nLab tensor product of functors



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



Just like a right AA-module can be tensored (over AA) with a left AA-module to obtain an abelian group, a functor T:π’ž op→𝒱T : \mathcal{C}^\op \to \mathcal{V} can be tensored (over π’ž\mathcal{C}) with a functor S:π’žβ†’π’±S : \mathcal{C} \to \mathcal{V} to obtain an object of 𝒱\mathcal{V}. Here, 𝒱\mathcal{V} can be an arbitrary monoidal category. A simple case is 𝒱=Set\mathcal{V} = Set with the cartesian product as monoidal structure.


Let π’ž\mathcal{C} be a category. Let (𝒱,βŠ—)(\mathcal{V},{\otimes}) be a monoidal category. Let T:π’ž op→𝒱T : \mathcal{C}^\op \to \mathcal{V} and S:π’žβ†’π’±S : \mathcal{C} \to \mathcal{V} be functors. Then their tensor product TβŠ— π’žST \otimes_{\mathcal{C}} S is defined (if it exists) as the coend

TβŠ— π’žSβ‰”βˆ« cT(c)βŠ—S(c). T \otimes_{\mathcal{C}} S \coloneqq \int^c T(c) \otimes S(c).

The following slight variation is also important. Let π’ž\mathcal{C} be a category. Let π’Ÿ\mathcal{D} be a category with all coproducts, so that the copower Xβ‹…d=∐ x∈Xdβˆˆπ’ŸX \cdot d = \coprod_{x \in X} d \in \mathcal{D} exists for any set XX and any object dβˆˆπ’Ÿd \in \mathcal{D}. Let T:π’ž opβ†’SetT : \mathcal{C}^\op \to Set and S:π’žβ†’π’ŸS : \mathcal{C} \to \mathcal{D} be functors. Then their tensor product is (if it exists) the coend

TβŠ— π’žSβ‰”βˆ« cT(c)β‹…S(c)βˆˆπ’Ÿ. 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).


  • If T:π’ž opβ†’SetT : \mathscr{C}^{op} \to Set and S:π’žβ†’SetS : \mathscr{C} \to Set, then TβŠ— π’žST \otimes_{\mathscr{C}} S can be presented as set of formal symbols tβŠ— cst \otimes_c s (or tβŠ—st \otimes s for short) where c∈Ob(π’ž)c \in Ob(\mathscr{C}), s∈S(c)s \in S(c), and t∈T(c)t \in T(c) modulo the equivalence relation generated by the instances (tf)βŠ— cs∼tβŠ— d(fs)(t f) \otimes_c s \sim t \otimes_d (f s) when defined, where fβˆˆπ’ž(c,d)f \in \mathscr{C}(c,d), and the β€˜action’ of π’ž\mathscr{C} is the shorthand tf=T(f)(t)t f = T(f)(t) and fs=S(f)(s)f s = S(f)(s).

  • Recall that a ring AA can be considered as an Ab-enriched category and that a right and a left module gives rise to an additive functor A opβ†’AbA^\op \to Ab respectively Aβ†’AbA \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 AbAb, coincides with their usual tensor product.

  • Let XX be a simplicial set and st:Ξ”β†’Topst : \Delta \to Top be the functor which associates to [n][n] the topological standard nn-simplex. Then the geometric realization of XX can be expressed as the tensor product |X|=XβŠ— Ξ”st|X| = X \otimes_\Delta st.

  • Let π’ž\mathcal{C} be a small category and let F:π’žβ†’π’ŸF : \mathcal{C} \to \mathcal{D} be a functor into a cocomplete category π’Ÿ\mathcal{D}. Since the category PSh(π’ž)PSh(\mathcal{C}) of presheaves on π’ž\mathcal{C} is the free cocompletion of π’ž\mathcal{C}, the functor FF induces a functor F^:PSh(π’ž)β†’π’Ÿ\hat F : PSh(\mathcal{C}) \to \mathcal{D}. This functor can be explicitly described as F^(X)=XβŠ— π’žF\hat F(X) = X \otimes_\mathcal{C} F.

  • Let Y:π’žβ†’PSh(π’ž)Y : \mathcal{C} \to PSh(\mathcal{C}) denote the Yoneda embedding. Let F:π’ž opβ†’SetF : \mathcal{C}^\op \to Set be a presheaf on π’ž\mathcal{C}. Then FβŠ—Y=FF \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 nβŠ— AMβ‰…M nA^n \otimes_A M \cong M^n. Similarly,

    Hom π’ž(β‹…,c)βŠ— π’žS=S(c). \Hom_\mathcal{C}(\cdot, c) \otimes_\mathcal{C} S = S(c).

    This follows from some coend manipulations:

    Hom(Hom π’ž(β‹…,c)βŠ— π’žS,t) = ∫ cβ€²βˆˆπ’žHom(Hom π’ž(cβ€²,c)β‹…S(cβ€²),t) = ∫ cβ€²βˆˆπ’žHom(Hom π’ž(cβ€²,c),Hom π’Ÿ(S(cβ€²),t)) = Nat(Hom π’ž(β‹…,c),Hom π’Ÿ(S(β‹…),t)) = Hom π’Ÿ(S(c),t). \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 π’ž(β‹…,c)Hom_\mathcal{C}(\cdot, c) looks like a delta distribution concentrated at cc.

Intuition using gluing specifications

Recall that a presheaf F:π’ž opβ†’SetF : \mathcal{C}^\mathrm{op} \to \mathrm{Set} can be seen as β€œgluing specification”: If G:π’žβ†’π’ŸG : \mathcal{C} \to \mathcal{D} is some functor into a cocomplete category π’Ÿ\mathcal{D}, this gluing specification can be realized as colim s∈F(X)G(X)\operatorname{colim}_{s \in F(X)} G(X). This colimit can also be written as the coend

∫ Xβˆˆπ’žF(X)β‹…G(X), \int^{X \in \mathcal{C}} F(X) \cdot G(X),

that is as the tensor product FβŠ— π’žGF \otimes_\mathcal{C} G. The tensor product can therefore be pictured as the G(X)G(X)β€˜s, glued as specified by FF.

  • Since the Yoneda embedding YY includes an object into the category of formal gluing specifications, gluing the Y(X)Y(X)β€˜s as specified by FF simply yields in FF; thus FβŠ— π’žY=FF \otimes_\mathcal{C} Y = F.

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