tensor product of functors



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monoidal categories

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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 08:17:37. See the history of this page for a list of all contributions to it.