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Let $\mathbf{Cat}$ be the category of small categories and let $\mathbf{sCat}$ denote the category of simplicial objects in $\mathbf{Cat}$. We have a functor $$ \text{str}\colon \mathbf{Cat}\longrightarrow \mathbf{sCat} $$ taking a category $\mathcal{C}$ to the simplicial object whose category of $n$-simplices is given by $\mathcal{C}^{[n]}$ and whose faces and degeneracies are given by precomposition with the cofaces and the codegeneracies in the simplex category.

Question: is there a left adjoint for this functor? If so, how can it be described explicitly?

EDIT: by looking at the simplex category as a discrete 2-category, this may actually be an instance of the enriched version of the fact that presheaf categories are the free cocompletion of ordinary (small) categories. Still, I'd like to have an explicit description of the left adjoint, which I can't immediately see using enriched Kan extensions.

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This analogous to the adjuntion between the singular simplicial set functor and geometric realization, or to the adjunction between the nerve functor and the fundamental category functor.

Let $I : \Delta \to \mathrm{Cat}$ be the "inclusion" that sends $[n]$ to the category also called $[n]$ that you mentioned in your description of $\mathrm{str}$. Then $\mathrm{str}(\mathcal{C}) = \mathrm{Fun}(I(-), \mathcal{C})$. The left adjoint is then given by $- \otimes I$, the enriched functor tensor product with $I$, computed by an enriched coend $\mathcal{X} \mapsto \int^{[n] \in \Delta} \mathcal{X}_n \times [n]$. (Here, $\mathcal{X}$ is a simplicial object in categories, $\mathcal{X}_n$ is its category of $n$-simplices, and the $[n]$ that $\mathcal{X}_n$ is multiplied by is really the category $I([n])$.)

See the nLab article Nerve and Realization for the general construction covering all three cases I mentioned here, and for a proof of the adjunction.

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  • $\begingroup$ Thanks for your answer: I must have edited my post exactly when you were posting your answer, so maybe you could take a look at the edit indeed :) $\endgroup$ – Marco Vergura Jun 25 '16 at 19:40
  • $\begingroup$ Oh, I guess you mean the enriched coend doesn't count as explicit, @MarcoVergura. I guess I agree. I'll think a bit about what it looks like. $\endgroup$ – Omar Antolín-Camarena Jun 25 '16 at 19:45
  • $\begingroup$ Yeah, it would be nice to find a description analogous to the one of the fundamental category of a simplicial set as a quotient of the free graph generated by the 0 and the 1 simplices, rather than as a pointwise left Kan extension. $\endgroup$ – Marco Vergura Jun 25 '16 at 19:54

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