category theory

## Definition

###### Proposition

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

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 an idempotent comonad.

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

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

11. The adjunction factors through its fixed points as

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

where $F_2$ 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.

###### Definition

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:

###### 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

### General

###### Example

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.

###### Example

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.

###### Example

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.

###### Example

((co)comma construction)
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 $Y$ (Shulman 2016). 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 $Cospan(X,Y)$.

### Involving topological spaces

###### Example

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).

###### Example

For any topological space $X$, there is an idempotent adjunction between the category $[O(X)^{\op}, Set]$ of presheaves on $X$ and the slice category $Top_{/X}$ of TopologicalSpaces 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.

### Between topological and diffeological spaces

###### Proposition

(adjunction between topological spaces and diffeological spaces)

There is a pair of adjoint functors

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

between the categories of TopologicalSpaces and of DiffeologicalSpaces, where

• $Cdfflg$ takes a topological space $X$ to the continuous diffeology, namely the diffeological space on the same underlying set $X_s$ whose plots $U_s \to X_s$ are the continuous functions (from the underlying topological space of the domain $U$).

• $Dtplg$ takes a diffeological space to the diffeological topology (D-topology), namely the topological space with the same underlying set $X_s$ and with the final topology that makes all its plots $U_{s} \to X_{s}$ into continuous functions: called the D-topology.

Hence a subset $O \subset \flat X$ is an open subset in the D-topology precisely if for each plot $f \colon U \to X$ the preimage $f^{-1}(O) \subset U$ is an open subset in the Cartesian space $U$.

Moreover:

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

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

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

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

Caution: There was a gap in the original proof that $DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces$. 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).

###### Proof

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

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

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

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

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

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

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

given simply as the identity on the underlying functions of underlying sets. This makes it immediate that this hom-isomorphism is natural in $X$ and $Y$ 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

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

to find that the counit of the adjunction

$Dtplg(Cdfflg(X)) \overset{\epsilon_X}{\longrightarrow} X$

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

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

$Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces$

is an idempotent comonad, hence that

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

is a natural isomorphism. But, as before for the adjunction counit $\epsilon$, we have that also the adjunction unit $\eta$ 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: