The n-Category Café

Urs wrote:
the sequence of inclusions

……

of strict and strictly abelian into strict into general infty-groupoids is the Dold-Kan correspondence.

`Is’ ?? Do you mean that this sequence of inclusions gives PRECISELY what is written in Dold-Kan?

Thanks, Urs. But I don’t understand in what sense the overall inclusion $Ch_{\bullet}^{+} \hookrightarrow KanCplx$ can be the Dold-Kan correspondence. (This may or may not be the same as Jim’s query.)

My understanding of the Dold-Kan correspondence is that it’s an equivalence between chain complexes and simplicial abelian groups. You’ve written down an inclusion of chain complexes into Kan complexes.

The underlying simplicial set of any simplicial abelian group is a Kan complex, so we have a forgetful functor $U: \mathbf{Ab}^{\Delta^{op}} \to KanCplx$. At first I thought you might mean that $U$ is full, so that $\mathbf{Ab}^{\Delta^{op}}$ is a full subcategory of $KanCplx$, and that the inclusion $Ch \hookrightarrow KanCplx$ determines an equivalence of categories $Ch \simeq \mathbf{Ab}^{\Delta^{op}}$. But now I realize you can’t mean that, because $U$ certainly isn’t full.

I don’t understand in what sense the overall inclusion $Ch_\bullet^+ \hookrightarrow KanCplx$ can be the Dold-Kan correspondence.

As that remark by Ronnie says, the image of that inclusion of a chain complex is the Kan complex underlying the simplicial abelian group corresponding to the chain complex under Dold-Kan. The Dold-Kan correspondence identifies the image of that functor in $KanCplx$.

Do you mean that this sequence of inclusions gives PRECISELY what is written in Dold-Kan?

Yes, the composite functor is precisely the Dold-Kan map $Ch_\bullet \to sAb$ composed with the forgetful $sAb \to KanCplx$.

I mentioned a reference with all the details: Remark 9.10.6 of Brown’s Nonabelian Algebraic Topology.

Ah! Good. Thanks, Mike.

Now that that particular idea is consigned to the scrapheap, let me say something more about the general idea. Maybe someone can see a different way of making it work.

For any category $\mathcal{E}$ with finite limits, we may consider the category $\infty\mathbf{Cat}(\mathcal{E})$ of (strict) $\infty$-categories in $\mathcal{E}$ and the category $\mathcal{E}^{\Delta^{op}}$ of simplicial objects in $\mathcal{E}$.

When $\mathcal{E} = \mathbf{Ab}$, there is an equivalence between the two categories, by the Dold-Kan Theorem.

When $\mathcal{E} = \mathbf{Set}$, there is an adjunction between the two categories, arising from Street’s oriental construction.

I wondered whether the equivalence that happens for $\mathbf{Ab}$ was an analogue of the adjunction that happens for $\mathbf{Set}$. I still wonder—because although the particular idea I suggested was wrong, it might be true in some other sense.

But even if Street’s adjunction turns out to have nothing to do with the Dold-Kan equivalence, there’s still a more general question. I want to know whether it’s possible to understand the Dold-Kan Theorem by finding some relationship (e.g. an adjunction) between $\infty\mathbf{Cat}(\mathcal{E})$ and $\mathcal{E}^{\Delta^{op}}$ that works for some large class of categories $\mathcal{E}$, then specializing to the case $\mathcal{E} = \mathbf{Ab}$, then showing that in that case the relationship is an equivalence. (The approach that I suggested would have worked whenever $\mathcal{E}$ is the category of models for a finite product theory.)

Thoughts, anyone?

But even if Street’s adjunction turns out to have nothing to do with the Dold-Kan equivalence,

But it does: what Brown writes “$\Pi \Delta^n$” is the (crossed module version of the) strict $\infty$-groupoid version of the $n$-th oriental. Homming the $n$th oriental into a strict $\infty$-groupoid is the same as homming this “$\Pi \Delta^n$” into the $\infty$-groupoid.

His Remark 9.10.6 in Nonabelian algebraic topology says that the Dold-Kan map $Ch_\bullet \to sAb$ is under the identification of chain complexes with certain strict $\infty$-groupoids (via crossed complexes) just forming the $\omega$-nerve of these strict $\infty$-groupoids.

