nLab
adjoint equivalence

Context

Category theory

2-Category theory

Equality and Equivalence

Adjoint equivalences

Idea

An adjoint equivalence is a more “coherent” or “structured” notion of equivalence, in which the 2-isomorphisms relating composites to identities are required to satisfy coherence laws (the zigzag identities for an adjunction).

Definition

An adjoint equivalence between categories is an adjunction fgf\dashv g in which the unit η\eta and counit ε\varepsilon are natural isomorphisms. It follows that it is an equivalence of categories.

There is an identical definition internal to any 2-category, which reproduces the above notion when applied in Cat.

Properties

We work in any 2-category. First, we observe:

Lemma

If (f,g,η,ε)(f,g,\eta,\varepsilon) is an adjoint equivalence, then so is (g,f,ε 1,η 1)(g,f,\varepsilon^{-1},\eta^{-1}).

Therefore, in an adjoint equivalence, each functor is both the left and the right adjoint of the other (i.e. it is an ambijunction?).

The definition as given above is also redundant:

Lemma

If (f,g,η,ε)(f,g,\eta,\varepsilon) is any equivalence, then it satisfies one zigzag identity iff it satisfies the other.

Proof

Using string diagram notation, with strings progressing up the page and 1-morphisms progressing from left to right, we can draw the data of an equivalence (omitting labels for the regions denoting objects) as follows:

Layer 1 f f f f f f f f g g g g g g g g ε \varepsilon η \eta η 1 \eta^{-1} ε 1 \varepsilon^{-1} ε \varepsilon ε 1 \varepsilon^{-1} = = = = ε 1 \varepsilon^{-1} ε \varepsilon = = = = η \eta η \eta η 1 \eta^{-1} η 1 \eta^{-1}

If we now suppose that one zigzag identity holds:

Layer 1 ε \varepsilon η \eta = =

then we can verify the other as follows. (The first step uses the inverse of the first zigzag identity.)

Layer 1 = = = = = = = = = = = = = = = = = = ε \varepsilon η \eta ε \varepsilon η \eta η 1 \eta^{-1} ε 1 \varepsilon^{-1}

Furthermore, although an adjoint equivalence is a “stronger” or “more structured” notion than a mere equivalence, the property of “being adjoint equivalent” is no stronger a condition than “being equivalent,” since every equivalence may be refined to an adjoint equivalence by modifying one of the natural isomorphisms involved. More specifically:

Theorem

If f:XYf\colon X\to Y is a morphism which is an equivalence, then given any morphism g:YXg\colon Y\to X and any isomorphism η:1gf\eta\colon 1 \cong g f, there exists a unique 2-isomorphism ε:fg1\varepsilon\colon f g \cong 1 such that (f,g,η,ε)(f,g,\eta,\varepsilon) is an adjoint equivalence.

Proof

Since ff is an equivalence, there exists a gg' and isomorphisms fg1f g' \cong 1 and 1gf1\cong g' f. However, we also have ggfggg \cong g f g' \cong g', so the isomorphism fg1f g' \cong 1 also induces an isomorphism fg1f g\cong 1, which we denote ξ\xi. Now η\eta and ξ\xi may not satisfy the zigzag identities, but if we define ε\varepsilon as follows:

fgfgξ 1fgfgfη 1fgξ1 f g \xrightarrow{f g \xi^{-1}} f g f g \xrightarrow{f \eta^{-1}} f g \xrightarrow{\xi} 1

then we can verify, using string diagram notation as above, that ε\varepsilon satisfies one zigzag identity, and hence (by the previous lemma) also the other:

Layer 1 = = = = = =

Finally, if ε:fg1\varepsilon'\colon f g \to 1 is any other isomorphism satisfying the zigzag identities with η\eta, then we have

ε=ε.(εfg).(fηg)=ε.(fgε).(fηg)=ε\varepsilon' = \varepsilon' . (\varepsilon f g) . (f \eta g) = \varepsilon . (f g \varepsilon') . (f \eta g) = \varepsilon

using the interchange law and two zigzag identities. This shows uniqueness.

In Categories Work, IV.4, there is a different proof of the weaker fact that if a functor ff is part of an equivalence, then it is part of an adjoint equivalence. This proof is given in Cat, but can be applied representably to any 2-category.

Since adjoints are unique up to unique isomorphism when they exist, it follows that any adjunction involving one functor which is an equivalence must be an adjoint equivalence. Therefore, for a fixed morphism ff, the “category of adjoint equivalence data (f,g,η,ε)(f,g,\eta,\varepsilon)” is either empty (if ff is not an equivalence) or equivalent to the terminal category (if ff is an equivalence). In other words, it is a (-1)-category.

