Item 3 (02LZ) — The Clowder Project

12.4.1 The Tensor Product of Presheaves With Copresheaves

Let $\mathcal{C}$ be a category, let $\mathcal{F}\colon \mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$ be a presheaf on $\mathcal{C}$, and let $G\colon \mathcal{C}\to \mathsf{Sets}$ be a copresheaf on $\mathcal{C}$.

Definition 12.4.1.1.1 ▶ The Tensor Product of Presheaves With Copresheaves

The tensor product of $\mathcal{F}$ with $G$ is the set $\mathcal{F}\boxtimes _{\mathcal{C}}G$¹ defined by

\[ \mathcal{F}\boxtimes _{\mathcal{C}}G\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int ^{A\in \mathcal{C}}\mathcal{F}(A)\times G(A). \]

¹Further Notation: Also written simply $\mathcal{F}\boxtimes G$.

Remark 12.4.1.1.2 ▶ Unwinding Definition 12.4.1.1.1

In other words, the tensor product of $\mathcal{F}$ with $G$ is the set $\mathcal{F}\boxtimes _{\mathcal{C}}G$ defined as the coend of the functor

\[ \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\xrightarrow {\mathcal{F}\times G}\mathsf{Sets}\times \mathsf{Sets}\xrightarrow {\times }\mathsf{Sets}, \]

which is equivalently the composition

in $\mathsf{Prof}$.

Example 12.4.1.1.3 ▶ The Tensor Product of Presheaves With Copresheaves on One Object Categories

Proposition 12.4.1.1.4 ▶ Properties of Tensor Products of Presheaves With Copresheaves

Let $\mathcal{C}$ be a category.

1.
Functoriality. The assignments $\mathcal{F},G,(\mathcal{F},G)\mapsto \mathcal{F}\boxtimes _{\mathcal{C}}G$ define functors

\[ \begin{array}{ccc} \mathcal{F}\boxtimes _{\mathcal{C}}-\colon \mkern -15mu & \mathsf{PSh}(\mathcal{C}) \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ -\boxtimes _{\mathcal{C}}G\colon \mkern -15mu & \mathsf{CoPSh}(\mathcal{C}) \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ -_{1}\boxtimes _{\mathcal{C}}-_{2}\colon \mkern -15mu & \mathsf{PSh}(\mathcal{C})\times \mathsf{CoPSh}(\mathcal{C}) \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}. \end{array} \]
2.
As a Composition of Profunctors. Let $\mathcal{C}$ be a category and let:
- •
  $\mathcal{F}\colon \mathsf{pt}\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}\mathcal{C}$ be a presheaf on $\mathcal{C}$, viewed as a profunctor.
- •
  $F\colon \mathcal{C}\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}\mathsf{pt}$ be a copresheaf on $\mathcal{C}$, viewed as a profunctor.
We have a natural isomorphism of profunctors
natural in $\mathcal{F}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$ and $F\in \operatorname {\mathrm{Obj}}(\mathsf{CoPSh}(\mathcal{C}))$.

3.
Interaction With Representable Presheaves. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. We have a bijection of sets

\[ \mathcal{F}\boxtimes _{\mathcal{C}}h^{X}\cong \mathcal{F}(X), \]

natural in $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, giving a natural isomorphism of functors

Interaction With Corepresentable Copresheaves. Let $G$ be a copresheaf on $\mathcal{C}$. We have a bijection of sets

\[ h_{X}\boxtimes _{\mathcal{C}}G\cong G(X), \]

natural in $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, giving a natural isomorphism of functors

Interaction With Yoneda Extensions. Let $G\colon \mathcal{C}\to \mathsf{Sets}$ be a copresheaf on $\mathcal{C}$. We have a natural isomorphism

natural in $G\in \operatorname {\mathrm{Obj}}(\mathsf{CoPSh}(\mathcal{C}))$.

Interaction With Contravariant Yoneda Extensions. Let $\mathcal{F}\colon \mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$ be a presheaf on $\mathcal{C}$. We have a natural isomorphism