Well, seeing as you ask so nicely, here’s a taste of what Joyal said…

First, here’s something about the finite difference calculus. Begin with a sequence $$a_0, a_1, a_2, \ldots$$ of numbers. Define $(\delta a)_n = a_{n + 1} - a_n$. Then we get a new sequence $$a_0, (\delta a)_0, (\delta^2 a)_0, \ldots$$ of numbers. (In fact we get a triangle, which I won’t attempt to draw, involving all the terms $(\delta^k a)_n$.) In the finite difference calculus there are formulas relating these two sequences. In one direction, the formula is $$a_n = \sum_{k = 0}^n \binom{n}{k} (\delta^k a)_0.$$

Next, here’s something about the Dold-Kan theorem. This is about relating simplicial abelian groups $$A_0 --- A_1 --- A_2 --- \cdots$$ (where $---$ indicates various back and forth arrows that I’m too lazy to typeset) to chain complexes $$B_0 \leftarrow B_1 \leftarrow B_2 \leftarrow \cdots.$$ One half of the equivalence is the functor $K: \mathbf{Ch} \to \mathbf{Ab}^{\Delta^{op}}$, which is (partially) described as follows: for a chain complex $B$, $$K(B)_n = \bigoplus B_k$$ where the sum is over all $k$ and surjections $[n] \to [k]$ in $\Delta$. How many such surjections are there? $\binom{n}{k}$.

So, in this categorification, the sequence $(a_n)$ becomes a simplicial abelian group. The sequence of differences becomes a chain complex. Newton’s formula for the original sequence in terms of the sequence of differences becomes the formula for the functor $K$. There’s more to say (e.g. about the other direction), but I won’t say it.

Sorry, of course I mean the 0th face of the unique thin filler of the 0th horn as the source.

And this is why I doubt that DK factors through Street’s nerve.

Let’s see. I am thinking that the following is evident, but maybe I am making a mistake somewhere:

1. The strict $\omega$-groupoid corresponding to the fundamental crossed complex $\Pi \Delta^n$ of the $n$-simplex regarded as a filtered topological space is the free strict $n$-groupoid on the $n$-th oriental $O(n)$.

2. For a strict $\omega$-category $C$ that happens to be a strict $\omega$-groupoid we therefore have

   $$Hom(O(n), C) \simeq Hom(\Pi \Delta^n, C)$$

3. That the Dold-Kan map $Ch_\bullet \to sAb \to KanCplx$ factors as the composite of the $\Theta$-map $\Theta : Ch_\bullet \to Crs \simeq Str \omega Grpd$ and the $\omega$-nerve $N : Str \omega Grpd \to KanCplx$ follows immediately from the above and remark 9.10.6.

What do you think?

Dmitry wrote:

By saying that

abelian groups in $\infty\mathbf{-Cat}$ are chain complexes

you are implicitly using another, globular version of the Dold-Kan correspondence.

Ah, interesting. I like that point of view.

You also bring out another important point that’s recently helped me to understand this stuff a lot better, namely:

In $\mathbf{Ab}$, every reflexive globular object is a strict $\infty$-category in a unique way.

This is a souped up, infinite-dimensional version of the fact that in $\mathbf{Ab}$, every object is a monoid in a unique way.

(You can take both statements further, as you know: in $\mathbf{Ab}$, every reflexive globular object is a strictly symmetric strict monoidal strict $\infty$-groupoid in a unique way; and every object is an abelian group in a unique way.)

The equivalence between chain complexes in $\mathbf{Ab}$ and strict $\infty$-categories in $\mathbf{Ab}$ splits into two steps:

1. the equivalence between chain complexes and reflexive globular abelian groups
2. the fact that every reflexive globular abelian group is a strict $\infty$-category in $\mathbf{Ab}$ in a unique way.

The first of these is also, I think, a categorification of an ancient algebraic principle. Given a chain complex $B$, the corresponding reflexive globular abelian group $A$ is defined by $$A_n = B_0 \oplus \cdots \oplus B_n.$$ You describe the opposite process, which involves taking kernels. The fact that these processes are mutually inverse is analogous to the principle of telescoping sums.

What I claim is that the cosimplicial object $O : \Delta Ch(Ab)$ is the composite of this functor $L$ with Street’s oriental functor $\mathcal{O} : \Delta \to \omega-Cat$.

Thanks, so that’s another way to Brown’s, mentioned above, to see that simplicial Dold-Kan is globular/cubical Dold-Kan followed by simplicial nerve:

rephrasing what you just wrote, we have that the simplicial Dold-Kan map assigns to a chain complex $C_\bullet$ the simplicial set

$$[n] \mapsto Hom_{Ch_\bullet}(L O(n), C_\bullet) = Hom_{\omega Cat}(O(n), R C_\bullet) = N (R C_\bullet)_n \,,$$

where $R C_\bullet$ is the strict $\omega$-category obtained from $C_\bullet$ by the globular Dold-Kan correspondence.

Thanks, Richard. That’s surely an important story.