idempotent adjunction

Idempotent adjunctions

Idempotent adjunctions



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

  1. FηF \eta is a natural isomorphism.

  2. εF\varepsilon F is a natural isomorphism.

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

  4. GFη=ηGFG F \eta = \eta G F.

  5. GFηG=ηGFGG F \eta G = \eta G F G.

  6. GεG\varepsilon is a natural isomorphism.

  7. ηG\eta G is a natural isomorphism.

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

  9. FGε=εFGF G \varepsilon = \varepsilon F G.

  10. FGεF=εFGFF G \varepsilon F = \varepsilon F G F.

  11. The adjunction can be factored as a composite

    CG 1F 1EG 2F 2D, C \stackrel{\overset{F_1}{\longrightarrow}}{\underset{G_1}{\hookleftarrow}} E \stackrel{\overset{F_2}{\hookrightarrow}}{\underset{G_2}{\longleftarrow}} D \,,

    where F 2F_2 and G 1G_1 are fully faithful, i.e. F 1G 1F_1\dashv G_1 is a reflection and F 2G 2F_2 \vdash G_2 is a coreflection.

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


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


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.


  • 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 XX, there is an idempotent adjunction between the category [O(X) op,Set][O(X)^{\op}, Set] of presheaves on XX and the category Top/XTop/X of spaces over XX (the right adjoint gives the presheaf of sections of a space over XX). 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:Span(X,Y)Cospan(X,Y):comma cocomma \colon Span(X,Y) \; \rightleftarrows \; Cospan(X,Y)\colon comma

    between spans and cospans of categories whose feet are given by categories XX and YY. 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)Span(X,Y) and the coreflection from codiscrete cofibrations in Cospan(X,Y)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

Last revised on January 18, 2020 at 23:40:11. See the history of this page for a list of all contributions to it.