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

Definition

Let and be monoidal categories. A lax monoidal functor from \mathcal C to \mathcal D (which may also just be called a monoidal functor) consists of a functor together with a natural transformation between functors and a morphism called the coherence maps or structure morphisms, which are such that for every three objects A, B and C of \mathcal C the diagrams commute in the category \mathcal D. Above, the various natural transformations denoted using are parts of the monoidal structure on \mathcal C and \mathcal D.

Variants

Examples

Alternate notions

If and are closed monoidal categories with internal hom-functors (we drop the subscripts for readability), there is an alternative formulation of φAB commonly used in functional programming. The relation between ψAB and φAB is illustrated in the following commutative diagrams:

Properties

Monoidal functors and adjunctions

Suppose that a functor is left adjoint to a monoidal. Then F has a comonoidal structure (F,m) induced by (G,n), defined by and If the induced structure on F 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.

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