external tensor product



Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



The concept of external tensor product is a variant of that of tensor product in a monoidal category when the latter is generalized to indexed monoidal categories (dependent linear type theory).


Consider an indexed monoidal category given by a Cartesian fibration

Mod() H \array{ Mod(-) \\ \downarrow \\ \mathbf{H} }

over a cartesian monoidal category H\mathbf{H}.


Given X 1,X 2HX_1, X_2 \in \mathbf{H} the external tensor product over these is the functor

:Mod(X 1)×Mod(X 2)Mod(X 1×X 2) \boxtimes \;\colon\; Mod(X_1)\times Mod(X_2) \longrightarrow Mod(X_1 \times X_2)

given on A 1Mod(X 1)A_1 \in Mod(X_1) with A 2Mod(X 2)A_2 \in Mod(X_2) by

A 1A 2(p 1 *A 1) X 1×X 2(p 2 *A 2)Mod(X 1×X 2), A_1 \boxtimes A_2 \coloneqq (p_1^\ast A_1) \otimes_{X_1 \times X_2} (p_2^\ast A_2) \in Mod(X_1 \times X_2) \,,

where p 1,p 2p_1, p_2 denote the projection maps out of the Cartesian product X 1×X 2HX_1 \times X_2 \in \mathbf{H}.


The external tensor product constitutes a tensor product on the total category ModMod of the given Grothendieck fibration Mod()HMod(-)\to \mathbf{H}; and with respect to this it is a monoidal fibration.


Relation to fiberwise tensor product


The fiberwise (“internal”) tensor product over XHX\in \mathbf{H} is recovered form the external one via a natural equivalence

A 1 XA 2Δ X *(A 1A 2) A_1 \otimes_X A_2 \simeq \Delta_X^\ast (A_1 \boxtimes A_2)

for A 1,A 2Mod(X)A_1, A_2 \in Mod(X), where Δ:XX×X\Delta \colon X \longrightarrow X \times X is the diagonal in H\mathbf{H} on XX.

Generation of Mod(X 1×X 2)Mod(X_1 \times X_2) from external tensor products

Under suitable conditions on compact generation of Mod()Mod(-) then one may deduce that Mod(X 1×X 2)Mod(X_1 \times X_2) is generated under external product from Mod(X 1)Mod(X_1) and Mod(X 2)Mod(X_2).

(Bondal-vdBerg 03, BFN 08, proof of prop. 3.24)



For general abstract literature dealing with the external tensor products see the references at indexed monoidal category and at dependent linear type theory.

For discussion in the context of categories of quasicoherent sheaves in (derived) see for instance

  • Alexei Bondal and M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258.

  • David Ben-Zvi, John Francis, David Nadler, Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 (arXiv:0805.0157)

Last revised on September 12, 2018 at 14:41:15. See the history of this page for a list of all contributions to it.