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Stream: theory: algebraic topology

Topic: bar construction

Patrick Nicodemus (Oct 24 2021 at 19:55):

I have seen fairly general treatments of the bar construction that encapsulate many cases. The treatments i have seen involve using adjunctions and monads to construct "simplicial resolutions" of an object, and then one takes the associated chain complex, either the unnormalized or normalized version.

One of the examples in the textbook i'm studying doesn't immediately seem to be of this form, per se. I'm a bit perplexed.

Patrick Nicodemus (Oct 24 2021 at 19:59):

Suppose $V$ is an associative, unital algebra, say over a PID or a field $k$. We assume that $V$ is augmented, so it has a map $\varepsilon : V\to k$ which is left inverse to the unit map / structure map of the algebra which satisfies some coherence conditions wrt to the multiplication on $V$ (it should be a monoid homomorphism). Let $\overline{V}$ be the kernel of $\varepsilon$.

The bar construction of $V$ is the chain complex $BV$ with $BV_n = \overline{V}^{\otimes n}$. The differential is the alternating sum of multiplications $d(v_1\otimes\dots v_n) = v_1v_2\otimes v_3\otimes v_4\dots \otimes v_n - v_1\otimes v_2v_3\otimes v_4\dots$

Patrick Nicodemus (Oct 24 2021 at 20:00):

the differential $BV_1\to BV_0$ is just the zero map.

Patrick Nicodemus (Oct 24 2021 at 20:00):

My question is , is there a nice categorical characterization of this construction using monads or adjunctions or something of this ilk? It's not clear to me.

Patrick Nicodemus (Oct 24 2021 at 20:01):

I am looking at this in Loday-Quillen's book Algebraic Operads. The way they characterize the differential is that it is the unique graded derivation of degree $-1$ on this graded vector space which extends the multiplication map $BV_2\to BV_1$. This is nice, but not really what i'm looking for.

Patrick Nicodemus (Oct 24 2021 at 21:18):

Nvm lol. I think I figured it out. Mark as solved etc

Morgan Rogers (he/him) (Oct 25 2021 at 08:45):

Care to share what you concluded? It's always more satisfying to hear what people have learned in the end!

Patrick Nicodemus (Oct 25 2021 at 21:39):

Yeah, Morgan, I totally forgot that there are two things called the bar construction. One is used for computing nice resolutions of objects or fibrant/cofibrant replacement. and the other one is used to give a kind of "delooping". Of course they are very closely related but they seem to have pretty different properties, as far as I can tell

Patrick Nicodemus (Oct 25 2021 at 21:43):

I was reading the words "bar construction" in the textbook and thinking of the version of the bar construction where you have some monad or comonad and you iterate it repeatedly to give rise to a kind of simplicial or cosimplicial resolution of the object. The bar construction in this book i think is an example of the more general version given here

Patrick Nicodemus (Oct 25 2021 at 21:43):

https://ncatlab.org/nlab/show/two-sided+bar+construction

Patrick Nicodemus (Oct 25 2021 at 21:44):

You want to think of the unit $1$ as both a left and right module of $V$ and carry out the construction of $B(1,V,1)$, and that'll give you this complex after taking the Moore normalization.

Patrick Nicodemus (Oct 25 2021 at 22:25):

Actually I have a new question now.

Patrick Nicodemus (Oct 25 2021 at 22:26):

My question is: Intuitively, if $V$ is an augmented algebra, why should $BV$ be a coalgebra? The comultiplication map is fairly simple and straightforward but why would you expect it to be an interesting aspect, or why a priori would you think that $BV$ would be a coalgebra because of what it represents

Patrick Nicodemus (Oct 27 2021 at 01:34):

Patrick Nicodemus said:

For a while I have been studying a construction involving the Day convolution of presheaves, which can be understood as a Kan extension, and recently I tried to dualize it to get a similar construction for the right adjoint internal hom functor, but I had a lot of difficulty because of the mixed variance of Hom, contravariant in one variable and covariant in another. Some constructions I wanted to carry out were clearly not functorial.
However, I realized how to solve my problem! The issues with covariance and contravariance can be solved by passing from the 2-category of categories to the bicategory of profunctors. There, I was able to prove that I can give the analogous construction for Hom as a right Kan extension in the category of profunctors :big_smile:

(This is closely related to the cobar construction, which i have been struggling to interpret categorically)

Patrick Nicodemus (Nov 17 2021 at 13:10):

I am studying the book on Algebraic Operads by Loday and Vallette and it treats everything in terms of chain complexes. Theoretically, by Dold-Kan, everything we say about chain complexes could be said in terms of simplicial vector spaces and vice versa, it is just a matter of computational convenience and translation. Can anyone recommend any books or papers where some of the same basic subject matter is treated in the category of simplicial vector spaces - for example, twisting morphisms, bar/cobar duality, etc, all treated without converting everything to chain complexes? I have not found such papers.

Patrick Nicodemus (Nov 17 2021 at 13:11):

I have been trying to translate it myself to get a better understanding for the constructions but it is tedious and there is lots of possibility for error

Tim Hosgood (Nov 17 2021 at 15:44):

I don't know any such references, but would be interested to hear an answer to this too!

Peter Arndt (Nov 18 2021 at 09:21):

You might want to look into the derived algebraic geometry literature. There they study derived schemes, which are like schemes, only not glued together from rings (i.e. commutative monoids in abelian groups with $\otimes$) but from $E_\infty$-algebras in simplicial abelian groups or in simplicial abelian k-algebras. In characteristic $0$ one can replace simplicial abelian groups by chain complexes, but not in characteristic >0.

Peter Arndt (Nov 18 2021 at 09:32):

If it is readable to you, you might also look at Lurie's Higher Algebra from Prop. 7.1.4.11 onwards.

Patrick Nicodemus (Apr 07 2022 at 13:37):

Suppose we have an adjunction $F: C\to D$, $U : D\to C$ with $F$ left adjoint to $U$. Write $G=FU$ for the comonad arising from the adjunction. If we use $\eta : \mathbf{id}_C \to UF$ and $\varepsilon : FU \to \mathbf{id}_D$ for the unit and counit arising from the multiplication, then write $\delta$ for the comultiplication $F\eta U$ of $G$.

If $X$ is an object in $D$, then the bar construction allows us to "resolve" $X$ by a simplicial object whose objects are all "free". The meaning of free here is up for debate, but we could take as the definition of "free" to mean: lying in the image of $F$, lying in the image of $G$, etc. I prefer to take as the definition of free, "is a $G$ coalgebra" as this is weak enough that it subsumes the other two cases but strong enough that we can derive interesting properties of free objects, for example a projective lifting property.

Thus I am tempted to say: The bar construction resolves an object in $D$ by a simplicial object composed of $G$ coalgebras. However, this is not a simplicial $G$ coalgebra, as the first face map $d_0 : GGX \to GX$ is $\varepsilon_G$, which is not a coalgebra map (where $GGX$ and $GX$ carry the expected free coalgebra structure.)

Therefore we have a simplicial object where all objects are $G$ coalgebras and all face and degeneracy maps are $G$ coalgebras, _except_ $d_0$. It is a bit of an awkward exception!

I am wondering if anyone can tell me if such objects are studied in the literature, or more generally, are there interesting properties of simplicial sets which require that all face and degeneracy maps except for $d_0$ satisfy a given property. I have come across some elsewhere, but I don't know if it is connected.

John Baez (Apr 07 2022 at 16:21):

My mind is not focused on this stuff now, but I think what nature is telling you is that it's wrong to think of a resolution the way you're doing. E.g. in the case of abelian groups, don't think of a resolution of $A$ as a chain complex

$\cdots \to A_2 \to A_1 \to A_0 \to A$

Think of it as a chain complex

$\cdots A_2 \to A_1 \to A_0$

which is equipped with a map of chain complexes to

$\cdots 0 \to 0 \to A$

This map is a quasi-isomorphism, and it's a cofibration, so its exhibiting

$\cdots A_2 \to A_1 \to A_0$

as a cofibrant replacement of the original $A$. This was the philosophy I was espousing earlier.

The same idea should also work in the more explicitly simplicial framework.

John Baez (Apr 07 2022 at 16:29):

Patrick Nicodemus said:

If $X$ is an object in $D$, then the bar construction allows us to "resolve" $X$ by a simplicial object whose objects are all "free". The meaning of free here is up for debate, but we could take as the definition of "free" to mean: lying in the image of $F$, lying in the image of $G$, etc. I prefer...

Yeah, I don't think "free" should mean "lies in the image of $F$" or "lies in the image of $G$" - those are properties of an object, but I think freedom should be a structure.

John Baez (Apr 07 2022 at 16:32):

For example, you can try to define a "free group" to be a group $H$ with the property that there exists some set $X$ such that $H$ is isomorphic to the free group on $X$, but I think this is less useful, in general, than the concept of a group being "free on the set $X$", which is a structure on a group, involving a universal property and a specific set $X$.

