With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
A monoidal functor is a functor between monoidal categories that preserves the monoidal structure: a homomorphism of monoidal categories.
A functor between strict monoidal categories and is called lax monoidal if it is equipped with a morphism
and a natural transformation
satisfying the following conditions
associativity For all we have a commuting diagram
unitality For all we have
Where and are respectively the left and right unitors. We do not explicitly mention the category in which they live, as it is clear from where the subscripted object lives.
Lax monoidal functors are the lax morphism for an appropriate 2-monad.
If and are isomorphisms then is called a strong monoidal functor. If they are even identities it is called a strict monoidal functor.
In contrast to this, a strong monoidal functor may also be called a weak monoidal functor. Sometimes the plain term “monoidal functor” is used to mean a strong monoidal functor, in which case the general situation is called a lax monoidal functor.
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.
If the monoidal categories are not strict one obtains correspondingly more coherence diagrams. One way to summarize these is to note that a monoidal category is equivalently its pointed delooping 2-category/bicategory (with a single object and as its hom-object), then a monoidal functor is equivalently a 2-functor/pseudofunctor . Using this one can infer the coherence diagrams as special cases from those discussed at pseudofunctor.
Lax monoidal functors send monoids to monoids.
If is a lax monoidal functor and
is a monoid in , then the object is naturally equipped with the structure of a monoid in by setting
This construction defines a functor
between the categories of monoids.
Similarly, an oplax monoidal functor sends comonoids to comonoids.
Just like monoidal categories, monoidal functors have a string diagram calculus; see these slides for some examples.