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Stream: learning: questions

Topic: Dold-Kan correspondence

John Baez (Jun 18 2024 at 07:08):

@John Onstead wrote:

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)?

Welcome to the Dold-Kan correspondence. This is worth getting to know well, especially since it holds a subtlety: the equivalence between chain complexes in $\mathsf{AbGp}$ and simplicial abelian groups is both lax monoidal and oplax monoidal in both directions, but not strong monoidal.

John Baez (Jun 18 2024 at 07:09):

I think the Dold-Kan correspondence, if anyone wants to talk about it, deserves its own thread. So here's that thread!

Oscar Cunningham (Jun 18 2024 at 13:33):

The forgetful functor $\mathsf{AbGp}\to\mathsf{Set}$ induces a commutative monad on $\mathsf{Set}$ whose algebras are the abelian groups. This automatically gives us the tensor product of algebras over a commutative monad and internal hom of algebras over a commutative monad, which are the usual tensor product and hom of abelian groups.

But I know that the internal hom of two bounded chain complexes can be unbounded. So what goes wrong when we try to repeat this argument with $\mathsf{AbGp}^{\Delta^\mathrm{op}}\to\mathsf{Set}^{\Delta^\mathrm{op}}$?

Kevin Carlson (Jun 18 2024 at 17:05):

Well, (1), as John already signaled us to be ready for, the equivalence of simplicial abelian groups with chain complexes bounded below is not strong monoidal! So the categories are the same but the monoidal categories aren’t and we shouldn’t conflate them. And, (2), I don’t think that’s true with the definition of tensor product of complexes I know, where the $n$ th term of the tensor is the sum of $A_i\otimes B_j$ for $i+j=n.$ maybe you’re thinking of the derived tensor product?

John Baez (Jun 18 2024 at 17:09):

"I don't think that's true..." - what don't you think is true? I guess you mean the claim that the internal hom of two bounded chain complexes can be unbounded? Yeah, I don't see how that could be true, not with the usual tensor product of chain complexes.

Kevin Carlson (Jun 18 2024 at 17:10):

Right, that’s what I meant, sorry!

Oscar Cunningham (Jun 18 2024 at 17:18):

To be clear, by 'bounded' I mean 'bounded below'. The ones that Dold-Kan applies to.

I agree that the tensor product of bounded complexes is bounded, but I don't think that's true for the internal hom. The nLab says it's given by

$[X,Y]_n = \prod_{i\in\mathbb{Z}}\mathrm{Hom}(X_i,Y_{i+n})$

It looks to me that if $X$ and $Y$ aren't bounded above, then $[X,Y]$ isn't bounded above or below.

Oscar Cunningham (Jun 18 2024 at 17:25):

@Kevin Carlson Regarding (1), another way to say that would be that the category of chain complexes is equipped with two inequivalent monoidal products. The usual tensor product and one obtained from translation of the tensor product on simplicial abelian groups along the Dold-Kan correspondence. Can we say concretely what that other one is? My first guess would be the levelwise tensor product.

John Baez (Jun 18 2024 at 17:25):

I'm a bit confused. First, I'd say Dold-Kan applies to "nonnegatively graded" chain complexes: collections of vector spaces $X_i$ for $i \in \mathbb{N}$, with maps $d: X_{i+1} \to X_i$ obeying $d^2 = 0$. I'd use "bounded below" for a chain complex consisting of vector spaces $X_i$ for $i \in \mathbb{Z}$ such that $X_i = \{0\}$ for $i$ sufficiently large and negative.

John Baez (Jun 18 2024 at 17:26):

So I don't know why we're talking about chain complexes that are "bounded below".

Oscar Cunningham (Jun 18 2024 at 17:27):

Okay good point

John Baez (Jun 18 2024 at 17:27):

Second, for more or less the same reason, I don't know why you're writing $i \in \mathbb{Z}$ in this formula:

$[X,Y]_n = \prod_{i\in\mathbb{Z}}\mathrm{Hom}(X_i,Y_{i+n})$

if we're talking about the chain complexes to which Dold-Kan applies.

John Baez (Jun 18 2024 at 17:28):

I would write $i \in \mathbb{N}$ or $i \ge 0$.

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

Then the formula for the internal hom would be right, if we're talking about nonnegatively graded chain complexes, and we're assuming $n \in \mathbb{N}$ as well.

Kevin Carlson (Jun 18 2024 at 17:35):

Oscar Cunningham said:

Kevin Carlson Regarding (1), another way to say that would be that the category of chain complexes is equipped with two inequivalent monoidal products. The usual tensor product and one obtained from translation of the tensor product on simplicial abelian groups along the Dold-Kan correspondence. Can we say concretely what that other one is? My first guess would be the levelwise tensor product.

I’m not quite sure what the “usual” one is supposed to be, but I think the commutative monad will induce the levelwise product!

John Baez (Jun 18 2024 at 17:49):

What I call the "usual" tensor product of chain complexes, the one they teach us as children, is

$(X \otimes Y)_i = \bigoplus_{j+k = i} X_j \otimes Y_k$

Kevin Carlson (Jun 18 2024 at 17:50):

Oops, yeah, I misread that as the usual tensor product of simplicial abelian groups. Yeah, probably the tensor on simplicial abelian groups transfers to the level wise tensor on chain complexes.