John Baez (Apr 07 2022 at 17:56):

Btw @Patrick Nicodemus, if you had read my post about "decalage" you'll see now that I've replaced it by something better about how to think of a free resolution as a simplicial object, namely not having the original object $A$ as the object of 0-simplices, but the free object $A_0$.

John Baez (Apr 07 2022 at 17:58):

If you want to understand this stuff really thoroughly I recommend this:

• Todd Trimble, Bar constructions.

John Baez (Apr 07 2022 at 19:32):

The punchline is the main theorem. It's stated in huge generality, but the basic idea is that starting with an algebra X of a monad M, the bar construction produces a simplicial algebra of that monad, characterized by the fact that it's the "initial X-acyclic" simplicial algebra of M.

John Baez (Apr 07 2022 at 19:35):

The "X-acyclic" structure acts like a deformation retraction from this simplicial algebra to X (which can be thought of as a simplicial algebra as well in a trivial sort of way).

Naso (Apr 08 2022 at 01:17):

John Baez said:

Btw Patrick Nicodemus, if you read my tweet about "decalage" you'll see now that I've replaced it by something better about how to think of a free resolution as a simplicial object, namely not having the original object $A$ as the object of 0-simplices, but the free object $A_0$.

Can I have a link to that tweet please @John Baez ?

John Baez (Apr 08 2022 at 05:33):

I said "tweet" but I meant "post here on Zulip". The post about "decalage" is gone; the post I'm talking about is a few posts above yours, starting with "My mind is not focused on this stuff now". It may only make sense in context of the whole conversation we've been having here, which is about the bar construction.

Patrick Nicodemus (Apr 08 2022 at 13:45):

John Baez said:

My mind is not focused on this stuff now, but I think what nature is telling you is that it's wrong to think of a resolution the way you're doing. E.g. in the case of abelian groups, don't think of a resolution of $A$ as a chain complex

$\cdots \to A_2 \to A_1 \to A_0 \to A$

Think of it as a chain complex

$\cdots A_2 \to A_1 \to A_0$

which is equipped with a map of chain complexes to

$\cdots 0 \to 0 \to A$

This map is a quasi-isomorphism, and it's a cofibration, so its exhibiting

$\cdots A_2 \to A_1 \to A_0$

as a cofibrant replacement of the original $A$. This was the philosophy I was espousing earlier.

The same idea should also work in the more explicitly simplicial framework.

John I may be wrong but I think the problem persists in each degree, not only at degree $0$ or $-1$ where the augmentation lives. So I don't think it's an issue of viewing it as one simplicial object or two.

If we want to resolve $A$ by the (unaugmented) simplicial object $\dots A_2\to A_2 \to A_0$ where we take $A_n = G^{n+1}(A)$, we have an unaugmented simplicial object

$\to G^3A\to G^2A\to GA$

In each degree we have a map $d_0 : A_{n+1}\to A_n$ given by $\varepsilon_{G^n}$, and this is not a coalgebra map.

I have read Todd's letter, that's what started me in this direction. It's a great letter.

Hmm let me explain where I'm coming from. Recall the basic lemma from homological algebra that says: if $A, B$ are modules, and we have $P_\bullet$ a chain complex of projectives augmenting $A$ (not necessarily exact) and $Q_\bullet$ an exact chain complex resolving $B$ (not necessarily projective) then for a map $f: A\to B$ one has a chain map $f_\bullet : P_\bullet\to Q_\bullet$, unique up to homotopy, extending $f$. Here there is a notion of "exact" and also a notion of "complex of projectives" and they're distinct. I'm trying to prove a similar theorem for the bar construction and it requires disentangling these two notions of "exact" and "complex of projectives". So when is a simplicial algebra "exact" and when is it a "complex of projectives"? I think that in Todd's letter, being $U$-acyclic arguably captures what it means for a simplicial $T$ algebra to be "exact". Regardless of whether a simplicial $T$ algebra is $U$ acyclic, I want to know when we can call a simplicial $T$ algebra "free". Is it simply that each object in the complex is a free $T$ algebra, or do we need the face and degeneracy maps to satisfy some kind of coherence condition with respect to the free structures?

I hope that this helps to clarify my question.

Patrick Nicodemus (Apr 08 2022 at 14:13):

I have my own concept of the analogous definition of a "complex of projectives", which I think will end up letting me prove my theorem, but it's awkward. Namely a simplicial $T$ algebra is said to be "presentable" if every object in the complex is a $G$ coalgebra (where $G$ is the comonad arising from the adjunction $C \iff C^T$) and all face and degeneracy maps are coalgebra maps except possibly for $d_0$. This is admittedly a strange definition. Note that the coalgebras for the comonad of the adjunction between Sets and Ab are exactly the free Abelian groups, so in this context a $G$ presentable simplicial abelian group is one where all groups are free and all face and degeneracy maps carry basis elements to basis elements except for $d_0$, which can be an arbitrary homomorphism.

