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 functor is a functor between monoidal categories that preserves the monoidal structure: a homomorphism of monoidal categories.
Let and be two monoidal categories. A lax monoidal functor between them is a functor:
together with coherence maps:
a morphism
for all
satisfying the following conditions:
(associativity) For all objects the following diagram commutes
where and denote the associators of the monoidal categories;
(unitality) For all the following diagrams commute
and
where , , , denote the left and right unitors of the two monoidal categories, respectively.
If and all are isomorphisms, then is called a strong monoidal functor. (Note that βstrongβ is also sometimes applied to βmonoidal functorβ to indicate possession of a tensorial strength.) If they are even identity morphisms, then is called a strict monoidal functor.
In the literature often the term βmonoidal functorβ refers by default to what in def. is called a strong monoidal functor. With that convention then what def. calls a lax monoidal functor is called a weak monoidal functor.
Lax monoidal functors are the lax morphisms for an appropriate 2-monad.
An oplax monoidal functor (with various alternative names including comonoidal), is a monoidal functor from the opposite categories to .
A monoidal transformation between monoidal functors is a natural transformation that respects the extra structure in an obvious way.
(Lax monoidal functors send monoids to monoids)
If is a lax monoidal functor and
is a monoid object in , then the object is naturally equipped with the structure of a monoid in by setting
and
This construction defines a functor
between the categories of monoids in and , respectively.
More generally, lax functors send enriched categories to enriched categories, an operation known as change of enriching category. See there for more details.
Similarly:
(oplax monoidal functors sends comonoids to comonoids)
For a monoidal category write for the corresponding delooping 2-category.
Lax monoidal functor correspond to lax 2-functor
If is strong monoidal then this is an ordinary 2-functor. If it is strict monoidal, then this is a strict 2-functor.
Just like monoidal categories, monoidal functors have a string diagram calculus; see these slides for some examples.
monoidal functor, strong monoidal functor
lax monoidal functor
Samuel Eilenberg, G. Max Kelly, p. 473 in: Closed Categories, in: S. Eilenberg, D. K. Harrison, S. MacLane, H. RΓΆhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]
Saunders MacLane, Β§XI.2 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Β§2.4 in: Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) [ISBN:978-1-4704-3441-0, pdf]
(discussed what we call strong monoidal functors)
Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras. (pdf)
Exposition of basics of monoidal categories and categorical algebra:
Last revised on September 2, 2023 at 12:15:47. See the history of this page for a list of all contributions to it.