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.

  • 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
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    It'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/closed-monoidal-category (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
up vote 10 down vote accepted

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.

  • 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
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    Probably 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 teeny-tiny instance. – Todd Trimble Jan 27 '16 at 5:59
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    I'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 Paul-André 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

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