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.
and a morphism
called the coherence maps or structure morphisms, which are such that for every three objects , and of the diagrams
commute in the category . Above, the various natural transformations denoted using are parts of the monoidal structure on and .
- 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 is a monoidal functor whose coherence maps 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 in :
- A symmetric monoidal functor is a braided monoidal functor whose domain and codomain are symmetric monoidal categories.
- The underlying functor from the category of abelian groups to the category of sets. In this case, the map sends (a, b) to ; the map sends * to 1.
- An important example of a symmetric monoidal functor is the mathematical model of topological quantum field theory, which has been recently developed. Let 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 is a symmetric monoidal functor
Monoidal functors and adjunctions
Suppose that a functor is left adjoint to a monoidal . Then has a comonoidal structure induced by , defined by
If the induced structure on 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.
- Kelly, G. Max (1974), "Doctrinal adjunction", Lecture Notes in Mathematics, 420, 257–280