idempotent adjunction

Idempotent adjunctions

Idempotent adjunctions



(equivalent conditions for idempotency)
Let F:CD:GF \,\colon\, C \rightleftarrows D \,\colon\, G be an adjunction with unit η\eta and counit ε\varepsilon. Then the following conditions on their whiskering 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 an idempotent monad.

  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 an idempotent comonad.

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

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

  11. The adjunction factors through its fixed points as

    CG 1F 1EG 2F 2D, C \underoverset {\underset{G_1}{\hookleftarrow}} {\overset{F_1}{\longrightarrow}} {\;\;\; \bot \;\;\;} E \underoverset {\underset{G_2}{\longleftarrow}} {\overset{F_2}{\hookrightarrow}} {\;\;\; \bot \;\;\;} 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 \dashv G_2 is a coreflection.


(idempotent adjunction)
When the equivalent conditions from Prop. hold, the adjunction is said to be idempotent.

An original reference for the equivalence of all but the last of these conditions is MacDonald & Stone 1982, Prop. 2.8; a textbook account is in Grandis 2021, Thm. 3.8.2. The full statement including the (co)reflective factorization through the fixed points is made explicit in the proof of Grandis 2021, Thm. 3.8.8, which also makes explicit that:


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 in Prop. . This fact arises when constructing generalized kernels.


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.


((co)comma construction)
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 (Shulman 2016). 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).

Involving topological spaces


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 slice category Top /XTop_{/X} of TopologicalSpaces 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.

Between topological and diffeological spaces


(adjunction between topological spaces and diffeological spaces)

There is a pair of adjoint functors

(1)TopSpAAAACdfflgDtplgDifflgSp TopSp \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DifflgSp

between the categories of TopologicalSpaces and of DiffeologicalSpaces, where


  1. the fixed points of this adjunction XX \inTopologicalSpaces (those for which the counit is an isomorphism, hence here: a homeomorphism) are precisely the Delta-generated topological spaces (i.e. D-topological spaces):

    XisΔ-generatedDtplg(Cdfflg(X))ϵ XX X \;\,\text{is}\;\Delta\text{-generated} \;\;\;\;\; \Leftrightarrow \;\;\;\;\; Dtplg(Cdfflg(X)) \underoverset{\simeq}{\;\;\epsilon_X\;\;}{\longrightarrow} X
  2. this is an idempotent adjunction, which exhibits Δ\Delta-generated/D-topological spaces as a reflective subcategory inside diffeological spaces and a coreflective subcategory inside all topological spaces:

(2)TopologicalSpacesAAAACdfflgDTopologicalSpacesAAAADtplgDiffeologicalSpaces TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DTopologicalSpaces \underoverset { \underset{ }{\hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces

Finally, these adjunctions are a sequence of Quillen equivalences with respect to the:

classical model structure on topological spacesmodel structure on D-topological spacesmodel structure on diffeological spaces

Caution: There was a gap in the original proof that DTopologicalSpaces QuillenDiffeologicalSpacesDTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces. The gap is claimed to be filled now, see the commented references here.

Essentially these adjunctions and their properties are observed in Shimakawa, Yoshida & Haraguchi 2010, Prop. 3.1, Prop. 3.2, Lem. 3.3, see also Christensen, Sinnamon & Wu 2014, Sec. 3.2. The model structures and Quillen equivalences are due to Haraguchi 13, Thm. 3.3 (on the left) and Haraguchi-Shimakawa 13, Sec. 7 (on the right).


We spell out the existence of the idempotent adjunction (2):

First, to see we have an adjunction DtplgCdfflgDtplg \dashv Cdfflg, we check the hom-isomorphism (here).

Let XDiffeologicalSpacesX \in DiffeologicalSpaces and YTopologicalSpacesY \in TopologicalSpaces. Write () s(-)_s for the underlying sets. Then a morphism, hence a continuous function of the form

f:Dtplg(X)Y, f \;\colon\; Dtplg(X) \longrightarrow Y \,,

is a function f s:X sY sf_s \colon X_s \to Y_s of the underlying sets such that for every open subset AY sA \subset Y_s and every smooth function of the form ϕ: nX\phi \colon \mathbb{R}^n \to X the preimage (f sϕ s) 1(A) n(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n is open. But this means equivalently that for every such ϕ\phi, fϕf \circ \phi is continuous. This, in turn, means equivalently that the same underlying function f sf_s constitutes a smooth function f˜:XCdfflg(Y)\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y).

In summary, we thus have a bijection of hom-sets

Hom(Dtplg(X),Y) Hom(X,Cdfflg(Y)) f s (f˜) s=f s \array{ Hom( Dtplg(X), Y ) &\simeq& Hom(X, Cdfflg(Y)) \\ f_s &\mapsto& (\widetilde f)_s = f_s }

given simply as the identity on the underlying functions of underlying sets. This makes it immediate that this hom-isomorphism is natural in XX and YY and this establishes the adjunction.

Next, to see that the D-topological spaces are the fixed points of this adjunction, we apply the above natural bijection on hom-sets to the case

Hom(Dtplg(Cdfflg(Z)),Y) Hom(Cdfflg(Z),Cdfflg(Y)) (ϵ Z) s (id) s \array{ Hom( Dtplg(Cdfflg(Z)), Y ) &\simeq& Hom(Cdfflg(Z), Cdfflg(Y)) \\ (\epsilon_Z)_s &\mapsto& (\mathrm{id})_s }

to find that the counit of the adjunction

Dtplg(Cdfflg(X))ϵ XX Dtplg(Cdfflg(X)) \overset{\epsilon_X}{\longrightarrow} X

is given by the identity function on the underlying sets (ϵ X) s=id (X s)(\epsilon_X)_s = id_{(X_s)}.

Therefore η X\eta_X is an isomorphism, namely a homeomorphism, precisely if the open subsets of X sX_s with respect to the topology on XX are precisely those with respect to the topology on Dtplg(Cdfflg(X))Dtplg(Cdfflg(X)), which means equivalently that the open subsets of XX coincide with those whose pre-images under all continuous functions ϕ: nX\phi \colon \mathbb{R}^n \to X are open. This means equivalently that XX is a D-topological space.

Finally, to see that we have an idempotent adjunction, it is sufficient to check (by this Prop.) that the comonad

DtplgCdfflg:TopologicalSpacesTopologicalSpaces Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces

is an idempotent comonad, hence that

DtplgCdfflgDtplgηCdfflgDtplgCdfflgDtplgCdfflg Dtplg \circ Cdfflg \overset{ Dtplg \cdot \eta \cdot Cdfflg }{\longrightarrow} Dtplg \circ Cdfflg \circ Dtplg \circ Cdfflg

is a natural isomorphism. But, as before for the adjunction counit ϵ\epsilon, we have that also the adjunction unit η\eta is the identity function on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case.


The characterization of idempotent adjunctions is proven/due to:

Textbook accounts:

See also:

The example of D-topological spaces inside topological/diffeological spaces is due to:

The example of the (co)comma adjunction is also mentioned in:

Last revised on October 5, 2021 at 03:37:38. See the history of this page for a list of all contributions to it.