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Stream: theory: category theory

Topic: simplicial methods & co

Matteo Capucci (he/him) (Jul 30 2021 at 10:20):

I'm following up from the wonderful messages @Patrick Nicodemus introduced himself with back at #general > Introduce Yourself! (cc @Tim Hosgood )
Patrick, you write

So, by purely computational motivations, i.e. trying to compute the "realization" of a simplicial Abelian group, we are naturally led to replace it by its Moore normalization, and this can give some a priori justification for the notion of chain complex (it is a reduction of a simplicial Abelian group which suffices for the purpose of gluing together monoids in additive categories and their higher tensor powers)

So my question is, why would one want to glue monoids together in that way? Is it related to the connection with the bar complex associated to a monad for an algebra?
I always suspected too there's something going on there, and your remarks look very insightful. I'm not really working on it but it certainly captures my interest.

Fawzi Hreiki (Jul 30 2021 at 11:22):

I don't know if this is related, but some approaches to generalising algebraic geometry define affine spaces as the duals of monoids in a given monoidal category. This seems to be one way of getting at the elusive $\mathbb{F}_1$. See for example https://arxiv.org/abs/1005.0287v1 and https://arxiv.org/abs/1410.1716.

Patrick Nicodemus (Jul 30 2021 at 22:33):

@Matteo Capucci (he/him) that's a great question. I don't have a good answer but I've been thinking about that for a while. I'd love to hear anyone else's contribution if they have something similar to say. It has been easier to find empirical examples of interesting patterns of this than to articulate a general philosophy of what is happening. In the non-Abelian case there's an obvious place we want to glue together monoids: where the monoidal category is Top equipped with the join of spaces as monoidal product, the distinguished monoid is the singleton space, and the higher tensor powers of the singleton are exactly the n-simplices for various n. Then gluing together these simplices is just building a "simplicial complex" in the sense of taking the geometric realization of a simplicial set, and of course we know that simplicial complexes are very important and basic spaces. In the non-Abelian case it's thus tempting to view these geometric realizations of simplicial Abelian groups as some kind of algebraic simplicial complexes where instead of n-simplexes we work with n-fold tensor powers of a module. But I can't think of good examples of this. Ironically the only examples I understand where is the case where our Abelian category is itself the category of chain complexes, which seems to be dodging the question.
For one thing the Dold-Kan functor SAb -> Ch(Ab) is itself an example of one of these "geometric realizations". That is, there is a distinguished monoid in Ch(Ab) - the chain complex $0\to \mathbb{Z}\to\mathbb{Z}\to 0$ concentrated in degrees 0 and 1. Think of this as being the chain complex version of the singleton simplicial complex - it has one 0-simplex, which is the convex hull of a set w 1 element, and one (-1)-simplex, which is the convex hull of the set with 0 elements, giving generators in degrees 1 and 0 respectively. We can view this as a pointed object, or as a monoid, or as a commutative monoid in a symmetric monoidal category; each of these gives functors from the free monoidal category generated by a pointed object, or the free monoidal category generated by a monoid, or the free symmetric monoidal category generated by a commutative monoid. Then with respect to each of these diagrams we can give a "geometric realization" for a presheaf of Abelian groups over said free monoidal category - in each case, respectively, we get the alternating-sum-of-face maps construction of a chain complex, or the Moore normalization of a simplicial Abelian group, or in the last case the so called "oriented chain complex" constructed in algebraic topology textbooks (see Spanier's book, for example). This accounts for many of the basic chain complexes encountered in singular homology.

Another interesting example - the Koszul complex. When V , M are two vector spaces and V acts on M by a map $V\otimes M\to M$ in a "commutative way" so that the map $V\otimes V\otimes M\to V\otimes M\to V\otimes M$ commutes with the symmetry isomorphism $V\otimes V\cong V\otimes V$, we can form the free symmetric monoidal category generated by $V, M$ and the map between them; this diagram can then be "realized" in the category of chain complexes with respect to one of the embeddings from earlier, giving the so-called Koszul complex of the action. But unfortunately like I said, the only concrete examples of gluings I know of are ways to build chain complexes. I don't have good examples in other categories. Maybe a category of sheaves of Abelian groups would be a good place to look for examples.

I am hoping that as I gain familarity with the homotopy literature I'll be able to answer some of these questions. For example I haven't yet developed a good intuition for cofibrant replacement, which seems like it would be helpful heere.

Patrick Nicodemus (Jul 30 2021 at 22:46):

@Matteo Capucci (he/him) There's also the interesting observation that we can sometimes hook the bar construction and the construction I'm describing together head to tail. For example, let A be an object in some category C equipped with a comonad G. Then we can give a homological resolution of A by the bar construction for T. If we Hom this complex T^\bullet A into some Abelian group object B in the category we give a simplicial Abelian group Hom(T*A, B) and we can take the Moore normalization and look at the cohomology groups of this cochain complex, which we think of as the cohomology groups of A with coefficients in B (wrt T). This is the theory developed substantially by Barr and Beck. Not a very deep observation but i've been trying to find other places where these constructions are hooked together head to tail. A basic question is here is why does one want to replace the object by its resolution in the first place? In good cases one can reconstruct the original object by taking "connected components" of the simplicial object. Again cofibrant replacement is probably a helpful answer here but I can't speak intelligently on that subject.