Therefore, in any 2-category, the following data are all equivalent (i.e. form equivalent categories):

  • A morphism f:XYf\colon X\to Y with the property of being an equivalence.

  • A morphism f:XYf\colon X\to Y with the structure of a morphism g:YXg\colon Y \to X and an isomorphism η:1gf\eta\colon 1 \cong g f, together with the property that there exists an isomorphism fg1f g \cong 1.

  • A morphism ff together with the structure of adjoint equivalence data (f,g,η,ε)(f,g,\eta,\varepsilon).

In other words, adjoint equivalences are the way to make the property of “being an equivalence” completely into “algebraic” structure. However, they are not equivalent to the category of the following data:

  • A morphism ff together with the structure of a morphism g:YXg\colon Y \to X and arbitrary isomorphisms η:1gf\eta\colon 1 \cong g f and ε:fg1\varepsilon\colon f g \cong 1.

Applications

Intervals in homotopy theory

One instance of the usefulness of adjoint equivalences is that the “walking adjoint equivalence” 2-category is equivalent to the point. Thus, it can be used as an interval object in 2Cat2Cat, and in fact it is one of the generating cofibrations for the canonical (Lack) model structure on 2Cat2Cat. This is not true of the “walking non-adjoint equivalence.”

Defining tricategories

The original definition of tricategory by Gordon-Power-Street involved coherence 2-morphisms with the property of being equivalences in the relevant hom-bicategories. This is fine for most purposes, but for others it is insufficient, such as the following.

  • Since “being an equivalence” is not algebraic structure, the GPS definition of tricategory, taken literally, is not an algebraic structure. In particular, it is not monadic over 3-globular sets, nor is it the algebras for a globular operad. Such monadicity is important if one wants to state coherence theorems as properties of free structures.

  • The definition of 3-functors and higher transfors between tricategories include data and axioms that involve composites incorporating not just the coherence equivalences, but their pseudo-inverses. Therefore, strictly speaking these definitions are not well-defined unless the definition of tricategory comes with chosen pseudo-inverses for these coherence equivalences—in which case one should certainly also choose full adjoint equivalence data in order that the space of choices be contractible.

These problems are, of course, easy to remedy by simply requiring adjoint equivalence data rather than merely single equivalence morphisms. This change was first written down by Gurski.

Cartesian closed 2-categories

In a cartesian closed category with equalizers, for any two objects XX and YY one can construct the “object of isomorphisms from XX to YY” as the following equalizer:

Iso(X,Y)X Y×Y XX X×Y Y Iso(X,Y) \to X^Y \times Y^X \;\rightrightarrows\; X^X \times Y^Y

where the top arrow on the right side is (composition, reversed composition) and the bottom arrow factors through (id,id):1X X×Y Y(id,id)\colon 1 \to X^X \times Y^Y. One can then prove that the maps Iso(X,Y)X YIso(X,Y)\to X^Y and Iso(X,Y)Y XIso(X,Y)\to Y^X are monic, so that Iso(X,Y)Iso(X,Y) can be regarded either as “the object of maps XYX\to Y which are isomorphisms” or “the object of maps YXY\to X which are isomorphisms” (or, as is most evident from its construction, “the object of pairs of maps XYX\to Y and YXY\to X which are inverse isomorphisms”).

In a cartesian closed 2-category?, however, the analogous “2-equalizer” Eqv(X,Y)Eqv(X,Y), does not have similar properties: the projections Eqv(X,Y)X YEqv(X,Y)\to X^Y and Eqv(X,Y)Y XEqv(X,Y)\to Y^X will not in general be fully faithful. Thus, we can only regard Eqv(X,Y)Eqv(X,Y) as “the object of not-necessarily-adjoint equivalence data (f,g,η,ε)(f,g,\eta,\varepsilon).” However, if we use a further equifier to construct its “subobject of adjoint equivalence data” AdjEqv(X,Y)AdjEqv(X,Y), then the projections AdjEqv(X,Y)X YAdjEqv(X,Y)\to X^Y and AdjEqv(X,Y)Y XAdjEqv(X,Y)\to Y^X will be fully faithful, so that AdjEqv(X,Y)AdjEqv(X,Y) can also be regarded as “the object of maps XYX\to Y which are equivalences” and dually.

In higher category theory

In higher category theory, one expects to have a similar “fully coherent” notion of “adjoint equivalence” in any n-category or infinity-category, and one hopes to prove a similar theorem that any equivalence can be refined to an adjoint equivalence. This is known to be true at least in the following cases:

Revised on September 18, 2012 21:17:50 by Urs Schreiber (82.169.65.155)