nLab enriched functor




Enriched functors are used in place of functors in enriched category theory: like functors they send objects to objects, but instead of mapping hom-sets to hom-sets they assign morphisms in the enriching category between hom-objects, while being compatible with composition and units in the obvious way.


Given two categories C,DC, D enriched in a monoidal category VV, an enriched functor F:CDF: C \to D consists of

  • A function F 0:C 0D 0F_0: C_0 \to D_0 between the underlying collections of objects;

  • A (C 0×C 0)(C_0 \times C_0)-indexed collection of morphisms of VV,

    F x,y:C(x,y)D(F 0x,F 0y)F_{x, y}: C(x, y) \to D(F_0x, F_0y)

    where C(x,y)C(x, y) denotes the hom-object in VV,

  • such that the following diagrams commute for all a,b,cC 0a, b, c \in C_0:

    • respect for composition:

      C(b,c)C(a,b) a,b,c C(a,c) F b,cF a,b F a,c D(F 0(b),F 0(c))D(F 0(a),F 0(b)) F 0(a),F 0(b),F 0(c) D(F 0(a),F 0(c)) \array{ C(b,c) \otimes C(a,b) &\stackrel{\circ_{a,b,c}}{\to}& C(a,c) \\ \downarrow^{F_{b,c} \otimes F_{a,b}} && \downarrow^{F_{a,c}} \\ D(F_0(b), F_0(c)) \otimes D(F_0(a), F_0(b)) &\stackrel{\circ_{F_0(a),F_0(b), F_0(c)}}{\to}& D(F_0(a), F_0(c)) }
    • respect for units:

      I j a j F 0(a) C(a,a) F a,a D(F 0(a),F 0(a)) \array{ && I \\ & {}^{j_a}\swarrow && \searrow^{j_{F_0(a)}} \\ C(a,a) &&\stackrel{F_{a,a}}{\to}&& D(F_0(a), F_0(a)) }

      VV‘s unit object II and the family of identity elements J iJ_i from it are defined in enriched category.



Relation to strong functors

For closed monoidal categories VV there is a close relation between VV-enriched functors and VV-strong functors.

For the moment see at enriched monad – relation to strong monads for more.


  • linear functor

  • smooth functor

  • topologically enriched functor

  • Consider the enriching category with [0,][0,\infty] the set of objects, maps given by \geq, and tensor by ++. Then VV-categories are Lawvere metric spaces, and VV-functors are distance-decreasing maps.

  • Categories enriched in the poset of truth values with conjunction as monoidal product are posets, and the corresponding enriched functors are precisely order-preserving maps.


For more references see at enriched category theory.

Last revised on August 23, 2023 at 10:06:25. See the history of this page for a list of all contributions to it.