nLab external tensor product

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

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

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

Definition

Given $X_1, X_2 \in \mathbf{H}$ the external tensor product over these is the functor

$$\boxtimes \;\colon\; Mod(X_1)\times Mod(X_2) \longrightarrow Mod(X_1 \times X_2)$$

given on $A_1 \in Mod(X_1)$ with $A_2 \in Mod(X_2)$ by

$$A_1 \boxtimes A_2 \coloneqq (p_1^\ast A_1) \otimes_{X_1 \times X_2} (p_2^\ast A_2) \in Mod(X_1 \times X_2) \,,$$

where $p_1, p_2$ denote the projection maps out of the Cartesian product $X_1 \times X_2 \in \mathbf{H}$.

Remark

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

Properties

Relation to fiberwise tensor product

Proposition

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

$$A_1 \otimes_X A_2 \simeq \Delta_X^\ast (A_1 \boxtimes A_2)$$

for $A_1, A_2 \in Mod(X)$, where $\Delta \colon X \longrightarrow X \times X$ is the diagonal in $\mathbf{H}$ on $X$.

Generation of $Mod(X_1 \times X_2)$ from external tensor products

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

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

Examples

• external tensor product of vector bundles

Related concepts

• direct product group representations

References

Textbook accounts:

• Werner Greub, Stephen Halperin, Ray Vanstone, p. 84 of: Connections, Curvature, and Cohomology Volume 1: De Rham Cohomology of Manifolds and Vector Bundles, Academic Press (1973) [ISBN:978-0-12-302701-6]

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:

• Alexei Bondal, Michel Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 1 (2003) 1-36 [arXiv:math/0204218]

• 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)