nLab category enriched in a bicategory



Category theory

Enriched category theory

2-Category theory



The notion of category enriched in a bicategory is the many-object-generalization of the notion of an enriched category enriched in a monoidal category via regarding a monoidal category as a bicategory with a single object.

Originally Bénabou called these polyads.


Let BB be a bicategory, and write \otimes for horizontal (1-cell) composition (written in Leibniz order). A category enriched in the bicategory BB consists of a set XX together with

  • A function p:XB 0p: X \to B_0,
  • A function hom:X×XB 1\hom: X \times X \to B_1, satisfying the typing constraint hom(x,y):p(x)p(y)\hom(x, y): p(x) \to p(y),
  • A function :X×X×XB 2\circ: X \times X \times X \to B_2, satisfying the constraint x,y,z:hom(y,z)hom(x,y)hom(x,z)\circ_{x, y, z}: \hom(y, z) \otimes \hom(x, y) \to \hom(x, z),
  • A function j:XB 2j: X \to B_2, satisfying the constraint j x:1 p(x)hom(x,x)j_x: 1_{p(x)} \to \hom(x, x),

such that the associativity and unitality diagrams, as written above, commute. Viewing a monoidal category MM as a 1-object bicategory ΣM\Sigma M, the notion of enrichment in MM coincides with the notion of enrichment in the bicategory ΣM\Sigma M.

Equivalently this is simply a lax functor from the codiscrete category on XX into BB. In particular if XX is the singleton set then this is the same as a monad.

If XX, YY are sets which come equipped with enrichments in BB, then a BB-functor consists of a function f:XYf: X \to Y such that p Yf=p Xp_Y \circ f = p_X, together with a function f 1:X×XB 2f_1: X \times X \to B_2, satisfying the constraint f 1(x,y):hom X(x,y)hom Y(f(x),f(y))f_1(x, y): \hom_X(x, y) \to \hom_Y(f(x), f(y)), and satisfying equations expressing coherence with the composition and unit data \circ, jj of XX and YY. (Diagram to be inserted, perhaps.)


Discussion of Kleisli objects (collages) for monads generalized to categories enriched in bicategories is in section 15.9 of

Last revised on February 12, 2019 at 04:03:57. See the history of this page for a list of all contributions to it.