Excellent monoidal model categories admit enriched fibrant replacement functors?

Question

Let $\mathbf{S}$ be an excellent model category in which all objects are cofibrant, viewed as an $\mathbf{S}$-enriched category by its canonical self-enrichment. Then we know that there is an obvious enriched full embedding of the subcategory of fibrant objects $\mathbf{S}^\circ \hookrightarrow \mathbf{S}$.

Since $\mathbf{S}$ is a combinatorial model category, the small object argument provides us with a fibrant replacement functor $\mathbf{S}\to \mathbf{S}^\circ$ that is not necessarily enriched.

First question: If $\mathcal{A}$ is an $\mathbf{S}$-enriched combinatorial model category (maybe in which all objects are cofibrant), is there an enriched version of the small object argument that provides us with a fibrant replacement functor $\mathcal{A}\to \mathcal{A}^\circ$ that is an enriched functor?

Second question: If the first question is not true in general for $\mathbf{S}$-enriched combinatorial model categories, is it at least true for the special case $\mathcal{A}=\mathbf{S}$ itself?

Answer 1

I think the answer is yes (to both questions). In Emily Riehl's book "Categorical homotopy theory", chapter 13 is all about the enriched small object argument. Theorem 13.2.1 on page 177 (I hope I'm looking at a version close to the final one) explains when the enriched small object argument works. A condition Riehl calls $(\star)$ is required. She remarks (13.2.3) that this condition is satisfied if, for every $X$ in $\mathcal{A}$, $-\otimes X$ preserves cofibrations and trivial cofibrations. Clearly, this is satisfied in your setting. If you define your fibrant replacement functor via the enriched weak factorization system (factoring $X\to \ast$ into a trivial cofibration $X\to RX$ and then a fibration $RX\to \ast$), then the functor $R$ is $\mathcal{S}$-enriched. This is the content of Riehl's Corollary 13.2.4.

Answer 2

First question: I couldn't find anything.

Second question: I just found a sufficient condition under some strong finiteness assumptions: According to the paper of Dundas, Röndigs, and Østvær, the enriched fibrant replacement functor exists when $\mathcal{S}$ satisfies two properties:

• The Schwede-Shipley "monoid axiom"
• The model structure is "weakly finitely generated".

The monoid axiom is automatically satisfied when $\mathbf{S}$ is excellent with all objects cofibrant, but the finiteness assumptions appear highly restrictive.

These finiteness conditions (modified by me for combinatorial model categories) are:

• There exists a small set of generating cofibrations $\mathcal{I}$ such that for each $f\in \mathcal{I}$, the domain and codomain of $f$ are both finitely presentable.
• There exists a small set $\mathcal{J}^\prime$ of trivial cofibrations with finitely presentable domains and codomains such that if $p:A\to B$ is a map with $B$ fibrant, then $p$ is a fibration if and only if it has the RLP with respect to $\mathcal{J}^\prime$.

For example, these conditions are satisfied by the Joyal model structure on simplicial sets, where we take $\mathcal{I}$ to be the set of boundary inclusions and $\mathcal{J}^\prime$ to be the set of inner horn inclusions union the inclusion $\ast \hookrightarrow J_{\operatorname{fin}}$ where $J_{\operatorname{fin}}$ is the finite fragment of the big Joyal interval $J$ (that is, $J_{\operatorname{fin}}$ the usual quotient of $\Delta[3]$).

If there are any newer papers that remove some of these strong finiteness assumptions, that is even better, but I thought I would report what I found this morning.