Patrick Nicodemus (Aug 02 2021 at 16:56):

Tom Leinster has a nice blog post here - https://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html

Where does it come from? Since the nerve functor is induced by the inclusion $\Delta \to \mathbf{Cat}$, we’re really asking where $\Delta$ and its inclusion into $\mathbf{Cat}$ come from.

One answer might be that $\Delta$ is the free monoidal category on a monoid. But that’s just wrong: $\Delta$, as defined above and in most of the literature, does not include the empty set, whereas this free monoidal category $\mathbf{D}$ is a skeleton of the category of finite, possibly empty, totally ordered sets. Had you been shooting for $\mathbf{D}$, you’d have hit it — but presheaves on $\mathbf{D}$ are not the same as simplicial sets.

I think this question is very interesting! But I don't understand the rest of the blog post unfortunately. I tried to read Weber's paper and made it through ~10 pages but it is just too long and I don't have time to read 70
pages of dense mathematics right now. Does anyone want to discuss what's going on here, and help me understand the high level picture, or contribute their own opinion as to why we care about unaugmented simplicial sets rather than augmented simplicial sets?

John Baez (Aug 03 2021 at 02:36):

I think the "Theorem" Tom states explains what's going on much more simply than Weber's paper. For any monad $T$ on a sufficiently nice category, you automatically get a concept of "$T$-simplicial set" which has a bunch of nice properties, such that any algebra of $T$ has a "nerve" that's a $T$-simplicial set. When you take $T$ to be the monad whose algebras are categories, a $T$-simplicial set is just a simplicial set, and you get the usual concept of the nerve of a category.

John Baez (Aug 03 2021 at 02:36):

I think this basic message is reassuring even if one doesn't care about the precise hypotheses of the theorem.

John Baez (Aug 03 2021 at 02:38):

It's saying that plain old simplicial sets, which are widely used as the nerves of categories, arise as a special case of a very general systematic construction for defining "nerves".

John Baez (Aug 03 2021 at 02:39):

You can ignore all the more fancy examples of nerves that Tom uses to illustrate this construction.

Ben MacAdam (Aug 03 2021 at 05:37):

Maybe I'm coming at this from the wrong angle (I'm certainly not an algebraist), but it feels like it may be a bit confusing to use the simplicial language when talking about the more general concept at play in the nerve construction, which is density. A subcategory $\iota: \mathcal{A} \hookrightarrow \mathbb{C}$ is dense whenever the functor:
$N_{\iota}: \mathbb{C} \to \widehat{\mathcal{A}}; C \mapsto \mathbb{C}(\iota -, C)$
is fully faithful (and really, $\iota$ may be replaced with any functor $K:\mathcal{A} \to \mathbb{C}$ here). Density is the subject of Chapter 5 in Kelly's Basic Concepts of Enriched Category Theory, which is quite readable if you know your way around kan extensions or coends and just assume $\mathcal{V} = \mathsf{Set}$.

Looking at categories, for example, I believe you can use the arrow category to find a fully faithful embedding into cubical sets (although, this may depend on what definition of cubical set you're using).

John Baez (Aug 03 2021 at 05:57):

Ben MacAdam said:

Maybe I'm coming at this from the wrong angle (I'm certainly not an algebraist), but it feels like it may be a bit confusing to use the simplicial language when talking about the more general concept at play in the nerve construction, which is density.

If you're saying what Tom calls a "$T$-simplicial set" may look very different from a simplicial set - so different that the word "simplicial" is a bit odd - I agree.

John Baez (Aug 03 2021 at 06:08):

While you may be able to find a fully faithful embedding of Cat into cubical sets, the idea of Weber's theorem is to find, for a large class of monads $T$, a canonical choice of category $C_T$ such that $T\mathrm{Alg}$ is naturally equipped with a fully faithful embedding into the category of presheaves on $C_T$.

Ben MacAdam (Aug 03 2021 at 22:26):

Maybe the problem is I haven't spent much time with Weber's original paper. My experience with Weber's Nerve theorem is mostly via the Berger-Mellies-Weber paper and Bourke-Garner's treatments of "nervous" monads. I think it's possible for a monad to be nervous with respect to different dense subcategories of your base category, and this would lead to fairly different presentations of $C_T$. But it would make sense if there was a maximal dense subcategory for which the monad is nervous gives a canonical $C_T$, and that would jive with what I remember from Clemens-Mellies-Weber.