Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: smooth version of Dold-Kan correspondence

ADITTYA CHAUDHURI (Aug 04 2021 at 20:29):

Does there exists an appropriate smooth version of Dold-Kan correspondence in the existing literature?

Ben MacAdam (Aug 04 2021 at 20:35):

Smooth in what sense?

ADITTYA CHAUDHURI (Aug 04 2021 at 20:37):

Smooth in the sense of simplicial abelian Lie groups.

Fawzi Hreiki (Aug 04 2021 at 20:37):

Well the correspondence works for any abelian category of which there are 'smooth versions' such as sheaves of modules on manifolds, if thats what you want.

ADITTYA CHAUDHURI (Aug 04 2021 at 20:39):

Basically, I am interested to know what does the simplicial abelian Lie groups correspond to "in terms of chain complexes"?

Ben MacAdam (Aug 04 2021 at 20:41):

Yes, following on @Fawzi Hreiki if you take a reasonable embedding of smooth manifolds into a locally presentable category (like the Dubuc topos or the microlinear objects in the Dubuc topos), you'll get the usual Dold-Kan correspondence. I suppose you could then take the pointwise Lie derivative and get a simplicial Lie algebra?

Fawzi Hreiki (Aug 04 2021 at 20:43):

ADITTYA CHAUDHURI said:

Smooth in the sense of simplicial abelian Lie groups.

Hmm, I'm not sure if the category of abelian Lie groups actually is abelian. I think this comes down to the fact that the category of smooth manifolds isn't all that nice.

ADITTYA CHAUDHURI (Aug 04 2021 at 20:46):

ok.. But is there a "good" category which captures "Abelian property" and "smoothness" simultaneously? For example we often incorporate cocompleteness and completenss into the category of smooth manifolds by embedding them into the category of generalized smooth spaces like diffeological spaces?

Fawzi Hreiki (Aug 04 2021 at 20:46):

Sure - any category of abelian group objects in a topos

ADITTYA CHAUDHURI (Aug 04 2021 at 20:47):

Ohh! yes

Fawzi Hreiki (Aug 04 2021 at 20:47):

Or any category of modules over a ring object in a topos

ADITTYA CHAUDHURI (Aug 04 2021 at 20:49):

So, my original question makes sense only after embedding into an appropriate category.. Am i right?

ADITTYA CHAUDHURI (Aug 04 2021 at 21:17):

According to https://ncatlab.org/nlab/show/simplicial+Lie+algebra it seems that at least in the Lie algebra level, simplicial Lie algebras corresponds to dg-Lie algebras? Though I am not sure whether I am understanding it correctly!!

John Baez (Aug 05 2021 at 00:46):

Yes, you can turn a simplicial Lie algebra into a differential graded Lie algebra, and conversely.

John Baez (Aug 05 2021 at 00:54):

There's an adjunction between the category of simplicial Lie algebras and the category of dg Lie algebras. But it's not an equivalence of categories. With some extra conditions you can get a "Quillen equivalence" - the nLab states a theorem along these lines due to Quillen.

John Baez (Aug 05 2021 at 01:26):

Fawzi Hreiki said:

Hmm, I'm not sure if the category of abelian Lie groups actually is abelian. I think this comes down to the fact that the category of smooth manifolds isn't all that nice.

Right, the category of abelian Lie groups is not an abelian category. Take a Lie group homomorphism

$f: \mathbb{R} \to S^1 \times S^1$

that wraps the real line around the torus $S^1 \times S^1$ with an irrational slope. This is a monomorphism in the category of abelian Lie groups, and also an epimorphism (because its range is dense), yet not an isomorphism - which is impossible in an abelian category.

