# Acyclic categories related to structures in algebraic topology

An acyclic category (also called loopfree category or scwol (small category without loops)) is a small category where only identity morphisms have inverses, and any morphism from an object to itself is the identity.

Every poset P can be regarded as an acyclic category by identifying the set of objects with the elements of P. And saying there is a morphism from x to y, if and only if $x\le y$. Hence we can regard acyclic categories as a generalization of posets.

Posets play a crucial role in combinatorial algebraic topology, e.g. in form of intersection lattices related to hyperplane arrangements, or face posets of simplicial complexes. I'm looking for examples where we get acyclic categories as generalized posets encoding information about some structure (e.g.: Is there some kind of generalized hyperplane arrangement which yields an intersection category).

I'm aware of the face category of a polytopal complex and the salvetti categori of a complex, toric arrangement, but I'd be glad if anyone knew some further applications for acyclic categories in the field of algebraic topology

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Since it's my current object of obsession, I might as well mention the large body of work on Bousfield lattices and lattices of (co)localising subcategories. Don't know if acyclic categories are really applicable, but there are certainly a lot of posets around (which we can realize trivially as sites in the way you mentioned). –  Jon Beardsley Aug 24 '12 at 14:40

One thing that has come up a little bit in my work and I believe also e.g. in work of Dmitry Feichtner-Kozlov is such an extension of the widely used fact (popularized by Gian-Carlo Rota) that the Moebius function $\mu_P(x,y)$ of a poset $P$ may be interpreted as the reduced Euler characteristic of a simplicial complex called the order complex associated to a subposet of $P$, namely associated to the open interval from $x$ to $y$ (i.e. the subposet comprised of those $z\in P$ satisfying $x < z < y$); the faces in the order complex are the chains of comparable elements in the open interval, so e.g. vertices are individual elements.

There can be situations where one may wish to use a deformation of the usual Moebius function in an inclusion-exclusion counting formula and/or where a variant of the usual Moebius function could have more pleasant formulas, including ones where it is possible to interpret this as the reduced Euler characteristic of the nerve of a small category obtained from a poset by letting each cover relation $u\prec v$ have a multiplicity counting the number of different maps from $u$ to $v$. The deformed Moebius function is the inverse function in the incidence algebra to this weighted incidence relation. This hasn't been explored very widely though, at least to my knowledge. For me this sort of weighted inclusion-exclusion arose naturally in calculating a symmetric function by an inclusion-exclusion of quasisymmetric functions.

As one example, the usual Moebius function for the poset of set partitions of a multiset gets quite messy, resulting from the fact that it is not multiplicative in terms of decomposition into blocks in a partition; 10 or 15 years ago, Robert Kleinberg and I looked at a multiplicative deformation of it which satisfied much nicer formulas than the traditional Moebius function in this case and interpreted this deformed Moebius function as such a reduced Euler characteristic for the nerve of a small category. Some references in this general direction are:

1. E. Babson and D. Kozlov, Group actions on posets, J. Algebra, 285 (2005), no. 2, 439--450.

2. P. Hersh, Chain decomposition and the flag f-vector, J. Combin. Theory Ser. A, 103 (2003), 27--52

3. P. Hersh and R. Kleinberg, A multiplicative deformation of the Moebius function for the poset of partitions of a multiset, Communicating Mathematics (special volume in honor of Joe Gallian's 65th birthday), 113--118, Contemp. Math., 479, Amer. Math. Soc., Providence, RI, 2009.

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This is not a generalization of posets, but an illustration of how they come up unexpectedly. There is a notion of subdivision of a (small) category, and the second subdivision of any category is a poset. This was folklore in the early 1960's. It plays a role in Thomason's Quillen equivalence between the category of small categories and the category of simplicial sets.

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Is there a good reference for this? Thanks! –  Patricia Hersh Jul 27 '12 at 2:39
Yes, I'm already aware of the notion of a subdivision of a small category. However, I didn't know it was used by Quillen that way. @Patricia: Matias L. Del Hoyo, "On The Subdivision of Small Category" would be a good reference to learn about the subdivision of a small category –  Roman Bruckner Jul 27 '12 at 7:08
I have (prompted by Peter May's answer) posted a question related to this construction, see mathoverflow.net/questions/103281/…. –  Jonathan Chiche Jul 27 '12 at 10:07
Roman, it was Thomason, not Quillen who used it. Jonathan, I've answered your question, if only with an advertisement for a book I'm writing. Patricia, I like your answer. There is no really good reference yet, to my mind, since there is much more of interest to be said than is in any published reference. As an aside, I'd like to put in a good word for your first reference, Babson and Kozlov: that explains when you can hope that the nerve functor on categories with G-actions commutes with colimits, such as passage to orbits. That came up in recent work on equivariant classifying spaces. –  Peter May Jul 27 '12 at 17:55
As for the Quillen equivalence between the category of small categories and the category of simplicial sets, the standard references include Fritsch and Latch "Homotopy inverses for nerve" and Thomason's "Cat as a closed model category". Another reference is Cisinski's Astérisque "Les préfaisceaux comme modèles des types d'homotopie". However, while the Quillen adjunction is beautifully explained by Thomason, the Quillen equivalence part is somewhat less clear. Cisinski's Master's thesis might (I do not have it handy) explain the details, but it is not publicly available as far as I know. –  Jonathan Chiche Jul 29 '12 at 3:28

Ralph Cohen, John Jones, and Graeme Segal found an interesting "construction" of a topological acyclic category $C(f)$ from a Morse-Smale function $f : M\to \mathbb{R}$ in a preprint in early 90's. Objects are critical points and morphism spaces are given by moduli spaces of gradient flows.

+1, I was going to mention CJS but of course Dai beat me to it. It is important to note that that this work was never published and so the results (particularly the hard part about Morse-Smale $f$ implying homeomorphism between $M$ and the classifying space of $C(f)$) should be carefully checked before being used. –  Vidit Nanda Aug 25 '12 at 0:18