John Baez (Jun 18 2024 at 17:55):

I always get confused about the one coming from the tensor product of simplicial abelian groups. Is it really just the "levelwise" tensor product of chain complexes:

$(X \boxtimes Y)_i = X_i \otimes Y_i$ ?

The thing is, when getting a chain complex $X$ from a simplicial abelian group, in Dold-Kan we don't just say $X_i$ is the abelian group of $i$-simplices - that would be too easy. It's the subgroup that's the intersection of the kernels of all but the last face map.

But that's not proving your guess is wrong!

John Baez (Jun 18 2024 at 17:56):

So you might be right, but it's not instantaneously obvious to me.

Kevin Carlson (Jun 18 2024 at 17:56):

Yeah, definitely a loose guess though, just because the Dold-Kan map at least just takes a subgroup at one level, so it’s not obvious how you’d do anything more clever.

John Baez (Jun 18 2024 at 18:00):

It's a bit frustrating that in all the wonderful material about this on the nLab, e.g. at monoidal Dold-Kan correspondence, I don't see anything that comes out and describes this other tensor product of nonnegatively graded chain complexes!

John Baez (Jun 18 2024 at 18:03):

And oh no - apparently two ways to turn a simplicial abelian group $A$ into a chain complex of abelian groups are both relevant here: the "obvious" way where we let $C_n(A)$ be just the abelian group of $n$-simplices, and the way I thought was the "clever" way, the normalized Moore complex, where we let $N_n(A)$ be the intersection of the kernels of all but the last face map!

John Baez (Jun 18 2024 at 18:04):

Both of them give you a functor from simplicial abelian groups to nonnegatively graded chain complexes of abelian groups that's lax monoidal, oplax monoidal, but not strong monoidal, and so on.

John Baez (Jun 18 2024 at 18:08):

So the question of what we get when we transfer the tensor product of simplicial abelian groups to the category of nonnegatively graded chain complexes of abelian groups is actually two questions.

Kevin Carlson (Jun 18 2024 at 18:10):

Also just saw what I hadn't remembered, that the quasi-inverse of the normalized Moore complex functor involves a bunch of different levels of the chain complex, so I no longer particularly suspect the transferred structure is just levelwise!

John Baez (Jun 18 2024 at 20:30):

A fun baby version of this problem is to compare

• categories internal to $\mathsf{AbGp}$

and

• 2-term chain complexes of abelian groups, $A_0 \xleftarrow{d} A_1$

John Baez (Jun 18 2024 at 20:33):

A category internal to $\mathsf{AbGp}$ has an abelian group of objects $C_0$ and an abelian group of morpisms $C_1$, and source and target maps $s, t: C_1 \to C_0$, etc. To get a 2-term chain complex starting from this (following the normalized Moore complex idea) we let

$A_0 = C_0$
$A_1 = \ker(s) \subseteq C_1$
$d = t|_{A_1}$

Interestingly composition in $C$ provides no new extra information, so it's okay that this recipe for getting the chain complex ignores composition!

John Baez (Jun 18 2024 at 20:36):

Conversely, to get a category $C$ internal to $\mathsf{AbGp}$ from a 2-term chain complex $A_0 \xleftarrow{d} A_1$, we let

$C_0 = A_0$
$C_1 = A_0 \oplus A_1$
$s(a_0, a_1) = a_0$
$t(a_0, a_1) = a_0 + d(a_1)$

and we can define composition too, using addition in $A_1$.

John Baez (Jun 18 2024 at 20:37):

The formula $C_1 = A_0 \oplus A_1$ is a baby version of how

the quasi-inverse of the normalized Moore complex functor involves a bunch of different levels of the chain complex

John Baez (Jun 18 2024 at 20:41):

The whole tensor product issue shows up in this baby version of the story, too. In some ways it's simpler than version for arbitrary positively graded chain complexes, but the restriction to 2-term chain complexes adds extra wrinkles, too.

Patrick Nicodemus (Jun 19 2024 at 02:22):

John Baez said:

What I call the "usual" tensor product of chain complexes, the one they teach us as children, is

$(X \otimes Y)_i = \bigoplus_{j+k = i} X_j \otimes Y_k$

this corresponds to the join of simplicial sets, i.e. day convolution

Patrick Nicodemus (Jun 19 2024 at 02:28):

This is worth getting to know well, especially since it holds a subtlety: the equivalence between chain complexes in AbGp and simplicial abelian groups is both lax monoidal and oplax monoidal in both directions, but not strong monoidal.

I've been meaning to write this up in detail but imo the usual story about Dold-Kan is not the most categorically natural one. I would go so far as to say that the historical convention chosen by Moore for the Moore normalization functor is in some sense "wrong" - not wrong in an objective sense but just an unfortunate convention in the sense that we get weird theorems which are difficult to interpret (both lax and oplax monoidal but not strong monoidal? what?)

Patrick Nicodemus (Jun 19 2024 at 02:28):

Let me advocate my alternative story of the Dold-Kan correspondence.

Patrick Nicodemus (Jun 19 2024 at 02:29):

The augmented simplex category is distinguished up to isomorphism by its universal property: it is the free strict monoidal category with distinguished monoid.

Patrick Nicodemus (Jun 19 2024 at 02:30):

If $C$ is a small category, then the presheaf category $\hat{C}$ is distinguished up to equivalence by its universal property: it is the free cocompletion of $C$.

Patrick Nicodemus (Jun 19 2024 at 02:35):

When $C=\Delta$, these two universal properties combine with each other in a pleasing way: the category of augmented simplicial sets is the free cocomplete monoidal category with distinguished monoid. What I mean by this is that $\mathbf{SSet}$ (I will here consider augmented simplicial sets) is cocomplete, equipped with a tensor product (Day convolution, aka the join of simplicial sets) which is cocontinuous in both arguments, and it is equipped with a distinguished monoid (the terminal object). It is universal among cocomplete categories equipped with a monoidal product, cocontinuous in both arguments, and a distinguished monoid. The most important such category is the category $\mathbf{Top}$ whose tensor product is the topological join $X\ast Y = X\coprod X\times I \times Y\coprod Y/\sim$, where $x\sim (x,0, y)$ and $(x,1,y)\sim y$ and whose distinguished monoid is the singleton.

Patrick Nicodemus (Jun 19 2024 at 02:38):

One checks that this all goes through in the $\mathbf{Ab}$-enriched setting, and that the category of simplicial Abelian groups is universal in the 2-category $\mathbf{Ab}-\mathbf{cat}$ among cocomplete categories (in the Ab-enriched sense, so including the tensor product of an object with an Abelian group) equipped with a tensor product cocontinuous in both arguments (the join again, but with Cartesian product of sets replaced by tensor product of Abelian groups.) The distinguished monoid is the constant presheaf at $\mathbb{Z}$

Patrick Nicodemus (Jun 19 2024 at 02:43):

The category of chain complexes of Abelian groups is also an Ab-enriched category with a tensor product (the usual tensor product of chain complexes mentioned earlier in this thread) and it has a distinguished monoid, $0\to \mathbb{Z}\to\mathbb{Z}\to 0$, concentrated in degrees zero and one. Now, note that if this were concentrated in degrees 0 and -1 instead of 0 and 1, we might squint our eyes and say, the generator in degree 0 corresponds to a point, and the generator in degree -1 corresponds to the empty simplex - one might take it or leave it out depending on convention, and this leads to the distinction between ordinary homology and reduced homology. However, our complex is concentrated in degrees 0 and 1, not 0 and -1. This distinction can be explained by choosing to index by the number of vertices in the simplex, not the dimension of the simplex - the 0 simplex is the convex hull of 1 point, so we place the generator in dimension 1. the -1 simplex is the convex hull with zero points, so we place the generator in dimension 0.

Patrick Nicodemus (Jun 19 2024 at 02:44):

So, I prefer to think of this complex $0\to\mathbb{Z}\to\mathbb{Z}\to 0$ as being a formal analogue of the singleton space.

Patrick Nicodemus (Jun 19 2024 at 02:46):

By the universal property of $\mathbf{SAb}$ in $\mathbf{Ab}-\mathbf{cat}$ which we stated previously, there is an Ab-enriched cocontinuous functor preserving the monoidal product and the distinguished monoid, unique up to natural isomorphism, $F : \mathbf{SAb}\to \mathbf{Ch}(\mathbf{Ab})$, and it turns out that this functor is an equivalence of categories

Patrick Nicodemus (Jun 19 2024 at 02:48):

Now, I wish we could say "and this is precisely the Dold-Kan correspondence!"

However, the devil is in the details. I have described an equivalence between augmented simplicial Abelian groups and chain complexes, not between unaugmented simplicial Abelian groups and chain complexes.

The functor $F$ I have described is, by construction, a (strong) monoidal equivalence

Patrick Nicodemus (Jun 19 2024 at 02:49):

The difference is in the indexing, and that is a big difference indeed. Reindexing the groups of a chain complex by one is called "suspension" and it is a nontrivial geometric operation. In particular, suspension gives an equivalence between $\mathbf{Ch}_{\geq 0}(\mathbf{Ab})$ and $\mathbf{Ch}_{\geq 1}(\mathbf{Ab})$, but not a monoidal equivalence.

Patrick Nicodemus (Jun 19 2024 at 02:52):

The usual Dold-Kan correspondence can be described as follows:

1. Given an unaugmented simplicial Abelian group, regard it as an augmented simplicial Abelian group by extending it with the zero group as augmentation
2. Apply the functor $F$ described above
3. Shift everything down by one

Patrick Nicodemus (Jun 19 2024 at 02:58):

If we return to John's original comment that

the equivalence is... both lax monoidal and oplax monoidal in both directions, but not strong monoidal.

I would prefer to say this as follows:
The suspension functor $\Sigma : (\mathbf{SAb}, \boxtimes, \mathbb{Z}) \to (\mathbf{SAb}, \otimes_{\mathbf{Day}}, \mathbf{1}_{\mathbf{Day}})$ is lax monoidal and oplax monoidal, but not strong monoidal.

($\mathbf{1}_{\mathbf{Day}}$ is the augmented simplicial Abelian group which is $\mathbb{Z}$ in degree zero and $0$ in all higher degrees.)

Patrick Nicodemus (Jun 19 2024 at 03:00):

There is no need to bring chain complexes into it anywhere, in my view - fundamentally, this is a theorem about the relationship between the degreewise tensor product (which corresponds geometrically to the Cartesian product of simplicial complexes) and the "total" tensor product (which corresponds geometrically to the join of simplicial complexes)

Patrick Nicodemus (Jun 19 2024 at 03:02):

And this theorem clearly corresponds to standard theorems in topology -
https://math.stackexchange.com/questions/3709847/prove-that-the-reduced-join-of-2-spaces-is-homeomorphic-to-the-reduced-suspensio
the reduced join of two spaces is homeomorphic to the reduced suspension of their smash product.

