nLab envelope of an adjunction




Let LRL \dashv R be a pair of adjoint functors (an adjunction in Cat).

The envelope of the adjunction, denoted Env(LR)Env(L \dashv R), is the category whose objects are quadruples

(cC,dD,f:Lcd,g:cRd) (c \in C, d \in D, f : L c \to d, g \colon c \to R d)

such that ff and gg are each other’s mate, and whose morphisms are pairs

(p:cc,q:dd) (p \colon c \to c', q \colon d \to d')

such that (Rq)f=fp(R q) \circ f = f' \circ p or equivalently qg=g(Lp)q \circ g = g' \circ (L p).


See Lemma 4.1 of Pavlovic and Hughes.

In particular, Inv(FG)Inv(F \dashv G) is the full subcategory of Env(FG)Env(F \dashv G) whose objects (c,d,f,g)(c, d, f, g) have ff and gg isomorphisms. (See Lemma 12.1 of Avery and Leinster.)

  • The envelope construction (respectively the fixed point construction) can be seen as 2-adjunctions between Cat, and the 2-category of adjunctions (respectively the wide sub-2-category of adjunctions whose morphisms (P,Q,α,β)(P, Q, \alpha, \beta) have α\alpha and β\beta invertible)).

(See Remark 12.2 of Avery and Leinster.)



  • Tom Avery, Tom Leinster. Isbell conjugacy and the reflexive completion. Theory and Applications of Categories, 36 12 (2021) 306-347 [tac:36-12, pdf]

  • Dusko Pavlovic, and Dominic JD Hughes. Tight limits and completions from Dedekind-MacNeille to Lambek-Isbell. (arXiv)

Last revised on March 31, 2023 at 07:19:07. See the history of this page for a list of all contributions to it.