nLab envelope of an adjunction

Definition

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

$$(c \in C, d \in D, f : L c \to d, g \colon c \to R d)$$

such that $f$ and $g$ are each other’s mate, and whose morphisms are pairs

$$(p \colon c \to c', q \colon d \to d')$$

such that $(R q) \circ g = g' \circ p$ or equivalently $q \circ f = f' \circ (L p)$.

Properties

• 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.

• Denoting by $Inv(F \dashv G)$ the fixed point of $F \dashv G$, the following square of fully faithful functors forms a 2-pullback in Cat.

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.)

• An adjunction that furthermore factors through $Inv(F \dashv G)$ is precisely an idempotent adjunction.

• 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 $(P, Q, \alpha, \beta)$ have $\alpha$ and $\beta$ invertible)).

(See Remark 12.2 of Avery and Leinster.)

Examples

• 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 $A$.

Related concepts

• fixed point of an adjunction

References

• 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)