Patrick Nicodemus (Jun 19 2024 at 05:04):

John Baez said:

And oh no - apparently two ways to turn a simplicial abelian group $A$ into a chain complex of abelian groups are both relevant here: the "obvious" way where we let $C_n(A)$ be just the abelian group of $n$-simplices, and the way I thought was the "clever" way, the normalized Moore complex, where we let $N_n(A)$ be the intersection of the kernels of all but the last face map!

this "obvious" way is the categorically natural way to convert an augmented semisimplicial Abelian group into a chain complex. The category of semisimplicial Abelian groups is the free Ab-enriched cocomplete monoidal category with a distinguished pointed object $e : 1_{\mathbf{Day}}\to\mathbb{Z}$. If you apply this universal property to $\mathbf{SSAb}\to \mathbf{Ch(Ab)}$ you get the alternating sum of face maps functor. By construction this is again strong monoidal with respect to the Day convolution.

Once you realize this, it seems somewhat obvious that the alternating sums definition is not the "right" one for a simplicial Abelian group as it doesn't incorporate the degeneracies. The Moore complex, on the other hand, can be defined in terms of quotienting out by the degeneracies, which makes it feel a bit more natural.

John Baez (Jun 19 2024 at 08:07):

Wow, this is a great story, @Patrick Nicodemus - a substantial improvement on all the stories I've heard about this subject. It's so much nicer to use universal properties, as you're doing, rather than to check tricky formulas. And admitting the importance of how the augmented simplex category is the free monoidal category on a monoid object seems like the way to use this. But this introduces a reindexing issue (since the augmented simplex category has "(-1)-simplices" in the usual numbering system). But you address this using suspension.

John Baez (Jun 19 2024 at 08:19):

Patrick Nicodemus said:

And this theorem clearly corresponds to standard theorems in topology -
https://math.stackexchange.com/questions/3709847/prove-that-the-reduced-join-of-2-spaces-is-homeomorphic-to-the-reduced-suspensio
the reduced join of two spaces is homeomorphic to the reduced suspension of their smash product.

I'll need to think about that and see if I can visualize it. I'm not sure I ever thought about the reduced [[join of topological spaces]]. I'm giving a link that explains the join, but not the reduced join. Heuristically we get the join $X \star Y$ of $X$ and $Y$, by drawing an interval between each point of $X$ and each point of $Y$, and then I guess we get the reduced join by taking $X \star Y$ and collapsing the original copy of $X$ to a point and collapsing the original copy of $Y$ to a point.

John Baez (Jun 19 2024 at 08:31):

Anyway, I hope you make your story available to the world at large. If writing a paper is enough work that you tend to put it off, you could at least put what you've just written now into the nLab. (You might not want to do that.) It would be great if the rather intricate story about how the [[Eilenberg-Zilber map]] is the laxator for the functor from simplicial abelian groups to chain complexes, and how the [[Alexander-Whitney map]] is the oplaxator, could be clarified using your thoughts. But I think if you get the basic ideas out into the world, other people may use them to clarify these related constructions.

Todd Trimble (Jun 19 2024 at 15:39):

Yes, it is indeed a very nice story; thank you very much Patrick. In spirit, in its emphasis on monoidal doctrines that distribute over free cocompletion (in an enriched sense), it reminds me very much of the story behind Kelly's On the Operads of J.P. May, as Kelly taught me in brilliantly clear form when I was a post-doc at Macquarie.

Patrick Nicodemus (Jun 19 2024 at 16:06):

Just a quick note on Eilenberg Zilber and Alexander Whitney while we are still having this conversation.
I'm going off old memories right now but I'll provide details when I get a chance. Hopefully this is correct. Let $\mathbb{Z}[\Delta]$ be the free Ab enriched category on the augmented simplex category. (Take free Abelian groups on Hom sets and extend composition by bilinearity). By Yoneda it should be clear that many natural operations on presheaves can be reduced to operations in this category. In particular it should be the case that Eilenberg-Zilber and Alexander-Whitney should both be describable somehow without using any presheaves. I believe it is something like

Let $P$ be the functor $\mathbb{Z}[\Delta]\otimes \mathbb{Z}[\Delta]\to\mathbb{Z}[\Delta]\otimes \mathbb{Z}[\Delta]$
defined by $(n,m)\mapsto (n+m,n+m)$.

I believe that the identity functor is a retract (in the category of functors and additive natural transformations) of $P$.
One of the natural transformations here is essentially the Eilenberg-Zilber map and the other is the Alexander Whitney map.

Other interesting things about chain complexes also can be observed in this category for Yoneda reasons. For example $\mathbb{Z}[\Delta]$ must be a symmetric monoidal category because the tensor product of chain complexes is symmetric monoidal. Even though $\Delta$ is of course not symmetric monoidal, we can define a symmetry isomorphism by taking formal linear combinations.

Patrick Nicodemus (Jun 19 2024 at 16:16):

(Obviously some missing details here - where does the suspension enter into it? I have to do some work here.)

Patrick Nicodemus (Jun 19 2024 at 16:18):

but thanks for the encouragement! I'll try to write it up rigorously/completely!

