With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A symmetric monoidal closed category is
which as such is also:
For monads on symmetric monoidal closed categories there is a close relation between structures of monoidal monads and of strong monads.
For the moment see at enriched monad – Relation to strong and monoidal monads for more on this.
Any cartesian closed category is symmetric monoidal closed.
A Bénabou cosmos (a good base for enrichment in enriched category theory) is defined to be a bicomplete symmetric monoidal closed category.
Samuel Eilenberg, G. Max Kelly, §III and specifically §III.6 in: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]
Max Kelly, §1.4, 1.5 in: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]
Francis Borceux, §6.1 of: Categories and Structures, Vol 2 of: Handbook of Categorical Algebra, Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]
Proof that the funny tensor product of categories is the only other symmetric closed monoidal structure on Cat besides the cartesian monoidal structure:
Last revised on November 15, 2023 at 16:32:28. See the history of this page for a list of all contributions to it.