Extension of a monoidal structure

Question

Let $\mathcal D$ be a dense subcategory of a category $\mathcal C$. Suppose that $\mathcal D$ admits a monoidal structure. Can the monoidal structure be extended from $\mathcal D$ to $\mathcal C$?

Answer 1

There's a well known extension of $\mathcal D$'s monoidal structure $\otimes$ to its category of presheaves $\hat{\mathcal D}$ known as the Day convolution product $\hat\otimes$. It's uniquely specified by the requirements that the Yoneda embedding be strong monoidal and that $\hat\otimes$ be cocontinuous in each variable. Of course, the Yoneda embedding is the ur-example of a dense functor; we can try extend to your more general case from here. The density of $\mathcal D$ in $\mathcal C$ is equivalent to the full-faithfulness of the restricted Yoneda embedding $\mathcal C\to \hat{\mathcal D}$: in other words, pairs $\mathcal D\subset \mathcal C$ are equivalent to full subcategories of presheaf categories containing the representables.

Now it becomes clear that the Day convolution $\hat\otimes$ may not restrict to a monoidal product on $\mathcal C$, since there's no reason for a nearly arbitrary full subcategory to be $\hat\otimes$-closed. For instance, we could take $\mathcal D$ to be a discrete monoidal category, essentially just a monoid $D$. Then $\hat{\mathcal D}$ is the category of sets over $D$ with monoidal structure $(F\hat\otimes G)_d=\coprod_{d_1d_2=d}F_{d_1}\times F_{d_2}$. Then the full subcategory of sets over $D$ with at most one nonempty fiber is very far from $\hat\otimes$-closed.

It would be sufficient that $\mathcal C$ be coreflective in $\hat D$, but the only coreflective subcategory of a presheaf category containing the representables is the whole presheaf category, so this will never occur except in the simplest case. More interestingly, we can ask to reflect $\hat\otimes$ back into $\mathcal C$: if we have a reflection $L:\hat{ \mathcal D}\to \mathcal C$, then define $\otimes'$ on $\mathcal C$ as $L((-)\hat\otimes (-))$. This is how the tensor product of sheaves can be constructed. But sheaves are not just a reflective subcategory of presheaves, they're also closed under internal homs, and you will not be able to prove that $\otimes'$ is associative without requiring that $\mathcal C$ be closed under internal homs in $\hat D$. (Internal homs for the Day product are defined by cocontinuity, just as for $\hat\otimes$ itself.) Day himself published a theorem characterizing reflective subcategories of monoidal closed categories closed under internal homs here.

Unfortunately I know no general techniques by which to check his conditions for a non-Cartesian monoidal structure. However, in case $\mathcal D$ is Cartesian, $\hat{\mathcal D}$ is as well, and then we can reflect $\hat\otimes=\times$ into $\mathcal C$ if and only if the reflection $L$ preserves finite products, which brings us back again to the world of sheaves. In this case, of course, we're not impressed: we've just reflected the Cartesian product into a reflective subcategory, which we knew was closed under limits anyway! However, this argument does show that we'll be able to reflect $\hat\otimes$ into $\mathcal C$ only if $\mathcal C$ is also closed, with the same homs as in $\hat{\mathcal D}$, which was certainly not obvious to begin with; furthermore, one can use the argument on non-Cartesian monoidal products derived from the Cartesian one, such as, of course, the tensor product of abelian group objects.

In short, the answer is: not in general, but there's a natural condition you can check in particular cases.

Answer 2

You can if you have a bit more information than just the monoidal structure. This is Proposition 9 from "Categorical and combinatorial aspects of descent theory ", by Street, where he attributes this result to Day. With your notation the result is the following:

Suppose $J : \mathcal D \to \mathcal C$ is a dense functor from a small monoidal category $\mathcal D$ into a complete and cocomplete category $\mathcal C$. The formula $$ c_1 \otimes c_2 := \int^{d_1,d_2} (\mathcal C(Jd_1,c_1) \times \mathcal C(c_2,Jd_2)) \cdot J(d_1\otimes d_2) $$ defines a (left and right) closed monoidal structure on X with J strong monoidal if and only if there exist functors $H$ and $H′ : \mathcal D^{op} \times \mathcal C \to \mathcal C$ and isomorphisms $$\mathcal C(Jd_1, H(d_2,c)) \cong \mathcal C(J(d_1 \otimes d_2),c) \cong \mathcal C(Jd_2,H'(d_1,c)) $$ natural in objects $d_1$ and $d_2$ of $\mathcal D$ and $c$ of $\mathcal C$.

This result is reformulated as follows in "Joint et tranches pour les $\infty$-catégories strictes" by Ara and Maltsiniotis (in French, Theorem 6.9):

Let $\mathcal C$ be a complete and cocomplete category, and $\mathcal D$ be a full subcategory of $\mathcal C$ equipped with a monodical structure. Suppose there exists a small full subcategory of $\mathcal D$ dense in $\mathcal C$, and functors $H$ and $H′ : \mathcal D^{op} \times \mathcal C \to \mathcal C$, together with bijections $$\mathcal C(Jd_1, H(d_2,c)) \cong \mathcal C(J(d_1 \otimes d_2),c) \cong \mathcal C(Jd_2,H'(d_1,c)) $$ natural in objects $d_1$ and $d_2$ of $\mathcal D$ and $c$ of $\mathcal C$.

Then there exists a unique monodical structure on $\mathcal C$ (up to unique monodical isomorphism) given by a product which commutes to small colimits in each variable, and for which the inclusion functor $\mathcal D \to \mathcal C$ extends into a monoidal functor. Moreover, this monodical structure is biclosed.

Still in this last article, the authors then extend this result to construct products that only commute to small connected colimits (Corollary 6.14). However that only works if the unit of the monoidal product on $\mathcal D$ is an initial object of $\mathcal C$. If you are interested I can translate the result to English but it is a pretty niche case.