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 monoidal category-structure is strict if its associator and left/right unitors are identity natural transformations. By the coherence theorem for monoidal categories, every monoidal category is monoidally equivalent to a strict one.
Explicitly, this means that:
A strict monoidal category is a category equipped with an object and a bifunctor such that for every objects and morphisms , we have:
The skeletal version of the symmetric groupoid is the free strict symmetric monoidal category on a single object.
Saunders MacLane, ยงXI.3 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
Peter Schauenburg, Turning Monoidal Categories into Strict Ones, New York Journal of Mathematics 7 (2001) 257-265 [nyjm:j/2001/7-16, eudml:121925]
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