category theory

## Definition

###### Definition

Let $F: C \rightleftarrows D : G$ be an adjunction with unit $\eta$ and counit $\varepsilon$. Then the following conditions are equivalent:

1. $F \eta$ is a natural isomorphism.

2. $\varepsilon F$ is a natural isomorphism.

3. $G \varepsilon F$ is a natural isomorphism — i.e. the monad induced by the adjunction is idempotent.

4. $G F \eta = \eta G F$.

5. $G F \eta G = \eta G F G$.

6. $G\varepsilon$ is a natural isomorphism.

7. $\eta G$ is a natural isomorphism.

8. $F \eta G$ is a natural isomorphism — i.e. the comonad induced by the adjunction is idempotent.

9. $F G \varepsilon = \varepsilon F G$.

10. $F G \varepsilon F = \varepsilon F G F$.

11. The adjunction can be factored as a composite

$C \stackrel{\overset{F_1}{\longrightarrow}}{\underset{G_1}{\hookleftarrow}} E \stackrel{\overset{F_2}{\hookrightarrow}}{\underset{G_2}{\longleftarrow}} D \,,$

where $F_2$ and $G_1$ are fully faithful, i.e. $F_1\dashv G_1$ is a reflection and $F_2 \dashv G_2$ is a coreflection.

When these conditions hold, the adjunction is said to be idempotent.

###### Remark

For an idempotent adjunction as in def. , the functors $F$ and $G$ restrict to an equivalence of categories between the full images of $F$ and of $G$ (which are, respectively, a coreflective subcategory of $D$ and a reflective subcategory of $C$, both equivalent to the $E$ in the last item above). In other words, for an idempotent adjunction, the category of fixed points has a particularly simple form.

###### Remark

If an idempotent adjunction is monadic, then (up to equivalence) it consists of the inclusion and reflection of a reflective subcategory (i.e. the algebras for an idempotent monad). Dually, if it is comonadic, it consists of the inclusion and coreflection of a coreflective subcategory. Thus, the primary interest in isolating the notion of idempotent adjunction is when considering adjunctions which are neither monadic nor comonadic.

## Examples

• Any adjunction between posets is idempotent. This is a central fact in the theory of Galois connections. Thus, in a sense, non-idempotent adjunctions are an important new idea arising by the “groupoidal” form of vertical categorification.

• More generally, an adjunction in which the full image of either functor is a poset must be idempotent. This follows from conditions 4, 5, 9, and 10 above. This fact arises when constructing generalized kernels.

• The “frame of opens” and “space of points” functors between topological spaces and locales form an idempotent adjunction. The resulting equivalence of categories is between sober spaces (which are reflective in Top) and spatial locales (which are coreflective in Loc).

• For any topological space $X$, there is an idempotent adjunction between the category $[O(X)^{\op}, Set]$ of presheaves on $X$ and the category $Top/X$ of spaces over $X$ (the right adjoint gives the presheaf of sections of a space over $X$). The resulting equivalence of categories is between sheaves in the modern sense of presheaves satisfying descent, and sheaves in the original sense as étalé spaces. See this blog post.

• The material-structural adjunction between material set theories and structural set theories is idempotent. The fixed categories consist of the models satisfying appropriate versions of the axiom of foundation or anti-foundation.

• The comma category construction forms part of an adjunction

$cocomma \colon Span(X,Y) \; \rightleftarrows \; Cospan(X,Y)\colon comma$

between spans and cospans of categories whose feet are given by categories $X$ and $Y$. A construction of cocomma categories can be found in the MathOverflow post in the references. This adjunction is idempotent and factors into the reflection into discrete two-sided fibrations in the category $Span(X,Y)$ and the coreflection from codiscrete cofibrations in $Cospan(X,Y)$.

• A categorification of the notion of idempotent adjunction is that of lax-idempotent 2-adjunction.

• In an adjoint triple, one of the adjunctions is idempotent if and only if the other one is; see there for the proof.

• An explicit description of cocomma categories? MathOverflow