nLab tensor product of functors

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 ‘action’ 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 “gluing specification”: 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 $G(X)$‘s, glued as specified by $F$.

• Since the Yoneda embedding $Y$ includes an object into the category of formal gluing specifications, gluing the $Y(X)$‘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 $|F|$.

Related concepts

• tensor product of modules
• coend
• weighted colimit

References

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