category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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
over a cartesian monoidal category .
Given the external tensor product over these is the functor
given on with by
where denote the projection maps out of the Cartesian product .
The external tensor product constitutes a tensor product on the total category of the given Grothendieck fibration ; and with respect to this it is a monoidal fibration.
The fiberwise (“internal”) tensor product over is recovered form the external one via a natural equivalence
for , where is the diagonal in on .
Under suitable conditions on compact generation of then one may deduce that is generated under external product from and .
(Bondal-vdBerg 03, BFN 08, proof of prop. 3.24)
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:
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)
Last revised on July 15, 2022 at 17:22:45. See the history of this page for a list of all contributions to it.