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Stream: learning: questions

Topic: the Yoneda embedding into a cocomplete monoidal category

Emily (Sep 29 2021 at 08:51):

These days I've been meaning to add some things missing from the nLab page on Day convolution, but am currently a bit confused about a few issues, mostly around the notion of a "Yoneda embedding into an arbitrary cocomplete monoidal category".

In that page (and in the original paper and thesis by Day), one speaks of Day convolution in PSh(C)=Fun(C^op,Sets) for C a small monoidal category. However, we can more generally define Day convolution in Fun(C,D) whenever D is a cocomplete symmetric monoidal category whose tensor product preserves colimits in both variables (henceforth a "nice cocomplete symmetric monoidal category"): we just replace products by cotensors, for example writing "$\mathrm{Hom}_{\mathcal{C}}(A\otimes B,-)\odot$" instead of "$\mathrm{Hom}_{\mathcal{C}}(A\otimes B,-)\times$".

In this more general context, the (contravariant) Yoneda embedding $\mathcal{C}^\mathsf{op}\hookrightarrow\mathsf{Fun}(\mathcal{C},\mathsf{Sets})$, $A\mapsto\mathrm{Hom}(A,-)$ is replaced by the functor $\mathcal{C}^\mathsf{op}\to\mathsf{Fun}(\mathcal{C},\mathcal{D})$ given by $A\mapsto \mathrm{Hom}(A,-)\odot\mathbf{1}_{\mathcal{D}}$, where $\mathbf{1}_{\mathcal{D}}$ is the monoidal unit of $\mathcal{D}$.

Question 1. Is this "Yoneda embedding into a nice cocomplete symmetric monoidal category" discussed somewhere in the literature? (IIRC the original paper by Day only does things for D=V the base of enrichment)

Question 2. Theorem 5.1 of Im–Kelly states that the Day convolution monoidal structure on PSh(C) is the universal one satisfying the following conditions:

1. The Yoneda embedding becomes a strong monoidal functor.
2. The tensor product of PSh(C) preserves colimits in both variables.

Is the analogous statement true when we replace PSh(C) for Fun(C,D) and the Yoneda embedding for the functor $\mathcal{C}^\mathsf{op}\to\mathsf{Fun}(\mathcal{C},\mathcal{D})$ above?

Question 3. Yet another universal property for Day convolution states an isomorphism of categories of the form

$$\mathsf{Fun}^{\otimes}(\mathcal{C}\times\mathcal{D},\mathsf{Sets}) \cong \mathsf{Fun}^{\otimes}(\mathcal{C},\mathsf{Fun}(\mathcal{D},\mathsf{Sets})).$$

Is this still true when we replace $\mathsf{Sets}$ by a nice cocomplete symmetric monoidal category $\mathcal{D}$?

Question 4. Lastly, does Day convolution also make sense when $\mathcal{D}$ is just a nice cocomplete monoidal category, instead of a (nice cocomplete) symmetric one?

Mike Shulman (Sep 29 2021 at 09:05):

Isn't this just the enriched version where you regard $C$ as "discretely" enriched over $D$?

Emily (Sep 29 2021 at 10:12):

Oh it certainly is! Thanks, Mike. This solves everything :smile: