Non-Cartesian Monoidal Model Structure on a Slice Category

Question

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered products over $A$. However, it should admit another monoidal structure coming from the product on $A$. In particular, given two maps $X\to A$ and $Y\to A$, there should be an object $Y\otimes X\to A\otimes A\to A$. Moreover, it seems, that monoid objects in $M_{/A}$ should be precisely the monoid morphisms $A'\to A$ for $A'$ another monoid. Is this monoidal structure described anywhere? How does it interact with the slice-category model structure? The particular case I'm thinking of is the case when $M=sSet$ with the Quillen model structure, and $A$ is a strict monoid with respect to Cartesian product, but I'd be pretty happy with this kind of statement for any nice model category of topological spaces.

Answer 1

This construction came up in an Australian Category Seminar talk given by Ross Street last month, from which I will copy for 1. and 2. below. I'm afraid I don't know a reference.

1. (monoidal structure) If $\mathscr{F}$ is a monoidal category and $T$ is a monoid in $\mathscr{F}$, then $\mathscr{F}/T$ becomes monoidal, with tensor product as you suggest: $$(\theta \colon F \to T) \otimes (\varphi \colon G \to T) = \mu \circ (\theta \otimes \varphi) \colon F\otimes G\to T\otimes T \to T, $$ and unit $\eta \colon I \to T$. Note that the projection functor $\mathscr{F}/T \to \mathscr{F}$ is strict monoidal.

2. (internal hom) To your comment, if $\mathscr{F}$ is closed, then so is $\mathscr{F}/T$. The internal hom of $\varphi \colon G \to T$ and $\psi \colon H \to T$ in $\mathscr{F}/T$ is given by the pullback in $\mathscr{F}$ of $$[1,\psi] \colon [G,H] \to [G,T] \quad \text{along} \quad [\varphi,1] \circ \lambda \colon T \to [T,T] \to [G,T],$$ where $\lambda$ corresponds to $\mu \colon T\otimes T \to T$ under the tensor-hom adjunction.

3. (monoidal model structure) Finally, if $\mathscr{F}$ is a monoidal model category then so is $\mathscr{F}/T$, since $\mathscr{F}/T \to \mathscr{F}$ is strict monoidal and creates colimits, cofibrations, fibrations and weak equivalences.

P.S. A general perspective on this construction which may be of interest is that the monoidal category $\mathscr{F}/T$ is the oplax limit of the arrow $T \colon 1 \to \mathscr{F}$ in the 2-category MonCat$_l$ of monoidal categories, (lax) monoidal functors and monoidal natural transformations. The forgetful 2-functor MonCat$_l$ $\to$ Cat creates oplax limits, and the oplax limit of the underlying arrow $T \colon 1 \to \mathscr{F}$ in Cat is the slice category $\mathscr{F}/T$. See Steve Lack's papers 'Limits for lax morphisms' and the more recent 'Enhanced 2-categories and limits for lax morphisms' with Michael Shulman for the general 2-monad situation.