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Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying reflexive quiver. This functor evidently commutes with all limits, and since $X$ is a presheaf category, we obtain (by the adjoint functor theorem) a left-adjoint $F:X\to Cat$ sending a reflexive quiver $E$ to the free category on that reflexive quiver, $FE$.

The functor $FU:Cat\to Cat$ naturally has the structure of a comonad, and given any comonad $C$ on a category $T$, we may form an augmented simplicial endofunctor $\bar{C}: T\to T^{\Delta_+}$ where the structure maps are iterates of the comultiplication and counit. We define $\bar{C}^+$ to be the diagram restricted to the full subcategory $\Delta \subseteq \Delta_+$, $\bar{C}_{-1}$ to be the discrete simplicial endofunctor whose every structure map is the identity. The augmentation determines a natural transformation $\bar{C}^+\to \bar{C}_{-1}$, which we call the augmentation morphism.

In the case where $T=Cat$ and $C=FU$, we see that $\bar{FU}^+$ determines not only a simplicial object in $Cat$ but in fact determines a simplicially-enriched category, since the sets of objects, $Ob(\bar{C}_n X)$ are equal for all $n$.

I have heard that the transformation $\bar{C}^+\to \bar{C}_{-1}$ is an objectwise weak equivalence and also that $\bar{C}(A)^+$ is cofibrant in the Bergner model structure on simplicial categories for every category $A$. That is, $\bar{C}^+$ is precisely a cofibrant replacement for the functor $\bar{C}_{-1}$, which sends an ordinary category to its associated discrete simplicial category.

However, I have been unable to find a reference proving either the cofibrancy assertion or the that the augmentation map is an objectwise weak equivalence. I'd be happy to read it in a reference, if that is an option.

Another quick question: If we replace $C$ with the non-reflexive (Joyal calls this non-reduced) free-category comonad (in the first paragraph, replace the category $X$ of reflexive quivers with the category $Y$ of quivers, do the two properties in question still still hold?

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Don't communicate the title of this question to Times. The reputation of mathematicians should suffer... –  Denis Serre Jul 4 '11 at 6:35
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Hey Harry - did you run out of characters for a question mark in the title? :P –  David Roberts Jul 4 '11 at 6:35
    
From time to time, when the mood strikes me, I consciously omit question marks. This was not one of those times. –  Harry Gindi Jul 4 '11 at 7:40
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I don't know what the Bergner model structure is, but a possibly relevant fact is that the bar construction is acyclic in the sense that it is a coalgebra for the decalage comonad, and therefore equivalent to the constant simplicial object given by its object of path components which is $C_{-1}$. This doesn't depend on whether the quivers are reflexive or not; it works for any adjunction. –  Todd Trimble Jul 4 '11 at 10:57
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@Todd: Here's a link to the paper describing the model structure: citeseerx.ist.psu.edu/viewdoc/… –  Harry Gindi Jul 4 '11 at 11:16
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up vote 4 down vote accepted

In fact, applying the simplicial construction you describe to a category $\mathcal{C}$ gives the homotopy coherent thickening $\mathfrak{C} N\mathcal{C}$ (where $N$ is the nerve and $\mathfrak{C}$ is the left adjoint to the homotopy coherent nerve, as in HTT). This is described in Emily Riehl's paper "On the structure of simplicial categories associated to quasicategories" (see Theorem 6.7).

As a result, since $\mathfrak{C}$ of any simplicial set is cofibrant, $\mathfrak{C}N \mathcal{C}$ certainly is (in the Bergner model structure). The fact that $\mathfrak{C } N \mathcal{C} \to \mathcal{C}$ is a weak equivalence follows because $N \mathcal{C}$ is a fibrant object in the Bergner model structure, so to say that the above map is a weak equivalence is to say that $N \mathcal{C} \to N \mathcal{C}$ is one. Here I have used two facts: that the nerve of an ordinary category is the same as its homotopy coherent nerve and the (much harder) fact that $\mathfrak{C}$ and the homotopy coherent nerve determine a Quillen equivalence between the Joyal and Bergner model structures (or at least HTT 2.2.0.1 suffices).

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I'll take it, but using HTT 2.2.0.1 is kind of cheating =p. –  Harry Gindi Dec 18 '11 at 17:04
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Incidentally, there is a very elegant and short proof of the Joyal model structure in "The theory of quasi-categories and its applications" (or in his unpublished manuscript "The theory of quasi-categories I"), which makes it transparent that the fibrant objects are the $\infty$-categories. However, there the categorical equivalences are defined in a different manner, and it seems that proving that $\mathfrak{C}$ preserves them is then nontrivial. –  Akhil Mathew Dec 18 '11 at 17:09
    
(Elegant and short, at least if you grant the claim that if $K$ is a simplicial set and $\mathcal{C}$ an $\infty$-category, then the equivalences in $\mathrm{Fun}(K, \mathcal{C})$ are the "pointwise" ones, which follows more easily from the formalism of marked simplicial sets in HTT ch. 3.) Anyway, I'm not really sure yet whether Lurie's theorems can be proved in a shorter manner using this -- it seems that in any event there's some hard work to be done. –  Akhil Mathew Dec 18 '11 at 17:13
    
It's also easy to construct it using Cisinski's Asterisque 308, fwiw. –  Harry Gindi Dec 18 '11 at 21:55
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Really? How so? I haven't read Cisinski's thesis; do you have a reference to the relevant statement? –  Akhil Mathew Dec 19 '11 at 1:16
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