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.
Lax monoidal functors between monoidal categories are in correspondence with morphisms between their underlying (representable) multicategories.
Strong monoidal functors between monoidal categories are in correspondence with morphisms between their underlying (representable) colored PROs.
Strict monoidal functors between monoidal categories are in correspondence with morphisms between their underlying colored PROs that preserve the distinguished isomorphisms and for all .
The 1-category of strict monoidal categories and strict monoidal functors is not equivalent to the 1-category of monoidal categories and strong monoidal functors.
The former has an initial object, whereas the latter does not.
The inclusion from the 1-category of strict monoidal categories and strong monoidal functors into the 1-category of monoidal categories and strong monoidal functors is not an equivalence.
As mentioned at monoidal category, not every skeletal monoidal category is monoidally equivalent to a strict skeletal monoidal category. Therefore the inclusion is not essentially surjective.
The inclusion from the 2-category of strict monoidal categories and strict monoidal functors into the 2-category of monoidal categories and strong monoidal functors is not an equivalence.
Not every strong monoidal functor between strict monoidal categories is equivalent to a strict one. See for example this MathOverflow question.
The inclusion of the the 2-category of strict monoidal categories and strong monoidal functors into the 2-category of monoidal categories and strong monoidal functors is an equivalence.
By the coherence theorem for monoidal categories, every monoidal category is strong monoidally equivalent to a strict one.
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 March 27, 2024 at 05:24:40. See the history of this page for a list of all contributions to it.