Todd Trimble (Jun 19 2024 at 16:47):

John will also remember that we made use of a similar chain complex to $0 \to \mathbb{Z} \to \mathbb{Z} \to 0$ in section 7 of our Schur functors paper, but with $\mathbb{Z}$ replaced by the generator $x$ of the free 2-rig generated by $x$ (starting around page 48).

The suspension, and conjugation by suspension, plays an important role in many constructions. If one thinks of a superspace $(C_0, C_1)$ or a dg-superspace $(C_0, C_1, \partial_0, \partial_1)$ as a proxy for a virtual construction "$C_0 - C_1$" [where in the dg-case one has a quasi-isomorphism $(C_0, C_1) \sim (H_0(C), H_1(C))$ -- cf. Euler's formula "$H_0 - H_1 = C_0 - C_1$", a key principle in our section 7] then $\mathbb{Z}_2$-graded suspension $(C_0, C_1) \mapsto (C_1, C_0)$ plays the role of "taking negatives" at the virtual level. (That is the sort of categorified explanation for how to think of negatives, or "co-negation" for the biring we were constructing.)

Todd Trimble (Jun 19 2024 at 16:48):

One of my favorite papers where this is utilized is Joyal's Foncteurs Analytiques et Especes de Structures (I'm skipping any accent marks; sorry) where he first shows how to combinatorially invert the linear (i.e. vector space-enriched) species $\exp(X) - 1$ to a species $\log(1+X)$, and then replaces $X$ by $-X$ by applying suspension (this is closely tied to his "rule of signs", which involves tensoring with a sign representation). Actually he conjugates $\log(1+X)$ by suspension, so as to obtain $-\log(1-X)$. In other words, he gives a linear species meaning to

$\log \frac1{1-X} = \log (1+X + X^2 + \ldots) = \log TX$

where $TX$ is the tensor algebra (free monoid) on $X$. The species $\log TX$ he winds up with solves for $\mathrm{Lie}(X)$, the free Lie algebra on $X$, from the composition of species isomorphisms

$TX \cong \mathcal{U} \mathrm{Lie}(X) \cong \mathrm{Sym} \mathrm{Lie}(X) \cong \exp(\mathrm{Lie}(X)).$

Todd Trimble (Jun 19 2024 at 16:48):

Where

• The first isomorphism is just by composing left adjoints to forgetful functors, one from algebra objects in linear species to Lie algebra objects in linear species;

• The second isomorphism is meant to be interpreted as an isomorphism between graded vector spaces, of Poincare-Birkhoff-Witt theorem type, coming from the evident filtered vector spaces for the universal enveloping algebra $\mathcal{U}$ and the symmetric algebra $\mathrm{Sym}$;

• The third isomorphism is by viewing $\mathrm{Sym}^n X$ as the quotient $X^n/\mathbf{n!}$ of a tensor power modulo the action of the symmetric group $\mathbf{n!} := S_n$.

Todd Trimble (Jun 19 2024 at 16:48):

Anyway, Joyal computes in this way the species underlying the Lie algebra operad $\mathrm{Lie}(X)$ as this $\log TX$ interpreted in this ingenious way, where the result is that $\mathrm{Lie}(X)[n]$, as a linear $S_n$-representation, is isomorphic to the top homology group $H^\ast$ of a partition lattice $\Pi(n) - \{\bot, \top\}$ for a set with $n$ elements (all other homology groups being zero), twisted by a determinant or sign representation,

$\mathrm{Lie}(X)[n] = (-1)^{n+1} H^\ast(\Pi(n)) \otimes \mathrm{sign}_{S_n},$

if I'm remembering correctly.

Stefano Ronchi (Jul 05 2024 at 10:57):

Oscar Cunningham ha scritto:

It looks to me that if $X$ and $Y$ aren't bounded above, then $[X,Y]$ isn't bounded above or below.

This might be an old question by now, but if you are still wondering about this kind of bounds on the tensor and internal hom of chain complexes, and especially when these are the normalized complexes of simplicial objects, I recently uploaded a preprint that deals with some of these questions (arxiv:2407.03306). It is all from the point of view of a differential geometer so we only deal with simplicial vector spaces.
What might be interesting to this discussion is that by using the fact that the Eilenberg-Zilber theorem does not only give a lax and oplax monoidal structure on the functor $N$ but shows they form a deformation retract, this gives information about the bounds of the complex $N(V \otimes W)$.
This also means the functor is strong monoidal between the homotopy categories. We use this fact to "flip" the theorem into a homotopy equivalence between $N(\Hom(V, W))$ and $\Hom(N(V), N(W))$, giving a "lax exponential" (?) structure. This might also be a general property of lax-oplax monoidal functor forming an homotopy equivalence.

I hope this might also shed some light on the role of the Eilenberg-Zilber theorem in describing tensor and hom across the two categories as @John Baez was wondering about, though I don't see the link with @Patrick Nicodemus's discussion at the moment but I would be happy to discuss it further!

Oscar Cunningham (Jul 05 2024 at 11:08):

@Stefano Ronchi Thanks, that looks very useful.

John Onstead (Jul 19 2024 at 21:46):

Hello everyone! I hope you've been well these past few weeks. I just got finished with a major project at work that took a little longer than expected, so now I have more time to come back here and learn more category theory!

