nLab
idempotent adjunction

Definition

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

1. $F\eta$ is a natural isomorphism.
2. $\epsilon F$ is a natural isomorphism.
3. $G\epsilon F$ is a natural isomorphism — i.e. the monad induced by the adjunction is idempotent.
4. $GF\eta =\eta GF$.
5. $GF\eta G=\eta GFG$.
6. $G\epsilon$ 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. $FG\epsilon =\epsilon FG$.
10. $FG\epsilon F=\epsilon FGF$.
11. The adjunction can be factored as a composite $C\underset{G_1}{\overset{F_1}{\rightleftarrows }}E\underset{G_2}{\overset{F_2}{\rightleftarrows }}D$ where $F_2$ and $G_1$ are fully faithful, i.e. $F_1⊣G_1$ is a reflection and $F_2⊢G_2$ is a coreflection.

When these conditions hold, the adjunction is said to be idempotent. It then follows that $F$ and $G$ restrict to an equivalence of categories between the full images of $F$ and of $G$ (which are, respectively, a reflective subcategory of $D$ and a coreflective subcategory of $C$).

Note that 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 $[O(X{)}^{op},\mathrm{Set}]$ of presheaves on $X$ and the category $\mathrm{Top}/X$ 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.

References

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