Monoidal tensor product which preserves directed limits

Question

Given a symmetric monoidal category $Q$, is there a construction of a (preferably full and faithful strong) monoidal embedding of $Q$ into some symmetric monoidal closed category $M$ which has all filtered colimits for which the functor $A\otimes -$ preserves cofiltered limits (i.e., limits with a directed poset as shape, I do not mean colimits here) for each fixed $A\in M$?

It would be nice if $M$ is both complete and cocomplete, but not essential. However, the existence of filtered colimits and the monoidal closure of $M$ are essential.

A possible solution might be considering the Yoneda embedding of $Q$ into $M:=[Q^\mathrm{op},\mathbf{Set}]$, which becomes symmetric monoidal closed when equipped with a tensor product via the Day Convolution. Could it be that this tensor product preserves cofiltered limits? The Day convolution can be defined in terms of a colimit, which does not commute with limits in general. But could we have an escape route, since we require the limits to have a special shape, namely of a directed poset?

Alternatively, is there some other construction of an embedding into a symmetric monoidal closed category $M$ where the functor $A\otimes -$ for fixed $A\in M$ has a left adjoint, hence preserves all limits?

The reason why we are interested in this question is because we aim to model circuit programming languages, where the circuits are represented by morphisms in the category $Q$. In principle, we would like to have a construction for arbitrary symmetric monoidal categories $Q$, but maybe there is some construction that only works if we impose some extra conditions on $Q$? A specific choice for $Q$ might be a free symmetric monoidal category over a finite monoidal category/signature. Another specific choice for $Q$ might be the category of finite-dimensional C*-algebras with completely positive subunital maps. But a construction for an arbitrary monoidal categories $Q$ would be preferable.

Answer 1

Public Service Announcement!

It's very confusing -- I'd daresay incorrect -- to say "directed limit" to mean "limit indexed by a cofiltered diagram". Actually, historically the term "directed limit" has been used to mean "colimit indexed by a filtered diagram" or even just "colimit"-- this usage predates the introduction of the terms "limit" and "colimit", I believe. The term you're looking for is "inverse limit", or if you want to be 100% clear, say "cofiltered limit" (because "inverse limit" has sometimes been used to just mean "limit").

Answer to Question:

Now to your question. As mentioned in the comments, if $C$ is a symmetric monoidal category then the Day convolution is a universal way to turn $C$ into a cocomplete category with a symmetric monoidal product that preserves colimits in each variable. It's straightforward to check that the Day convolution restricts to a symmetric monoidal structure on the Ind-completion of $C$, denoted $Ind(C)$, with $\otimes$ preserving filtered colimits in each variable. Here $Ind(C) \subset [C^\mathrm{op},\mathsf{Set}]$ is the full subcategory of presheaves which are filtered colimits of representables. It is the universal category with filtered colimits generated by $C$, and so the Day convolution makes it the universal way to turn $C$ into a symmetric monoidal category with filtered colimits preserved in each variable by $\otimes$.

To get a similar universal construction for cofiltered limits, just dualize everything. The dual Day convolution is a symmetric monoidal structure on $[C,\mathsf{Set}]^\mathrm{op}$ preserving limits in each variable, and restricts to a symmetric monoidal structure the Pro-completion of $C$, $Pro(C) = Ind(C^\mathrm{op})^\mathrm{op} \subset [C,\mathsf{Set}]^\mathrm{op}$, which preserves cofiltered limits in each variable. $Pro(C)$ is the universal category with cofiltered limits on $C$, and it consists of those copresheaves on $C$ which are filtered colimits of representables -- with morphisms being natural transformations in the opposite direction.

Caveat emptor:

The $Ind$ and $Pro$ completions really are universal. In particular, the inclusion $C \to Ind(C)$ does not preserve filtered colimits that might happen to exist in $C$ / dually the inclusion $C \to Pro(C)$ does not preserve cofiltered limits that might happen to exist in $C$. So if you are interested in the cofiltered limits that already exist in $C$, then $Pro(C)$ will be unsatisfactory for you because the Pro construction simply ignores all existing cofiltered limits and freely adds in new ones. (This is just like how if $G$ is a group, then the free group $F(G)$ on the underlying set of $G$ will completely ignore the existing multiplication on $G$.)

However, if $C$ already has cofiltered limits, then the inclusion $C \to Pro(C)$ will have a right adjoint which takes an object of $Pro(C)$, which you can think of as a formal cofiltered limit, and evaluates that limit in $C$ (in the group analogy, if $G$ is a group then there is a canonical group homomorphism $F(G) \to G$ which multiplies together formal words in the group language over $G$). Then you can start asking questions like how this functor interacts with the Day convolution.

Of course, if you don't care about cofiltered limits that might happen to be present in $C$, then $Pro(C)$ is probably exactly what you want.

It's nice to know, though, that $C \to Ind(C)$ does preserve finite colimits / $C \to Pro(C)$ does preserve finite limits.

Supplementary note:

If you're not familiar with the $Ind$ and $Pro$ completions, I should probably tell you that $Ind(C)$ can alternately be described as follows. An object consists of a filtered category $I$ and a diagram $X: I \to C$. The homsets are $Hom_{Ind(C)}(X: I \to C, Y: J \to C) = \varprojlim_{i \in I} \varinjlim_{j \in J} Hom_C(X_i,Y_j)$. Dually, an object in $Pro(C)$ consists of a cofiltered category $P$ and a diagram $Z: P \to C$. The homsets are $Hom_{Pro(C)}(Z: P \to C, W: Q \to C) = \varprojlim_{q \in Q} \varinjlim_{p \in P} Hom_C(Z_p, W_q)$.

This is a quite explicit sense in which $Ind(C)$ consists of "formal filtered colimits" / $Pro(C)$ consists of "formal cofiltered limits" -- the objects literally are the diagrams you want to take the (co)limit of!