Item 1 (02M7) — The Clowder Project

Table of Contents
Part 4: Categories
Chapter 12: Presheaves and the Yoneda Lemma
Section 12.4: Functor Tensor Products
Subsection 12.4.2: The Tensor of a Presheaf With a Functor
Item 1 (cite)

12.4.2 The Tensor of a Presheaf With a Functor

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

Definition 12.4.2.1.1 ▶ The Tensor of a Presheaf With a Functor

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

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

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

Remark 12.4.2.1.2 ▶ Unwinding Definition 12.4.2.1.1

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

\[ \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\xrightarrow {\mathcal{F}\times G}\mathsf{Sets}\times \mathcal{D}\xrightarrow {\odot }\mathcal{D}. \]

Proposition 12.4.2.1.3 ▶ Properties of Tensors of Presheaves With Functors

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

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

\[ \begin{array}{ccc} \mathcal{F}\odot _{\mathcal{C}}-\colon \mkern -15mu & \mathsf{PSh}(\mathcal{C}) \mkern -17.5mu& {}\mathbin {\to }\mathcal{D},\\ -\odot _{\mathcal{C}}G\colon \mkern -15mu & \mathsf{Fun}(\mathcal{C},\mathcal{D}) \mkern -17.5mu& {}\mathbin {\to }\mathcal{D},\\ -_{1}\odot _{\mathcal{C}}-_{2}\colon \mkern -15mu & \mathsf{PSh}(\mathcal{C})\times \mathsf{Fun}(\mathcal{C},\mathcal{D}) \mkern -17.5mu& {}\mathbin {\to }\mathcal{D}. \end{array} \]