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?

Times. The reputation of mathematicians should suffer... – Denis Serre Jul 4 '11 at 6:35