nLab enriched functor

Idea

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.

Definition

Given two categories $C, D$ enriched in a monoidal category $V$, an enriched functor $F: C \to D$ consists of

• A function $F_0: C_0 \to D_0$ between the underlying collections of objects;

• A $(C_0 \times C_0)$-indexed collection of morphisms of $V$,

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

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

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

  • respect for composition:

    $$\array{ C(b,c) \otimes C(a,b) &\stackrel{\circ_{a,b,c}}{\to}& C(a,c) \\ \downarrow^{F_{b,c} \otimes F_{a,b}} && \downarrow^{F_{a,c}} \\ D(F_0(b), F_0(c)) \otimes D(F_0(a), F_0(b)) &\stackrel{\circ_{F_0(a),F_0(b), F_0(c)}}{\to}& D(F_0(a), F_0(c)) }$$

  • respect for units:

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

    $V$‘s unit object $I$ and the family of identity elements $J_i$ from it are defined in enriched category.

Properties

General

• Analogous to the functor category for ordinary functors between ordinary categories, enriched functors between two enriched categories form an enriched functor category.

Relation to strong functors

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

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

Examples

• linear functor

• smooth functor

• topologically enriched functor

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

Related concepts

• enriched natural transformation

• enriched functor category

• enriched (∞,1)-functor

References

• Max Kelly, Basic Concepts of Enriched Category Theory (web)

• Bill Lawvere (1973). Metric spaces, generalized logic and closed categories. Reprinted in TAC, 1986. Web.

• Emily Riehl, §3.5 in Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

For more references see at enriched category theory.