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\begin{document}

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\section*{adjoint equivalence}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\hypertarget{adjoint_equivalences}{}\section*{{Adjoint equivalences}}\label{adjoint_equivalences}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{intervals_in_homotopy_theory}{Intervals in homotopy theory}\dotfill \pageref*{intervals_in_homotopy_theory} \linebreak
\noindent\hyperlink{defining_tricategories}{Defining tricategories}\dotfill \pageref*{defining_tricategories} \linebreak
\noindent\hyperlink{cartesian_closed_2categories}{Cartesian closed 2-categories}\dotfill \pageref*{cartesian_closed_2categories} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An adjoint equivalence is a more ``coherent'' or ``structured'' notion of [[equivalence]], in which the [[2-morphism|2-]][[isomorphism]]s relating composites to identities are required to satisfy [[coherence law]]s (the [[zigzag identities]] for an [[adjunction]]).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An \textbf{adjoint equivalence} between [[categories]] is an [[adjunction]] $f\dashv g$ in which the [[unit of an adjunction|unit]] $\eta$ and [[unit of an adjunction|counit]] $\varepsilon$ 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]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

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

\begin{lemma}
\label{}\hypertarget{}{}
If $(f,g,\eta,\varepsilon)$ is an adjoint equivalence, then so is $(g,f,\varepsilon^{-1},\eta^{-1})$.

\end{lemma}
Therefore, in an adjoint equivalence, each functor is both the [[left adjoint]] and the [[right adjoint]] of the other (i.e. it is an [[ambidextrous adjunction]]).

The definition as given above is also redundant:

\begin{lemma}
\label{}\hypertarget{}{}
If $(f,g,\eta,\varepsilon)$ is any equivalence, then it satisfies one [[zigzag identity]] iff it satisfies the other.

\end{lemma}
\begin{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:

[[!include equivalence data - SVG]]

If we now suppose that one zigzag identity holds:

[[!include zigzag identity 1 - SVG]]

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

[[!include zigzag identity 1 implies 2 - SVG]]

\end{proof}
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:

\begin{theorem}
\label{}\hypertarget{}{}
If $f\colon X\to Y$ is a morphism which is an equivalence, then given any morphism $g\colon Y\to X$ and any isomorphism $\eta\colon 1 \cong g f$, there exists a unique 2-isomorphism $\varepsilon\colon f g \cong 1$ such that $(f,g,\eta,\varepsilon)$ is an adjoint equivalence.

\end{theorem}
\begin{proof}
Since $f$ is an equivalence, there exists a $g'$ and isomorphisms $f g' \cong 1$ and $1\cong g' f$. However, we also have $g \cong g f g' \cong g'$, so the isomorphism $f g' \cong 1$ also induces an isomorphism $f g\cong 1$, which we denote $\xi$. Now $\eta$ and $\xi$ may not satisfy the zigzag identities, but if we define $\varepsilon$ as follows:

\begin{displaymath}
f g
\xrightarrow{f g \xi^{-1}} f g f g
\xrightarrow{f \eta^{-1} g} f g
\xrightarrow{\xi} 1
\end{displaymath}
then we can verify, using string diagram notation as above, that $\varepsilon$ satisfies one zigzag identity, and hence (by the previous lemma) also the other:

[[!include adjointification zigzag identity - SVG]]

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

\begin{displaymath}
\varepsilon' = \varepsilon' . (\varepsilon f g) . (f \eta g) = 
\varepsilon . (f g \varepsilon') . (f \eta g) = \varepsilon
\end{displaymath}
using the [[interchange law]] and two zigzag identities. This shows uniqueness.

\end{proof}
In [[Categories Work]], IV.4, there is a different proof of the weaker fact that if a [[functor]] $f$ 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 $f$, the ``category of adjoint equivalence data $(f,g,\eta,\varepsilon)$'' is either empty (if $f$ is not an equivalence) or equivalent to the [[terminal category]] (if $f$ 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):

\begin{itemize}%
\item A morphism $f\colon X\to Y$ with the [[property]] of being an equivalence.


\item A morphism $f\colon X\to Y$ with the \emph{structure} of a morphism $g\colon Y \to X$ and an isomorphism $\eta\colon 1 \cong g f$, together with the \emph{property} that there exists an isomorphism $f g \cong 1$.


\item A morphism $f$ together with the structure of adjoint equivalence data $(f,g,\eta,\varepsilon)$.



\end{itemize}
In other words, adjoint equivalences are the way to make the property of ``being an equivalence'' completely into ``algebraic'' structure. However, they are \emph{not} equivalent to the category of the following data:

\begin{itemize}%
\item A morphism $f$ together with the structure of a morphism $g\colon Y \to X$ and arbitrary isomorphisms $\eta\colon 1 \cong g f$ and $\varepsilon\colon f g \cong 1$.

\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{intervals_in_homotopy_theory}{}\subsubsection*{{Intervals in homotopy theory}}\label{intervals_in_homotopy_theory}

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

\hypertarget{defining_tricategories}{}\subsubsection*{{Defining tricategories}}\label{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.

\begin{itemize}%
\item 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 functor|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 object|free]] structures.


\item 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.



\end{itemize}
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.

\hypertarget{cartesian_closed_2categories}{}\subsubsection*{{Cartesian closed 2-categories}}\label{cartesian_closed_2categories}

In a [[cartesian closed category]] with [[equalizers]], for any two objects $X$ and $Y$ one can construct the ``object of isomorphisms from $X$ to $Y$'' as the following equalizer:

\begin{displaymath}
Iso(X,Y) \to X^Y \times Y^X \;\rightrightarrows\; X^X \times Y^Y
\end{displaymath}
where the top arrow on the right side is (composition, reversed composition) and the bottom arrow factors through $(id,id)\colon 1 \to X^X \times Y^Y$. One can then prove that the maps $Iso(X,Y)\to X^Y$ and $Iso(X,Y)\to Y^X$ are monic, so that $Iso(X,Y)$ can be regarded either as ``the object of maps $X\to Y$ which are isomorphisms'' or ``the object of maps $Y\to X$ which are isomorphisms'' (or, as is most evident from its construction, ``the object of pairs of maps $X\to Y$ and $Y\to X$ which are inverse isomorphisms'').

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

\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{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:

\begin{itemize}%
\item For [[Gray-categories]], the statement and proof is in [[Steve Lack]]`s paper \href{http://arxiv.org/abs/1001.2366}{1001.2366} on the [[model structure for Gray-categories]]. See [[adjoint 2-equivalence]].


\item For [[tricategories]], the corresponding statement can be deduced from the Gray-categorical version using the [[coherence theorem for tricategories]]. A direct proof can also be found in [[Nick Gurski]]`s paper \href{http://www.tac.mta.ca/tac/volumes/26/14/26-14abs.html}{Biequivalences in tricategories}.


\item For [[strict omega-categories]], more or less this fact can be found in the study of ``generic squares'' in the paper \href{http://arxiv.org/abs/0712.0617}{0712.0617} on the [[model structure for strict omega-categories]].


\item For [[quasicategories]], the theorem is true, where an ``adjoint equivalence'' means simply a map out of the [[nerve]] of the [[interval groupoid]]; see [[equivalence in a quasicategory]].



\end{itemize}
[[!redirects adjoint equivalences]]

[[!redirects adjoint equivalence of categories]] [[!redirects adjoint equivalences of categories]]



\end{document}