In the meantime I did take a look at the Dold-Kan Correspondence and this thread on it, and I understand it at a "high level" (IE, about the equivalence of categories and what this entails). But I'm still confused about something that I think will help make it "click" more: I wanted to know how to construct the category of chain complexes in the first place. The obvious way to do so is to find the abelian/additive category Chain that is the "walking chain complex" such that every abelian/additive functor from it to another abelian/additive category "picks out" a chain complex in that category. In other words, Chain represents the functor Chain(-): AbelianCats -> Cat that sends an abelian category to the category of chain complexes in it. If I knew what Chain was, then I could more easily understand how this correspondence might work on a deeper level!

Kevin Carlson (Jul 19 2024 at 22:28):

That's a nice idea. I'm only guessing at where you might be stuck, but maybe you don't have much idea about pre-additive or Ab-enriched categories, yet? These are just categories whose homs are abelian groups and whose composition is bilinear, and their functors are just ordinary functors whose action on morphism sets gives homomorphisms of abelian groups. It's definitely easiest to find the pre-additive category Ch, pre-additive functors out of which into any pre-additive (eg additive or abelian) category are chain complexes. You need a (pre-additive) functor out of Ch to involve choosing objects $X_n$ for every $n\in \mathbb Z,$ maps $d_n:X_n\to X_{n-1}$ for every $n,$ satisfying $d_n\circ d_{n+1}=0...$ That's actually very close to a definition of Ch already, once you see how to think about it.

John Onstead (Jul 20 2024 at 00:39):

I am familiar with pre-additive categories from the point of view of enriched category theory. Let me see if I understand the suggestion. Let's start with a category that was one big chain ... X3 -> X2 -> X1 -> X0 almost like if we interpret the order of natural numbers as a category. A functor from this category would choose objects Xn for every n in Z as well as maps dn: Xn -> Xn-1 for every n. We know the composite of two consecutive maps is a zero-morphism. Nlab says a zero-map is one that factors through a zero-object, so 0: A -> 0 -> B. So for an example we have X3 -> X2 -> X1 composing to give 3,1: X3 -> X1 which somehow coincides with 0: X3 -> 0 -> X1. I guess this means that for every triple (Xn, Xn-1, Xn-2) we have a commutative square of the form Xn -> Xn-1 -> Xn-2; Xn -> 0 -> Xn-2. But here's the problem: this category with just a chain of morphisms ... X3 -> X2 -> X1 -> X0 lacks a zero object since X0 is a terminal but not initial object. So I'm not sure how to proceed from here!
Edit: I just remembered, even if our category has a zero object, there's no guarantee that a functor out of it will preserve the zero object. This makes the task even harder!

John Baez (Jul 20 2024 at 08:34):

@John Onstead - I bet pre-additive functors automatically preserve zero objects. I bet you can show a zero object in a pre-additive category is precisely an object $Z$ such that the identity morphism $1_Z : Z \to Z$ is the additive identity in the abelian group $\mathrm{hom}(Z,Z)$. And this condition is preserved by any pre-additive functor.

John Baez (Jul 20 2024 at 08:36):

(What I did just now is remember that zero objects are an example of [[absolute colimits]] in $\mathsf{Ab}$-enriched categories (aka pre-additive categories), meaning they're preserved by all $\mathrm{Ab}$-enriched functors (aka pre-additive functors). And then I guessed a proof.)

John Onstead (Jul 20 2024 at 11:04):

Ah thanks, I didn't know zero objects were absolute colimits in pre-additive categories. So with this concern out of the way it seems that so long as the X0 object in the category I mentioned above is a zero object, then a functor out of this category will map it to the zero object of the other category. This gives us a good notion of "walking chain complex": a category of form ... X3 -> X2 -> X1 -> X0 such that X0 is a zero object (maybe by just adding a unique morphism from X0 to each other objects) and every triple (Xn, Xn-1, Xn-2) forms a commutative square with X0 in the way described.

With this insight I can already understand better how the Dold-Kan correspondence might work. Both chain complexes and simplicial objects involve chains of the form ...X3 -> X2 -> X1 -> X0. So it makes sense that you can construct one from the other in a certain way. But I will still need to do more reviewing.

John Onstead (Jul 20 2024 at 11:06):

In the meantime. My next questions are a little embarrassing for me to ask, but... what is the point of a chain complex? Why do we care about having a whole chain of abelian groups strung together, and why this fixation that the composition of two consecutive maps in this chain give a zero morphism? What is the chain complex actually doing or representing? Lastly, why are maps in a chain complex called differentials when calculus is nowhere in sight? Any help in conceptualizing what is going on with chain complexes is much appreciated!

John Baez (Jul 20 2024 at 11:19):

John Onstead said:

In the meantime. My next questions are a little embarrassing for me to ask, but... what is the point of a chain complex? Why do we care about having a whole chain of abelian groups strung together, and why this fixation that the composition of two consecutive maps in this chain give a zero morphism?

Study algebraic topology and compute some homology groups of spaces! You'll see that while chain complexes are equivalent to simplicial abelian groups, they're a hell of a lot easier to compute with, because they don't have so much crap hanging around: wads of face maps and degeneracies. You can think of a chain complex as a streamlined version of a simplicial abelian group, where all that crap is compressed into a single map $d: X_n \to X_{n-1}$ in each dimension.

The key word here is compute: the virtues of chain complexes appear when you start to actually compute things. In all our conversations so far we have been polishing a lot of carburetors, but never actually getting in a car and driving it somewhere.

