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 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. Let ( C , ⊗ , I C ) {displaystyle ({mathcal {C}},otimes ,I_{mathcal {C}})} and ( D , ∙ , I D ) {displaystyle ({mathcal {D}},ullet ,I_{mathcal {D}})} be monoidal categories. A monoidal functor from C {displaystyle {mathcal {C}}} to D {displaystyle {mathcal {D}}} consists of a functor F : C → D {displaystyle F:{mathcal {C}} o {mathcal {D}}} together with a natural transformation between C × C → D {displaystyle {mathcal {C}} imes {mathcal {C}} o {mathcal {D}}} functors and a morphism called the coherence maps or structure morphisms, which are such that for every three objects A {displaystyle A} , B {displaystyle B} and C {displaystyle C} of C {displaystyle {mathcal {C}}} the diagrams commute in the category D {displaystyle {mathcal {D}}} . Above, the various natural transformations denoted using α , ρ , λ {displaystyle alpha , ho ,lambda } are parts of the monoidal structure on C {displaystyle {mathcal {C}}} and D {displaystyle {mathcal {D}}} . Suppose that a functor F : C → D {displaystyle F:{mathcal {C}} o {mathcal {D}}} is left adjoint to a monoidal ( G , n ) : ( D , ∙ , I D ) → ( C , ⊗ , I C ) {displaystyle (G,n):({mathcal {D}},ullet ,I_{mathcal {D}}) o ({mathcal {C}},otimes ,I_{mathcal {C}})} . Then F {displaystyle F} has a comonoidal structure ( F , m ) {displaystyle (F,m)} induced by ( G , n ) {displaystyle (G,n)} , defined by