Let $L \dashv R$ be a pair of adjoint functors (an adjunction in Cat).
The envelope of the adjunction, denoted $Env(L \dashv R)$, is the category whose objects are quadruples
such that $f$ and $g$ are each other’s mate, and whose morphisms are pairs
such that $(R q) \circ f = f' \circ p$ or equivalently $q \circ g = g' \circ (L p)$.
A functor $L \colon C \to D$ is left adjoint to a functor $R \colon D \to C$ if and only if there is an isomorphism of comma categories $L \downarrow D \cong C \downarrow R$ and this isomorphism commutes with the forgetful functors to the product category $C \times D$ (see there). In this case, the envelope $Env(L \dashv R)$ is also isomorphic to these comma categories over the evident forgetful functor to $C \times D$.
Every adjunction factors through its envelope via a reflective subcategory and a coreflective subcategory.
See Lemma 4.1 of Pavlovic and Hughes.
In particular, $Inv(F \dashv G)$ is the full subcategory of $Env(F \dashv G)$ whose objects $(c, d, f, g)$ have $f$ and $g$ isomorphisms. (See Lemma 12.1 of Avery and Leinster.)
(See Remark 12.2 of Avery and Leinster.)
The envelope of the Isbell duality adjunction associated to a category $A$ is the Isbell envelope of $A$ (called the category of gaps in $A$ by Pavlovic and Hughes).
The envelope of the nuclear adjunction associated to the Isbell duality adjunction for a small category $A$ is what Pavlovic and Hughes call the category of intervals in $A$.
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.