DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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Enriched functor

Definition

Let $A$ and $B$ be $V$ -categories (Enriched category over a monoidal base) over the same monoidal base $V$ .

A $V$ -functor $T : A \to B$ consists of

a function $T : Ob (A) \to Ob (B)$ ;
for every pair $A, B \in Ob (A)$ , a morphism in $V$ $T_{A, B} : A (A, B) ⟶ B (T A, T B);$

such that for every triple $A, B, C$ the composition square commutes:

$T_{A, C} \circ M_{A, B, C}^{A} = M_{T A, T B, T C}^{B} \circ (T_{B, C} \otimes T_{A, B}),$

and for every object $A$ the identity morphisms agree:

$T_{A, A} \circ j_{A}^{A} = j_{T A}^{B} .$

The $V$ -functor is fully faithful when every structure morphism $T_{A, B}$ is an isomorphism in $V$ .

Depends on

Enriched category over a monoidal base

Used by

Enriched adjunction Definition
Enriched natural transformation Definition
Enriched weighted limit Definition
Representable enriched functor Definition
Change of base extends to enriched functors and natural transformations as a 2-functor Theorem
Constant enriched functors need not exist Theorem
Set-object enriched categories, enriched functors, and enriched natural transformations form a strict 2-category Theorem
The free enriched category is left 2-adjoint to the underlying-category construction Theorem
The underlying-category construction is a 2-functor Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. M. Kelly, Basic Concepts of Enriched Category Theory, equations (1.5) and (1.6) (standard reference, not scraped)
Geoffrey Cruttwell, Normed Spaces and the Change of Base for Enriched Categories, Section 2.2.1 (standard reference, not scraped)