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 $C, D$ 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 $V$,
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:
respect for units:
$V$‘s unit object $I$ and the family of identity elements $J_i$ from it are defined in enriched category.
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.
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.
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.
Last revised on August 23, 2023 at 10:06:25. See the history of this page for a list of all contributions to it.