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
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.
A cocomplete posetal symmetric monoidal closed category is precisely a commutative quantale?.
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 March 27, 2025 at 08:47:51. See the history of this page for a list of all contributions to it.