Monoidal functor on arvau

Wikipedia / Monoidal functor

In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible.
The coherence maps of strong monoidal functors are invertible.
The coherence maps of strict monoidal functors are identity maps.

Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors.

Definition

be monoidal categories. A lax monoidal functor (which may also just be called a monoidal functor) consists of a functor together with a natural transformation

and a morphism

called the coherence maps or structure morphisms the diagrams

,
and

Variants

The dual of a monoidal functor is a comonoidal functor; it is a monoidal functor whose coherence maps are reversed. Comonoidal functors may also be called opmonoidal, colax monoidal, or oplax monoidal functors.
A strong monoidal functor are invertible.
A strict monoidal functor is a monoidal functor whose coherence maps are identities.
A braided monoidal functor is a monoidal functor between braided monoidal categories) such that the following diagram commutes for every pair of objects A, B :
A symmetric monoidal functor is a braided monoidal functor whose domain and codomain are symmetric monoidal categories.

Examples

from the category of abelian groups to the category of sets to 1.
is commutative).
is strongly monoidal.
An important example of a symmetric monoidal functor is the mathematical model of topological quantum field theory be the category of cobordisms of n-1,n-dimensional manifolds with tensor product given by disjoint union, and unit the empty manifold. A topological quantum field theory in dimension n
The homology.

Alternate notions

are closed monoidal categories (we drop the subscripts for readability), there is an alternative formulation

ψAB : F(A ⇒ B) → FA ⇒ FB

of φAB commonly used in functional programming. The relation between ψAB and φAB is illustrated in the following commutative diagrams:

Properties

is a monoid object.

Monoidal functors and adjunctions

is left adjoint, defined by

and

is strong, then the unit and counit of the adjunction are monoidal natural transformations, and the adjunction is said to be a monoidal adjunction; conversely, the left adjoint of a monoidal adjunction is always a strong monoidal functor.

Similarly, a right adjoint to a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is a strong monoidal functor.