With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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.
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
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
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).
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 βactionβ 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,
This follows from some coend manipulations:
From this perspective, the representable functor $Hom_\mathcal{C}(\cdot, c)$ looks like a delta distribution concentrated at $c$.
Recall that a presheaf $F : \mathcal{C}^\mathrm{op} \to \mathrm{Set}$ can be seen as βgluing 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
that is as the tensor product $F \otimes_\mathcal{C} G$. The tensor product can therefore be pictured as the $G(X)$βs, glued as specified by $F$.
Since the Yoneda embedding $Y$ includes an object into the category of formal gluing specifications, gluing the $Y(X)$βs 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 $|F|$.
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.