The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Question

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?

Answer 1

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).

Answer 2

Here's a direct way of seeing it, without using the Quillen equivalence to simplicial sets equipped with the Joyal model structure:

The cofibrant objects in the Bergner model structure are the 'simplicial computads', i.e. simplicial categories whose category of $n$-simplices is freely generated for all $n$, i.e. in the image of $F$ such that degenerate images of generators are generators. Knowing that the Bergner model structure is cofibrantly generated by $\{\emptyset\to\ast\}\cup\{\mathbb{2}[\partial\Delta^n]\to\mathbb{2}[\Delta^n]\}$, the proof is fairly straight forward. (See e.g. Lemma 16.2.2 in Emily Riehls book on categorical homotopy theory.) This settles the cofibrancy part of the claim. As it can be directly checked that the augmentation of the comonadic resolution the OP described is a DK-equivalence, we get the asserted claim.