take some of Todd's explanations and weave them into the nLab articles, especially [[bar construction]] but also others. I think it's currently rather hard to get certain key ideas from those articles. In particular, they could use a bit of talk about the cone monad, acyclicity, decalage and how we get all this good stuff not up in EM(T) but only after applying U.

Todd Trimble (Jun 10 2024 at 11:54):

As long as you're contemplating this, John, I'll add another item that I didn't fully explain above: how exactly do you get a splitting of that fork via the coalgebra axioms? I should say right away that my explanation above deviated from my usual preferred convention for the decalage comonad: I would much prefer to set $D = [1] + (-): \Delta^{op} \to \Delta^{op}$, not $D = (-) + 1: \Delta^{op} \to \Delta^{op}$. I don't think it matters much in the end -- the discrepancy has to do with how maps in $\Delta$ are ordered, and I like to order my face maps one way and not another -- anyway let's not worry right now about this detail, because it would be a distraction. Just go along with the definition $D = [1] + (-)$.

Todd Trimble (Jun 10 2024 at 11:55):

If you look up the definition of split coequalizer (or split fork) in some standard source like Categories for the Working Mathematician, you'll find a diagram with five maps

$X \overset{f}{\underset{g}{\rightrightarrows}} Y \overset{p}{\to} Z, \qquad Y \underset{i}{\leftarrow} Z, \qquad X \underset{j}{\leftarrow} Y$

satisfying three equations: $pi = 1_Z$, $fj = 1_Y$, and $gj = ip$. All three are encoded in the structure of a right $D$-coalgebra. To see this, let $!: [0] \to [1]$ be the unit of the monoid $[1]$ back in $\Delta$. Then as discussed above, there is a face map $X(!): X(1) \to X(0)$ playing the role of the augmentation, and there are two face maps from $X(2)$ to $X(1)$ which look like this:

$X(2) \overset{X(!+1)}{\underset{X(1+!)}{\rightrightarrows}} X(1)$.

Todd Trimble (Jun 10 2024 at 11:55):

Make the mental translation $X = X(2), Y = X(1), Z = X(0)$ and also $f = X(!+1)$ and $g = X(1+!)$ and $p = X(!)$. And in the presence of a right $D$-coalgebra structure $h: X \to XD$, mentally translate $i = h(0)$ and $j = h(1)$. The counit of our comonad on simplicial objects, whose value at a simplicial object $X$ is a simplicial object map of the form $XD \to X$, has components $X(!+n): X(1+n) \to X(0+n)$, and the counit axiom for the coalgebra structure $h$, which says that the composite

$X \overset{h}{\to} XD \to X$

is the identity, gives (upon setting $n = 1$ and $n = 0$) the two equations $fj = 1_Y$ and $pi = 1_Z$. The third equation, $gj = ip$, is a consequence of the naturality of $h$, namely it's the naturality square that says

$X(1+!) \circ h(1) = h(0) \circ X(!)$.

Todd Trimble (Jun 10 2024 at 11:55):

Finally, I strongly recommend seeing how the equations play out in the bar resolution, where these counit axioms correspond to instances of triangular equations, and the naturality square looks like an interchange. It looks especially nice in string diagram form, where these triangular equations look like "yanking moves", and the naturality/interchange looks like moving a cup past a cap.

John Baez (Jun 12 2024 at 09:57):

I added an overview of the bar construction and bar resolution to the top of this page:

• nLab, Bar construction: idea.

I hope that this will make life easier for future John Onsteads.

John Baez (Jun 12 2024 at 11:03):

Maybe @John Onstead knows places on the nLab that use "bar construction" to mean "bar resolution" or vice versa. Certainly the math literature as a whole doesn't seem to draw a sharp distinction between these two (as I mention at the above link). But I think it's a useful distinction, and it's made quite clearly on the page [[bar construction]], so it might be good to push it forward throughout the nLab.

Tim Hosgood (Jun 12 2024 at 11:53):

reading the new overview, the sentence

[...] systematically “puffs it up”, replacing it with a simplicial object  in which all equations in the original algebra are replaced by 1-simplices [...]

makes me think of constructing the groupoid $\mathbb{B}G$ from a group $G$, since the edges are precisely the "equations" $hg^{-1}\colon g\mapsto h$. is this delooping groupoid somehow related to the bar construction as well, as a sort of 1-coskeletal case?

John Baez (Jun 12 2024 at 12:27):

Actually the construction of this groupoid from a group $G$ is a special case of the bar construction.

John Baez (Jun 12 2024 at 12:28):

(deleted)

Tim Hosgood (Jun 12 2024 at 12:28):

(sorry, I accidentally wrote $\mathbb{B}G$ instead of $\mathbb{E}G$ in the above, but it would be cool if the map $\mathbb{E}G\twoheadrightarrow\mathbb{B}G$ shows up too!)

John Baez (Jun 12 2024 at 12:34):

You look at the monad $T$ on $\mathsf{Set}$ whose algebras are $G$-sets, namely the monad sending any set $X$ to $G \times X$:

$T(X) = G \times X$

Then you take the one-element set $\ast$ thought of as a $G$-set in a trivial way. So this lives in the category of algebras of $T$, called $\mathsf{Set}^T$.

The bar construction takes any algebra of any monad and turns it into a simplicial object in the category of algebras. So we apply this to $\ast$ and we get a simplicial $G$-set $E G$, where each equation like $gx = y$ is turned into a 1-simplex going from a 0-simplex called $(g,x)$ to a 0-simplex called $(1,y)$.

The general yoga of the bar construction, described in the nLab page, says that $E G$ is contractible.

John Baez (Jun 12 2024 at 12:36):

So, we've 'puffed up' the point $\ast$ to a big fat contractible blob on which $G$ acts freely.

John Baez (Jun 12 2024 at 12:38):

I guess another thing I should do is add more examples of the bar construction to the nLab page, so people can see how ubiqitous it is. Here are some @Tim Hosgood has mentioned:

• the Cech nerve of a covering
• the classifying space $B G$ of a group (thought of as a simplicial set).

John Baez (Jun 12 2024 at 14:17):

Okay, I added a bit about the classifying space $BG$. Would this have helped, @Tim Hosgood?

Tim Hosgood (Jun 12 2024 at 15:19):

this is really nice!

Tim Hosgood (Jun 12 2024 at 15:20):

my only remaining question would be if the bar construction somehow gives you either (a) the map from $EG\to BG$, or (b) a reason to consider the quotient $EG/G$

Tim Hosgood (Jun 12 2024 at 15:21):

as in, i know that i should care about this quotient (or the map into it) because i already care about bundles and descent, but does the bar construction itself tell me that i should care about this thing?

John Baez (Jun 12 2024 at 15:28):

I think it should, I think if we treat EG as a left G-space as in my writeup (mildly unorthodox) then we are encouraged to tensor it with a right G-space X and get the 'homotopy quotient' of X by G,

X//G := (X $\times$ EG)/G,

which is BG when X is a point.

John Baez (Jun 12 2024 at 15:35):

This homotopy quotient X//G is important for other G-spaces X too: for example, its cohomology is called the [[equivariant cohomology]] of X.

John Baez (Jun 12 2024 at 15:38):

But I also think this pattern: taking a rather dull algebra of a monad, taking its bar resolution, and then tensoring that or homming that into something, is a widespread pattern in math!

John Baez (Jun 12 2024 at 15:39):

I think this is how we define Tor and Ext groups, for example.

John Baez (Jun 12 2024 at 15:45):

Here however we are taking the monad for (left) G-sets, taking a boring algebra of this (a point), applying the bar construction to get EG, and then tensoring that with a right G-space X to get the homotopy quotient X//G = (X $\times$ EG)/G

John Baez (Jun 12 2024 at 15:48):

I feel I don't completely understand this, but I feel that understanding it would unify a lot of tricks people use.

Tim Hosgood (Jun 12 2024 at 15:54):

i've not before seen that the (homotopy) quotient could be thought of as given by tensoring with X, could you maybe explain this a tiny bit?

John Baez (Jun 12 2024 at 16:01):

Tensoring a right G-set and a left G-set gives a set just like tensoring a right R-module and a left R-module gives an abelian group. Does that make sense? I can write down the formula for it if not.

John Baez (Jun 12 2024 at 16:04):

If that makes sense, then maybe you can see that

(X $\times$ EG)/G

is tensoring the right G-space X with the left G-space X.

John Baez (Jun 12 2024 at 16:06):

And you may or may not know this as the homotopy quotient of X by G.

John Baez (Jun 12 2024 at 16:08):

I guess I don't know exactly what you want to know about!

Tim Hosgood (Jun 12 2024 at 16:23):

ah ok, no, this does make sense to me actually, I just blanked on the fact that $(X\times EG)/G$ is exactly what I usually see written as $X\times_G EG$

Mike Shulman (Jun 12 2024 at 16:26):

If you think about [[two-sided bar constructions]], then $EG = B(G,G,*)$ and $BG = B(*,G,*)$ and the map $EG\to BG$ is induced by the map $G\to *$ and functoriality.

John Baez (Jun 12 2024 at 16:26):

Okay, that's good.

John Baez (Jun 12 2024 at 16:27):

Tim Hosgood said:

ah ok, no, this does make sense to me actually, I just blanked on the fact that $(X\times EG)/BG$ is exactly what I usually see written as $X\times_G EG$

You mean $(X\times EG)/G$, but okay, good - I didn't want to write $X\times_G EG$ because fewer people know what $\times_G$ means! (It could even mean a pullback to some people, and I think that would just be wrong here, though I'd be happy if it actually is somehow.)

John Onstead (Jun 12 2024 at 20:25):

John Baez said:

I hope that this will make life easier for future John Onsteads.

Thanks for the writeup, it is very clarifying! It really does incorporate a lot of the things we were discussing above. If you are looking for more special cases I would next connect this idea to the Cech nerve, either on the bar resolution or Cech nerve page.

