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Stream: theory: algebraic topology

Topic: dold-kan correspondence for general abelian cats

Matteo Capucci (he/him) (Oct 13 2020 at 14:40):

On the nLab page about the Dold-Kan correspondence, the result is given as an equivalence between connective chain complexes and simplicial objects in any abelian category $A$. The given reference is a paper by Dold and Puppe, which unfortunately is in German.
Do you know a reference in English (or Spanish or Italian, for what matters)? Looking in the interweb I can only find expositions for the case $A = \mathbf{Ab}$.
If not, could someone explain to me what's the nerve/realization adjunction in the general case? In particular, how's the 'standard cosimplicial object' of $A$ obtained?

Reid Barton (Oct 13 2020 at 15:18):

Is Prop. 2.3 on that page not what you're looking for?

Matteo Capucci (he/him) (Oct 13 2020 at 16:19):

No that's for the case $A = \mathbf{Ab}$, not for a general abelian category

Reid Barton (Oct 13 2020 at 16:27):

It works as stated for any abelian category

Matteo Capucci (he/him) (Oct 13 2020 at 16:29):

Indeed, you're right for the realization functor $\Gamma$, I didn't notice. But what about the nerve functor? It says to be just the Moore complex functor, though on the corresponding page only the case for simplicial groups is covered

Reid Barton (Oct 13 2020 at 16:30):

Oh yeah, it doesn't give a formula for $N$.

Reid Barton (Oct 13 2020 at 16:31):

But once you have a formula for $N$, then you can just interpret it in any abelian category.

Reid Barton (Oct 13 2020 at 16:32):

https://ncatlab.org/nlab/show/Moore+complex#NormalizedChainComplexOnGeneralGroup

Reid Barton (Oct 13 2020 at 16:33):

For the "joint kernel", you can take the kernel of the map into the (finite) direct sum whose components are given by the various face maps.

Matteo Capucci (he/him) (Oct 13 2020 at 16:35):

Oh right! I was too quick to dismiss that definition as not general enough

Matteo Capucci (he/him) (Oct 13 2020 at 16:35):

Thanks :)

Reid Barton (Oct 13 2020 at 16:37):

If you want a less explicit description, then you can think of an arbitrary abelian category $A$ as a kind of module over {f.g. abelian groups}, so that you can tensor or cotensor an object of $A$ by a f.g. abelian group. For this purpose, you'll only need the formulas $\mathbb{Z} \otimes A = A$, $\mathrm{Hom}(\mathbb{Z}, A) = A$.

Reid Barton (Oct 13 2020 at 16:40):

Then by extension you can also compute what the object of maps from a f.g. simplicial abelian group (in particular, $\mathbb{Z}[\Delta^k]$) to a simplicial object of $A$ should be, as an object of $A$; it will be given by some finite limit

Reid Barton (Oct 13 2020 at 16:41):

er, I don't think this is quite the right example

Reid Barton (Oct 13 2020 at 16:44):

well, I'm getting confused by the presentation on the nLab but in the end the point is that the correspondence for Ab is given by some finite (co)limits, and we can interpret these equally well in any abelian category

Matteo Capucci (he/him) (Oct 13 2020 at 16:45):

I see :thinking: I was expecting a trick of this kind, though it's far from clear to me how to pull it off

Reid Barton (Oct 13 2020 at 16:55):

It's easy to get mixed up here because what would ordinarily be an adjunction in one direction is in fact an equivalence, and so also an adjunction in the other direction. As a result, both $N$ and $\Gamma$ can be described using either limits or colimits.

Reid Barton (Oct 13 2020 at 16:57):

In the case of $\Gamma$ there is a finite direct sum = finite coproduct = finite product. In the case of $N$, IIRC, there's a canonical splitting that splits off the joint kernel of the d_i, so it can also be described as a quotient.

Reid Barton (Oct 13 2020 at 16:59):

In the phrase "$N$ and $\Gamma$ are nerve and realization with respect to the cosimplicial chain complex" I think $N$ is supposed to be the realization, so $N$ is being described as a colimit/coend.

Reid Barton (Oct 13 2020 at 17:02):

Anyways, what I wanted to say originally is that while the object that tells you how to turn a simplicial abelian group into a chain complex of abelian group is a cosimplicial chain complex, the object that tells you how to turn a simplicial object of $A$ into a chain complex in $A$ is also (or can be) a cosimplicial chain complex of abelian groups.

Reid Barton (Oct 13 2020 at 17:02):

If $A$ is a general abelian category then it might not have infinite coproducts so we need some finiteness condition on the cosimplicial chain complex.