# nLab tensor product of functors

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

Just like a right $A$-module can be tensored (over $A$) with a left $A$-module to obtain an abelian group, a functor $T : \mathcal{C}^\op \to \mathcal{V}$ can be tensored (over $\mathcal{C}$) with a functor $S : \mathcal{C} \to \mathcal{V}$ to obtain an object of $\mathcal{V}$. Here, $\mathcal{V}$ can be an arbitrary monoidal category. A simple case is $\mathcal{V} = Set$ with the cartesian product as monoidal structure.

## Definition

Let $\mathcal{C}$ be a category. Let $(\mathcal{V},{\otimes})$ be a monoidal category. Let $T : \mathcal{C}^\op \to \mathcal{V}$ and $S : \mathcal{C} \to \mathcal{V}$ be functors. Then their tensor product $T \otimes_{\mathcal{C}} S$ is defined (if it exists) as the coend

$T \otimes_{\mathcal{C}} S \coloneqq \int^c T(c) \otimes S(c).$

The following slight variation is also important. Let $\mathcal{C}$ be a category. Let $\mathcal{D}$ be a category with all coproducts, so that the copower $X \cdot d = \coprod_{x \in X} d \in \mathcal{D}$ exists for any set $X$ and any object $d \in \mathcal{D}$. Let $T : \mathcal{C}^\op \to Set$ and $S : \mathcal{C} \to \mathcal{D}$ be functors. Then their tensor product is (if it exists) the coend

$T \otimes_{\mathcal{C}} S \coloneqq \int^c T(c) \cdot S(c) \in \mathcal{D}.$

Intuition for the tensor product of functors can be gained by relating it to the tensor product of modules (see the first example below) and by a picture involving gluing specifications (see below).

## Examples

• If $T : \mathscr{C}^{op} \to Set$ and $S : \mathscr{C} \to Set$, then $T \otimes_{\mathscr{C}} S$ can be presented as set of formal symbols $t \otimes_c s$ (or $t \otimes s$ for short) where $c \in Ob(\mathscr{C})$, $s \in S(c)$, and $t \in T(c)$ modulo the equivalence relation generated by the instances $(t f) \otimes_c s \sim t \otimes_d (f s)$ when defined, where $f \in \mathscr{C}(c,d)$, and the βactionβ of $\mathscr{C}$ is the shorthand $t f = T(f)(t)$ and $f s = S(f)(s)$.

• Recall that a ring $A$ can be considered as an Ab-enriched category and that a right and a left module gives rise to an additive functor $A^\op \to Ab$ respectively $A \to Ab$. Then their tensor product as functors, calculated in the Ab-enriched setting and using the ordinary tensor product of abelian groups as monoidal structure on $Ab$, coincides with their usual tensor product.

• Let $X$ be a simplicial set and $st : \Delta \to Top$ be the functor which associates to $[n]$ the topological standard $n$-simplex. Then the geometric realization of $X$ can be expressed as the tensor product $|X| = X \otimes_\Delta st$.

• Let $\mathcal{C}$ be a small category and let $F : \mathcal{C} \to \mathcal{D}$ be a functor into a cocomplete category $\mathcal{D}$. Since the category $PSh(\mathcal{C})$ of presheaves on $\mathcal{C}$ is the free cocompletion of $\mathcal{C}$, the functor $F$ induces a functor $\hat F : PSh(\mathcal{C}) \to \mathcal{D}$. This functor can be explicitly described as $\hat F(X) = X \otimes_\mathcal{C} F$.

• Let $Y : \mathcal{C} \to PSh(\mathcal{C})$ denote the Yoneda embedding. Let $F : \mathcal{C}^\op \to Set$ be a presheaf on $\mathcal{C}$. Then $F \otimes Y = F$. This fact is the co-Yoneda lemma (also referred to as the ninja Yoneda lemma in some circles).

• In some sense, representable functors generalize free modules: Recall $A^n \otimes_A M \cong M^n$. Similarly,

$\Hom_\mathcal{C}(\cdot, c) \otimes_\mathcal{C} S = S(c).$

This follows from some coend manipulations:

$\begin{array}{rcl} \Hom(\Hom_\mathcal{C}(\cdot, c) \otimes_\mathcal{C} S, t) &=& \int_{c' \in \mathcal{C}} \Hom(\Hom_\mathcal{C}(c',c) \cdot S(c'), t) \\ &=& \int_{c' \in \mathcal{C}} \Hom(\Hom_\mathcal{C}(c',c), \Hom_{\mathcal{D}}(S(c'), t)) \\ &=& Nat(\Hom_\mathcal{C}(\cdot, c), \Hom_\mathcal{D}(S(\cdot), t)) \\ &=& \Hom_\mathcal{D}(S(c),t). \end{array}$

From this perspective, the representable functor $Hom_\mathcal{C}(\cdot, c)$ looks like a delta distribution concentrated at $c$.

## Intuition using gluing specifications

Recall that a presheaf $F : \mathcal{C}^\mathrm{op} \to \mathrm{Set}$ can be seen as βgluing specificationβ: If $G : \mathcal{C} \to \mathcal{D}$ is some functor into a cocomplete category $\mathcal{D}$, this gluing specification can be realized as $\operatorname{colim}_{s \in F(X)} G(X)$. This colimit can also be written as the coend

$\int^{X \in \mathcal{C}} F(X) \cdot G(X),$

that is as the tensor product $F \otimes_\mathcal{C} G$. The tensor product can therefore be pictured as the $G(X)$βs, glued as specified by $F$.

• Since the Yoneda embedding $Y$ includes an object into the category of formal gluing specifications, gluing the $Y(X)$βs as specified by $F$ simply yields in $F$; thus $F \otimes_\mathcal{C} Y = F$.

• Let $\mathcal{C}$ be specifically the simplex category $\Delta$. Then $F$ is just a simplicial set, so the intuition of $F$ as a gluing specification is even more vivid. Tensoring with $st : \Delta \to Top$ realizes this specification in the category of topological spaces, using standard $n$-simplices as building blocks, and therefore yields the geometric realization $|F|$.

## References

A basic reference is Categories Work, Section IX.6.

Last revised on July 23, 2021 at 12:17:37. See the history of this page for a list of all contributions to it.