John Baez (Jun 12 2024 at 22:12):

Yes, I'll add the Cech nerve example to the list of examples on the [[bar construction]] page, and also another fun example: the associahedron! @Todd Trimble gave a nice way to build this using the bar construction.

John Onstead (Jun 18 2024 at 06:16):

The discussion on the topos theory blogs about "stitching together" things reminded me of why I originally started this thread, which is to eventually find a way to connect resolutions, descent, and local-global into one big story. I feel comfortable moving more in that direction now that the confusions about resolutions, how to define them, and the different kinds of resolution are all resolved. I think mainly my confusion there stemmed from the nlab (which is now being fixed, thanks!) and the fact that the concept of the bar construction was at the confluence of an enormous number of distinct concepts, but I'm a little more confident I have it all sorted out now.
Following John Baez's post about the historical motivations of resolutions (which we've been travelling backwards along, going from abstract to less abstract), the next stage is to discuss chain complexes and their relation to homology and cohomology. So I have these starter questions: first, what is the relationship between a chain complex and a simplicial object in an additive category (an Ab-enriched category with finite coproducts)? Does del or del^op have a natural Ab-enrichment that allows Ab-enriched functors from it to Ab-enriched categories, and if so, then are these Ab-enriched functors interpreted as the simplicial objects? Lastly, how does cohomology and chain complexes relate together (this last one might be a big question, but maybe an overview of the connection would help)?

Todd Trimble (Jun 18 2024 at 10:19):

which is now being fixed, thanks!

What needed fixing, exactly?

Todd Trimble (Jun 18 2024 at 10:19):

So I have these starter questions: first, what is the relationship between a chain complex and a simplicial object in an additive category (an Ab-enriched category with finite coproducts)?

There's a lot to say here as well, but the most basic relationship is that from a simplicial object $C = (C_n)_{n \geq 0}$ with values in an Ab-enriched category, you can construct a chain complex by taking alternating sums of face maps. With the normal grading used by topologists, there are $n+1$ face maps $C_n \to C_{n-1}$, say $d_n^i$ for $i = 0$ to $n$, and the differential $\partial_n: C_n \to C_{n-1}$ is defined by $d_n^0 - d_n^1 + d_n^2 - \ldots$. The standard face map identities ensure that $\partial_{n-1} \circ \partial_n = 0$, and so a chain complex is obtained this way.

I probably first encountered this when I was studying singular homology. The procedure is that given a topological space $X$, you (1) form a simplicial set as a composite

$\Delta^{op} \overset{\sigma^{op}}{\to} \mathsf{Top}^{op} \overset{\mathsf{Top}(-, X)}{\longrightarrow} \mathsf{Set}$

where $\sigma: \Delta \to \mathsf{Top}$ is the standard affine simplex functor. Then (2) follow this by the free abelian group functor $\mathsf{Set} \to \mathsf{Ab}$ (or it could be a free $R$-module functor; there are various options) to get a simplicial abelian group. Then (3) form the chain complex from that, as above.

Todd Trimble (Jun 18 2024 at 10:20):

Perhaps John will come by to sprinkle some sugar on this. But this is a bare-bones description of singular homology.

But steps (2) and (3) can be applied to lots of simplicial sets. For example, those that arise from bar constructions. This leads to things like group homology and cohomology.

Josselin Poiret (Jun 18 2024 at 13:05):

About this, I was just reading this introduction by Urs to see how others would introduce homological algebra from a high level viewpoint, and there's a very nicely written explanation of why you could consider homology as an abelianized version of homotopy.

Josselin Poiret (Jun 18 2024 at 13:06):

John Onstead said:

what is the relationship between a chain complex and a simplicial object in an additive category

This is the contents of the [[Dold-Kan correspondence]], although in this weaker case it might not exactly apply, you might want to consider the abelian category case first.

John Baez (Jun 18 2024 at 13:57):

Todd Trimble said:

So I have these starter questions: first, what is the relationship between a chain complex and a simplicial object in an additive category (an Ab-enriched category with finite coproducts)?

There's a lot to say here as well...

Maybe you (or @Morgan Rogers (he/him))could move your post on this to the thread Dold-Kan correspondence? That's where I put my reply. My theory is that it's better to have different threads on different topics rather than one massive thread on lots of different questions.

John Baez (Jun 18 2024 at 13:59):

(It's easy for the author of a post to move it, impossible for anyone else to move it - except a moderator.)

Todd Trimble (Jun 18 2024 at 16:26):

No, I'm not going to move it there, because I did not bring up Dold-Kan correspondence. I was maybe going to say something about that later, but I wanted to put down just the simplest idea here first, where I think it still belongs.

I will not complain though if someone else wants to move it over there.

John Onstead (Jun 30 2024 at 06:10):

Hi, wanted to give an update so it doesn't look like I've left anybody hanging! I've been real busy for the past two weeks but in a few days I'll be able to more fully get into this and the Dold-Kan correspondence. From what I've seen so far it looks interesting!