Reid Barton (Apr 08 2022 at 14:23):

One trivial comment: it would not make sense to ask for a "simplicial free object" in the sense that the objects are coalgebras and all the face & degeneracy maps are coalgebra maps, because then the colimit (= thing being resolved) would have to be free also.

Reid Barton (Apr 08 2022 at 14:24):

The "freeness as structure" idea is reminiscent of [[algebraic weak factorization systems]]--maybe there is some awfs on simplicial abelian groups (say) that corresponds to what you want to capture.

John Baez (Apr 08 2022 at 20:21):

Patrick wrote:

all face and degeneracy maps are coalgebra maps except possibly for $d_0$

As I said, I believe the right way to think about this stuff avoids this "except possibly for $d_0$" business. That clause is ugly. Ugliness has no place in math of this sort. I think the ugliness arises from taking a wrong viewpoint on what's going on. I tried to sketch the right viewpoint, but only in the special case of abelian groups, where the simplicial objects can be depicted as chain complexes. I again urge you to read Todd Trimble's notes for the general story.

Todd Trimble (Apr 08 2022 at 20:32):

I'm not sure I would go as far as John! Parts of what Patrick wrote do remind me of things that crop up in this business where one speaks of things like "extra degeneracies", which in the nLab article refers to the coalgebra structure over the decalage comonad, which puts the "resolution" in "bar resolution".

If I spend a little more time on it, I might come up with something more fully responsive to Patrick, but for now I'd like instead to sketch a picture which ought to be standard, but I'm not sure it is. It's of a certain 2-category with three objects $0, 1, 2$ that manages to package together the algebraist's simplex category $\Delta$ of finite ordinals (as $\hom(1, 1)$), its opposite $\Delta^{op}$ which can be realized concretely as the category of finite intervals = finite totally ordered sets with top and bottom (as $\hom(0, 2)$), and two more relevant categories: the category $\hom(0, 1)$ which consists of finite totally ordered nonempty sets and bottom-preserving monotone maps, and the category $\hom(1, 2)$ which consists of totally ordered nonempty sets with top-preserving monotone maps.

I would invite Patrick or anyone else to draw morphisms in this category using string diagrams, using the objects 0, 1, 2 to label planar regions.

One can think of this 2-category as the "walking 2-category with a monad $M$ (whose underlying object is $1$) together with a left $M$-module (underlying object $0$, according to one convention) and right $M$-module" (underlying object $2$). This picture is useful for contemplating 2-sided bar constructions, for example, as given by a functor of the form $\hom(0, 2) \to E$ which is an augmented simplicial object in $E$. I'm not sure where or even if this 2-category appears in the literature. @Mike Shulman might know.

$\Delta^{op}$ in some sense "sits" inside $\Delta$, insofar as the condition of preserving top and bottom is more restrictive -- adds another restriction -- to just monotonicity. By the same token, by taking opposites, $\Delta$ sits inside $\Delta^{op}$.

I have a vague sense that what Patrick is groping toward might fit somewhere within this picture, insofar as "missing morphisms" might refer to such "restrictions".

Todd Trimble (Apr 08 2022 at 20:36):

As something of an aside, but more in the direction of throwing some cold water on what Patrick is bringing up, I'll point out that if $C = Set$ and $U: D \to Set$ is monadic, then there is a very strong tendency for the category of $G$-coalgebras just to be $Set$ itself. For example, Mesablishvili proves (proposition 6.13) that if $F(!): F(0) \to F(1)$ is a regular isomorphism but not an isomorphism, then $F: Set \to D$ is comonadic. The "majority" of monads on $Set$ that occur in nature have this property.

Mike Shulman (Apr 08 2022 at 21:03):

I don't recall seeing that 3-object 2-category before, but I think if you coalesce your objects 0 and 2 you get the free 2-category containing an adjunction, which is studied very explicitly in section 3 of Riehl-Verity Homotopy coherent adjunctions and the formal theory of monads. In section 7 of All (∞,1)-toposes have strict univalent universes I sketched how this 2-category and its universal property gives rise to two-sided bar constructions.

Todd Trimble (Apr 09 2022 at 10:26):

Thanks, Mike; glad to have those references. The three object 2-category I had in mind is good for other things besides, such as the two-sided bar construction $B(1, G, 1)$ that gives the base space for the classifying bundle of a group $G$.

Mike Shulman (Apr 10 2022 at 15:01):

Yes, that makes sense.