Item 3 (02MG) — The Clowder Project

Table of Contents
Part 4: Categories
Chapter 12: Presheaves and the Yoneda Lemma
Section 12.4: Functor Tensor Products
Subsection 12.4.3: The Tensor of a Copresheaf With a Functor
Item 3 (cite)

12.4.3 The Tensor of a Copresheaf With a Functor

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

Definition 12.4.3.1.1 ▶ The Tensor of a Copresheaf With a Functor

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

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

¹Further Notation: Also written simply $F\odot G$.

Remark 12.4.3.1.2 ▶ Unwinding Definition 12.4.3.1.1

In other words, the tensor of $F$ with $G$ is the object $F\odot _{\mathcal{C}}G$ of $\mathcal{D}$ defined as the coend of the functor

\[ \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathcal{C}\times \mathcal{C}^{\mathsf{op}}\xrightarrow {F\times G}\mathsf{Sets}\times \mathcal{D}\xrightarrow {\odot }\mathcal{D}. \]

Proposition 12.4.3.1.3 ▶ Properties of Tensors of Copresheaves With Functors

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

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

\[ \begin{array}{ccc} F\odot _{\mathcal{C}}-\colon \mkern -15mu & \mathsf{CoPSh}(\mathcal{C}) \mkern -17.5mu& {}\mathbin {\to }\mathcal{D},\\ -\odot _{\mathcal{C}}\mathcal{G}\colon \mkern -15mu & \mathsf{Fun}(\mathcal{C}^{\mathsf{op}},\mathcal{D}) \mkern -17.5mu& {}\mathbin {\to }\mathcal{D},\\ -_{1}\odot _{\mathcal{C}}-_{2}\colon \mkern -15mu & \mathsf{Fun}(\mathcal{C}^{\mathsf{op}},\mathcal{D})\times \mathsf{CoPSh}(\mathcal{C}) \mkern -17.5mu& {}\mathbin {\to }\mathcal{D}. \end{array} \]
2.
Interaction With Corepresentable Copresheaves. We have an isomorphism

\[ h^{X}\odot _{\mathcal{C}}G\cong G(X), \]

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

\[ h^{(-)}\odot _{\mathcal{C}}G\cong G. \]

3.
Interaction With Contravariant Yoneda Extensions. We have a natural isomorphism

\[ \operatorname {\mathrm{Lan}}_{\style {display: inline-block; transform: rotate(180deg)}{よ}\mkern -2.5mu}(G)\cong G\odot _{\mathcal{C}}(-), \]

natural in $G\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(\mathcal{C}^{\mathsf{op}},\mathcal{D}))$.