nLab external tensor product

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

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).

Definition

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}.

Definition

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}.

Remark

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.

Properties

Relation to fiberwise tensor product

Proposition

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)

Examples

References

Textbook accounts:

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

Discussion in the context of categories of quasicoherent sheaves in (derived) algebraic geometry:

Last revised on July 15, 2022 at 17:22:45. See the history of this page for a list of all contributions to it.