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 slicecategory 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.

For instance, unless I'm mistaken, one immediately has problems even getting that this monoidal structure is closed? – Jonathan Beardsley Jan 26 '16 at 22:15

1It's funny, we were talking about this type of construction at the nForum and nLab very recently. If you take the slice $M/I$ over the monoidal unit, then you get the cofree semicartesian monoidal category over $M$. An example we were looking at was the category of vector spaces over a ground field $k$, where $\text{Vect}/k$ is a collage of vector spaces and affine spaces. I'm describing it in a hurry in this tiny comment box, but see here for the discussion: nforum.ncatlab.org/discussion/4383/closedmonoidalcategory (esp. starting around comment 8 ff). – Todd Trimble♦ Jan 27 '16 at 1:22

@ToddTrimble maybe I should get back over there! Haven't been on the forum in a really long time, but I seem to be wading into more and more categorical stuff recently (e.g. categories of operators, multicategories, the above). – Jonathan Beardsley Jan 27 '16 at 3:14
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 tensorhom 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 2category MonCat$_l$ of monoidal categories, (lax) monoidal functors and monoidal natural transformations. The forgetful 2functor 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 2categories and limits for lax morphisms' with Michael Shulman for the general 2monad situation.

Thanks so much Alexander! This is really great! I could find very little about this construction anywhere in the literature, but it seemed like something that should obviously be doable, or at least discussable. – Jonathan Beardsley Jan 27 '16 at 3:19

1Probably someone should ask Street about the history of the construction. I think instances have been recognized for a long time; for example, the thing denoted $Oper(M)$ here ncatlab.org/nlab/show/clique#monoidal_strictifications is something I think I saw in a 1971 paper by Kelly and Mac Lane. Of course this is just one teenytiny instance. – Todd Trimble♦ Jan 27 '16 at 5:59

1I'm also very interested in the history of this construction. Let me mention that (1) and (2) can also be formulated as follows: for any monoidal closed functor $p : \mathcal{E} \to \mathcal{B}$ which is also a Grothendieck bifibration, if $(M,\otimes,1) \in \mathcal{B}$ is a monoid in the basis, then its fiber $\mathcal{E}_M$ is monoidal closed. This approach is something PaulAndré Melliès and I have looked at in recent years. One advantage of this formulation is that it also covers the Day construction of a monoidal closed structure on the presheaf category over a (pro)monoidal category. – Noam Zeilberger Jan 27 '16 at 11:37