David Michael Roberts (Aug 05 2021 at 03:29):

While the functor from chain complexes to simplicial abelian groups lifts to work for everything internal to manifolds (and indeed, probably internal to any category with finite products), the functor the other way is less clear. One needs to take the intersection of a bunch of kernels, and it's not immediate this is a manifold (it may be, I haven't checked). Much as for Lie 2-groups, where one wants to make sure that the groupoid internal to manifolds is in fact a Lie groupoid (so that source and target are submersions), some niceness might need to be assumed. Further, even when dealing with crossed modules of Lie groups, people often want to assume extra niceness, like the existence of smooth local sections of $G\to H$, or even just the existence of a smooth section on a neighbourhood of the identity of $H$.

I'm pretty sure that one doesn't need an abelian category, though, for Dold–Kan to work. I mean, it works for the semiabelian context too, with simplicial groups and a nonabelian version of chain complexes. The question is what happens when internalising. Working out the 2- or 3-term version would be instructive, I think.

John Baez (Aug 05 2021 at 04:57):

Thanks, David! I dodged the main question because I wasn't up to figuring out exactly what was needed of a category $C$ for Dold-Kan to work internal to $C$. It's just a matter of going through the proof and seeing what one uses. But it takes a bit of time.

Zhen Lin Low (Aug 05 2021 at 05:04):

The fact that Dold–Kan works for abelian groups tells us that the simplex category and the "free cochain complex" category are Morita equivalent, so that should imply it works for any Cauchy-complete additive category whatsoever.

ADITTYA CHAUDHURI (Aug 05 2021 at 06:09):

@David Michael Roberts Thanks I will work through as you said. You mentioned "non abelian version of chain complexes", I am not much familiar with such objects. It seems they are pretty interesting. Can you suggest some references in that direction?

ADITTYA CHAUDHURI (Aug 05 2021 at 06:12):

Zhen Lin Low said:

The fact that Dold–Kan works for abelian groups tells us that the simplex category and the "free cochain complex" category are Morita equivalent, so that should imply it works for any Cauchy-complete additive category whatsoever.

Thank you ."The fact that Dold–Kan works for abelian groups tells us that the simplex category and the "free cochain complex" category are Morita equivalent" . This fact seems interesting. Can you suggest some references where such result is mentioned or anything in that direction?

ADITTYA CHAUDHURI (Aug 05 2021 at 06:13):

@John Baez Thank you!

David Michael Roberts (Aug 05 2021 at 06:28):

@ADITTYA CHAUDHURI See here: https://ncatlab.org/nlab/show/hypercrossed+complex (ignore the group(oid) stuff, just think about groups) and the links with Moore complexes

David Michael Roberts (Aug 05 2021 at 06:30):

like the existence of smooth local sections of $G\to H$, or even just the existence of a smooth section on a neighbourhood of the identity of $H$

whoops, I meant local sections of $G\to t(G)$, where $t\colon G\to H$ is the crossed module (and similarly for a nhd of the identity of $t(G)$). In general, of course, $t$ is not even surjective!

David Michael Roberts (Aug 05 2021 at 06:38):

But I agree that working with simplicial abelian groups internal a pretopos, say, or even perhaps just an exact category (https://ncatlab.org/nlab/show/exact+category), might be enough to get some form of an internal Dold–Kan equivalence. Maybe less.

Embedding the category of manifolds into that of diffeological spaces means all the nice constructions one wants will be available, as the latter is a quasitopos. But most people would just go all the way and embed simplicial abelian groups into oo-sheaves (on Mfld) of simplicial abelian groups, I guess.

ADITTYA CHAUDHURI (Aug 05 2021 at 06:46):

@David Michael Roberts Thanks a lot!

Dmitri Pavlov (Aug 18 2021 at 21:52):

The Dold–Kan correspondence works in any idempotent-complete additive category. For a proof, see Theorem 1.2.3.7 in Lurie's Higher Algebra. The category of abelian Lie groups is idempotent-complete. So are many other bigger categories, e.g., sheaves of abelian groups on the site of smooth manifolds.

ADITTYA CHAUDHURI (Aug 19 2021 at 21:39):

@Dmitri Pavlov Thanks a lot!