John Baez (Jul 20 2024 at 11:22):

Also, ponder the meaning of the mystical phrase "the boundary of a boundary is zero". And think about identities like

$\nabla \cdot (\nabla \times \vec{v}) = 0$

and

$\nabla \times (\nabla f) = 0$

which are examples.

Tim Hosgood (Jul 22 2024 at 13:10):

another reason that chain complexes might be preferred over the other side of Dold–Kan is how "naturally" they arise when you do geometry: you have widgets and those widgets have some notion of degree and you have a way of turning one widget into another that, in doing so, changes its degree by 1

Tim Hosgood (Jul 22 2024 at 13:11):

generally if i ever want to prove anything about collections of higher structures, i take my dg-categories coming from geometry and turn them into simplicial things and then do some more classical looking homotopy theory; but i ever want to calculate anything about one specific individual higher structure, then i'll stay in dg-world where i have the beautiful Maurer–Cartan equation to keep me company

Todd Trimble (Jul 22 2024 at 17:17):

but i ever want to calculate anything about one specific individual higher structure, then i'll stay in dg-world where i have the beautiful Maurer–Cartan equation to keep me company

Nice remark. I would like to make better acquaintance with Maurer-Cartan myself, to better understand the bar-cobar adjunctions which seem central to so many things.

Patrick Nicodemus (Jul 22 2024 at 17:19):

Todd Trimble said:

but i ever want to calculate anything about one specific individual higher structure, then i'll stay in dg-world where i have the beautiful Maurer–Cartan equation to keep me company

Nice remark. I would like to make better acquaintance with Maurer-Cartan myself, to better understand the bar-cobar adjunctions which seem central to so many things.

I dont personally understand Maurer Cartan myself as well as I'd like to but I encourage you to check out the "Sweedler theory" paper by Anel, I read a bit of it and I found it pretty interesting

Todd Trimble (Jul 22 2024 at 18:36):

I dont personally understand Maurer Cartan myself as well as I'd like to but I encourage you to check out the "Sweedler theory" paper by Anel, I read a bit of it and I found it pretty interesting

Oh, I'm well on it. He and Joyal have written a lot about it, in connection with measuring coalgebras (which btw Paige North discussed at CT 2024).

Tim Hosgood (Jul 22 2024 at 19:05):

if you're happy with the defining equation for the dg-nerve then this is a specific (but still very general) case of Maurer–Cartan

Tim Hosgood (Jul 22 2024 at 19:09):

but i will say that i "understand" Maurer–Cartan without having any real intuition for bar/cobar things, so it's likely the case that we're not quite coming from the same angle!

John Onstead (Jul 22 2024 at 19:16):

John Baez said:

Also, ponder the meaning of the mystical phrase "the boundary of a boundary is zero".

I see, so the whole axiom that two consecutive maps in a chain complex compose to the zero map is an abstraction of this "boundary of a boundary is zero" concept. I'm also starting to see where homology comes in, but I will need more time to go over this!

I also see that the use of chain complexes is that they are useful for computation. This is interesting to know but I do think in more abstract rather than calculation-based terms which might be why this isn't so easy for me to understand as simplicial objects!

John Baez (Jul 22 2024 at 20:44):

I hope you start doing some calculations. The beautiful abstract stories we're telling here are distillations of adventures where people met new phenomena and tried to understand them. If one wants to go on new adventures and discover brand new patterns not seen before by humankind, or even really understand the old stuff, one needs to do calculations.

Perhaps it's good that most of us who learned homological meet it in school where we are forced to compute lots of homology and cohomology groups, whether we want to or not!

Notification Bot (Jul 23 2024 at 08:50):

4 messages were moved from this topic to #learning: questions > Maurer-Cartan equation by John Baez.

John Onstead (Jul 26 2024 at 06:04):

I'm trying to do some calculations, but I'm finding it very difficult without that formal education you mentioned. These calculations are not very intuitive, especially when the only calculations I've ever done have involved real (and some complex) numbers. So the only problems I've worked on are the usual functions of real and complex values- my major didn't require high level math courses so the highest level math I got to was in high school calculus.

Aside from this I think I'm mostly clear on the Dold-Kan correspondence. I might start a new thread soon to continue the discussions we had before this aside, to continue trying to figure out how descent, cohomology, and local-global fit together. The only question I had left was about the connection between homotopy and homology. If I understand things correctly, there's a way to see how the simplicial homotopy of a space and the usual homology of the space are related under the Dold-Kan correspondence. But recently I found a theorem known as Hurewicz theorem that also seems to relate together homotopy and homology. My question is: are these two theorems related in any way at all, or are they two completely separate ways to connect homotopy and homology together?

John Baez (Jul 26 2024 at 09:36):

John Onstead said:

I'm trying to do some calculations, but I'm finding it very difficult without that formal education you mentioned.

I too find calculations difficult. I learned electromagnetism and quantum mechanics from The Feynman Lectures in high school, but there were essentially no problems in these books (just a small booklet with some very hard non-routine problems), so when I entered college and took classes I still had to work my butt off to do the homework: I understood the concepts, or so I thought, but there's an extra layer of understanding required to apply them. Instead of having the concepts laid out neatly for display purposes and admiring them, we need to grab ahold of the right one at the right time to accomplish a concrete goal, forcing us to examine everything we know and see what might help at each stage.

