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Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the identity morphisms. A poset is an acyclic category; here there is at most $1$ morphism between any two objects. The category $\mathbf{Ac}$ of acyclic categories lies (strictly) between the category $\mathbf{Pos}$ of posets and the category $\mathbf{Cat}$.

I am looking for references where categorical and homotopy theoretic properties of $\mathbf{Ac}$ are discussed.

A few references that I am aware of are the following:

  1. Combinatorial algebraic topology by D. Kozlov. Chapter 10 deals with basic constructions involving acyclic categories, their subdivisions, category of intervals and Mobius inversion. Chapter 14 is about group actions on posets and quotients in $\mathbf{Ac}$. Finally, in Chapter 15 he gives an explicit construction of homotopy colimit of diagram of spaces indexed over an acyclic category.
  2. Homotopy limits and colimits by R. Vogt. The paper mainly deals with topological categories (i.e., small categories whose morphism sets can be topologized), however, some constructions can be translated to $\mathbf{Ac}$.
  3. Cellular stratified spaces I and II by Dai Tamaki. This is an extensive account of various useful constructions involving (generalized) cell complexes whose face categories are acyclic. He describes a range of applications of acyclic categories; from configuration spaces to toric topology.

I am interested in knowing answers to the following:

  1. Homotopy category of $\mathbf{Ac}$.
  2. Model structures on $\mathbf{Ac}$.
  3. Grothendieck construction for diagrams in $\mathbf{Ac}$.
  4. Quillen's theorems A and B.
  5. The category of $\mathbf{Ac}$-diagrams.
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    $\begingroup$ The homotopy theory of $\mathbf{Ac}$ should be the same as that of $\mathbf{Cat}$: there is a "barycentric subdivision" that turns every category into a homotopy-equivalent acyclic one. $\endgroup$
    – Zhen Lin
    Oct 18, 2013 at 7:43
  • $\begingroup$ Barycentric subdivision turns any category into a poset, actually. $\endgroup$
    – Fernando Muro
    Oct 18, 2013 at 10:02
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    $\begingroup$ Fernando, it is the second subdivision of a category that is always a poset. $\endgroup$
    – Peter May
    Oct 18, 2013 at 15:45
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    $\begingroup$ Quillen's theorems are for any category. $\endgroup$
    – Benjamin Steinberg
    Oct 18, 2013 at 15:57
  • $\begingroup$ Peter, indeed, that's what I meant. $\endgroup$
    – Fernando Muro
    Oct 18, 2013 at 17:22

1 Answer 1

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Here is a cool new (and very readable) preprint which uses the second barycentric subdivision (as discussed in Zhen Lin, Fernando Muro and Peter May's comments) to construct a cofibrantly generated model structure on $\textbf{Ac}$ which is Quillen-equivalent to the Thomason model structure on $\textbf{Cat}$.

R Bruckner. A Model Structure On The Category Of Small Acyclic Categories, arXiv:1508.00992 [math.AT]

Caveat: this only applies to small acyclic categories.

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