Locally minimal simplicial categories

Question

Given a (fibrant) simplicially enriched category $\mathcal{C}$, I'm interested in the possibility of replacing it with a weakly equivalent one (in Bergner model structure) such that all the mapping spaces are minimal Kan complexes, that I read about for example in $\textit{Higher Topos Theory}$, section 2.3.3. The naive idea, that is, individually replacing every mapping space with a minimal model, is bound not to work, because the construction of minimal models for a Kan complex is overtly not functorial, since it relies on an inductive choice of representatives for homotopy classes of simplices in each dimension.

Is there a way to obviate this issue and somehow choose such representatives in a way that is compatible with compositions in $\mathcal{C}$? I expect the construction won't give a functor $s \mathcal{Cat} \to s \mathcal{Cat}$ even in case of a positive answer, but I would be quite content with performing this on a single given simplicially enriched category.

In case of a negative answer, is there any other way that I can obtain a simplicially enriched category with the desired property? Feel free to add any conditions you like on $\mathcal{C}$ if it helps. Thanks!

Answer 1

It's not possible in general to ensure that all the hom-spaces in a simplicial category are minimally fibrant. Here's a counterexample inspired by Isbell.

Consider $Set$ with its cartesian monoidal structure (the same approach, mutatis mutandis, will work with the cocartesian monoidal structure). Let $C \subset Set$ be the subcategory with two objects, $\{\emptyset\}$, $\mathbb N$, and with morphisms the bijections. Then after making a choice of bijection $\mathbb N \cong \mathbb N \times \mathbb N$, the groupoid $C$ inherits a monoidal structure equivalent to the restriction of the cartesian monoidal structure on $Set$. As Zhen Lin shows at the linked argument of Isbell, there is no strict monoidal structure on $C$ equivalent to this monoidal structure.

(Of course, in line with the strictification theorem, the monoidal structure can be strictified after passing to an equivalent groupoid which is not skeletal; the point is that the flexibility gained by having extra isomorphic copies of each object is essential; this monoidal category cannot be simultaneously skeletized and strictified.)

Let $BC$ be the 1-object bicategory (with groupoidal homs) associated to $C$, and let $bC$ be any simplicial category corresponding to $BC$. Suppose that $D$ were a simplicial category with minimally-fibrant hom-spaces equivalent to $bC$. By passing to a skeleton, we may assume that $D$ has one object.

Now, we know the homotopy type of the hom-space of $D$ -- it is the classifying space of $C$ -- which is a minimal Kan complex! By uniquness of minimal models, this implies that the hom-space of $D$ is in fact isomorphic to the classifying space of $C$. Therefore the composition on $D$ -- which is strictly associative and unital -- restricts to a strict monoidal structure on $D$ equivalent to the one we've chosen. This contradicts Isbell's theorem that no such strictification exists.

EDIT: I think Isbell's argument uses some non-invertible morphisms, so doesn't quite apply to $C$ as above. Here is an alternative argument to Isbell's in the cocartesian variation. That is, let $C' \subset Set$ be the subcategory with two objects $\emptyset, \mathbb N$, and morphisms the bijections. The cocartesian monoidal structure on $Set$ restricts to a monoidal structure on $C'$ after we choose a bijection $\mathbb N \cong \mathbb N \amalg \mathbb N$. Without loss of generality, this bijection corresponds to the partition of $\mathbb N$ into the even numbers and odd numbers. Such a monoidal structure cannot be strictified. To see this, observe that if $f: \mathbb N \to \mathbb N$ is any nontrivial bijection, then $(1 \otimes f) \otimes 1$ moves some number which is congruent to 2 mod 4, and fixes any number which is not 2 mod 4, whereas $1 \otimes (f \otimes 1)$ moves some number which is congruent to 1 mod 4, and fixes any number which is not 1 mod 4. Therefore these two bijections are distinct, contradicting the strictness of the monoidal structure.