John Baez (Jul 26 2024 at 09:43):

I think unless one can compute the homology and cohomology of a circle, a sphere, a torus, a Klein bottle, a projective plane, and a bunch of other spaces, one is missing some important ideas about what homology and cohomology mean. For example, how homology groups 'count holes'. For these computations it's easiest to use, not singular homology, but simplicial homology, where one takes a simplicial complex and constructs from it a very small, manageable chain complex, whose homology groups one can compute by hand.

John Baez (Jul 26 2024 at 09:47):

There are textbooks that explain this stuff, but also YouTube videos like this series by N J Wildberger. Once you see how these calculations work, they're actually very fun, because you can see how topology is turning into algebra before your very eyes.

John Baez (Jul 26 2024 at 09:51):

I'll try to stop talking about this, but I felt I had to, because while our conversations about simplicial objects and algebraic topology are a lot of fun, they feel very unbalanced to me, as if we were engaged in a long and exciting discussion of techniques for doing many interesting things, while carefully avoiding actually doing any of those things, or even talking about doing them.

John Baez (Jul 26 2024 at 13:50):

John Onstead said:

The only question I had left was about the connection between homotopy and homology. If I understand things correctly, there's a way to see how the simplicial homotopy of a space and the usual homology of the space are related under the Dold-Kan correspondence.

I'm a bit puzzled by the term "the simplicial homotopy of a space" - is that a thing people say? One can attach a simplicial set to a space $X$, whose simplices are all continuous maps from the geometric realizations of the standard simplices to $X$. The nLab calls this the [[singular simplicial complex]] of $X$. I'm not sure I've heard that term either! But is that what you're talking about?

Anyway, if you turn this simplicial set into a simplicial abelian group (using the free abelian group functor $\mathrm{Set} \to \mathrm{AbGp}$), and then convert that into a chain complex, and then take the homology groups of that chain complex, those are called the singular homology groups of $X$. And for most people these are the "usual" homology groups of $X$, denoted $H_n(X)$.

John Baez (Jul 26 2024 at 13:53):

The chain complex we get from this procedure is usually enormous, since there are usually lots of ways to map simplices into a space. So we don't calculate its homology groups directly - for direct computations, it's better to use the approach in the YouTube videos. Instead, we use theorems about singular homology to reduce the calculation to calculations we already know how to do.

John Baez (Jul 26 2024 at 13:56):

But recently I found a theorem known as Hurewicz theorem that also seems to relate together homotopy and homology.

The Hurewicz theorem starts by giving homomorphisms $\pi_n(X) \to H_n(X)$ where $H_n(X)$ is defined as above - using singular homology. Then it says more. For starters, it says that $H_1(X)$ is the abelianization of $\pi_1(X)$.

John Baez (Jul 26 2024 at 13:58):

So it's very related to the earlier mentioned ideas - I think that's what you wanted to know.

Notice that Dold-Kan doesn't need to enter the story, since we don't really need to know that simplicial abelian groups are equivalent to chain complexes: we just need a way to get a chain complex from a simplicial abelian group. People usually learn Dold-Kan much later, after the Hurewicz theorem. To some extent it serves to "justify" chain complexes by giving them a conceptual interpretation.

Todd Trimble (Jul 26 2024 at 14:49):

for direct computations, it's better to use the approach in the YouTube videos

There is also cellular homology for a CW-complex, which is also very calculational in nature but also, I would say, a lot less cumbersome than simplicial homology (for starters, sometimes it's a pain to triangulate a space, and sometimes you can't). I would say simplicial homology has a relatively concrete feel to it, so it's good way to begin visualizing what homology is all about. (In fact I've never computed using simplicial homology, although I know how it works. When I took algebraic topology in grad school, we went straight for cellular (co)homology to start, and then later learned about spectral sequences.)

Understanding why cellular homology gives the correct results is an interesting exercise that involves the theorems about singular homology that John was alluding to, especially the Eilenberg-Steenrod axioms. It's interesting and instructive to derive cellular homology straight away from those axioms.

John Baez (Jul 26 2024 at 17:50):

I did simplicial homology first in school, and since John Onstead likes simplicial things, that was on my mind. It provides visual insight into the differential $\partial$ for the chain complex coming from a (semi!)simplicial abelian group, and how it's an alternating sum of face maps, and how that fact leads to the cancellations that give $\partial^2 = 0$.

But you're right that cellular homology quickly becomes more efficient. It's great too!

John Onstead (Jul 27 2024 at 09:50):

Thanks for the help! I agree that there's a difference between conceptual understanding and hands-on experience. One of my favorite math Youtube channels 3blue1brown stressed this in his intro to calculus and intro to linear algebra series- that his videos, which gave conceptual understanding, were no substitute for doing the practice problems. I eventually hope to start something like David Egolf's topos theory blog workthrough where I can do calculations for various things and get feedback. But there's still plenty of conceptual understanding I am missing, so I hope to come back in a day or so to continue my ongoing mission of understanding locality from the category theory perspective!

In the meantime I will be sure to look into cellular homology and do some calculations on my own. Algebraic topology (homology, homotopy, and cohomology) has always been something I wanted to learn and so I'm excited to give it a try!

John Baez (Jul 27 2024 at 10:43):

Those 3 or 4 YouTube videos I recommended give a really clear and quick intro to simplicial homology, how you compute it, and what it means. The comments show some in the audience were grateful that they finally understood it.