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\section*{homotopy exact square}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-category theory}}\label{category_theory}

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

\hypertarget{homotopy_exact_squares}{}\section*{{Homotopy exact squares}}\label{homotopy_exact_squares}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{characterization}{Characterization}\dotfill \pageref*{characterization} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{comma_squares}{Comma squares}\dotfill \pageref*{comma_squares} \linebreak
\noindent\hyperlink{fully_faithful_functors}{Fully faithful functors}\dotfill \pageref*{fully_faithful_functors} \linebreak
\noindent\hyperlink{final_functors}{Final functors}\dotfill \pageref*{final_functors} \linebreak
\noindent\hyperlink{PullbacksOfFibrations}{Pullbacks of fibrations}\dotfill \pageref*{PullbacksOfFibrations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{homotopy exact square} is the analogue of an [[exact square]] which applies to [[homotopy Kan extensions]], or equivalently to [[(∞,1)-Kan extensions]]. It is especially important in the theory of [[derivators]], which provide a calculus for computing with homotopy Kan extensions whose primary tool is the use of homotopy exact squares.

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

Let $A$, $B$, $C$, and $D$ be small [[categories]], and consider a square of [[functors]]

\begin{displaymath}
\itexarray{A & \overset{f}{\to} & B\\
  ^u\downarrow & \swArrow & \downarrow^v\\
  C& \underset{g}{\to} & D}
\end{displaymath}
which is inhabited by a [[natural transformation]] (which might be an [[identity]]). Let $M$ be either

\begin{itemize}%
\item a [[model category]],
\item a [[simplicially enriched category]] (or a topologically enriched category, or a [[dg-category]]),
\item an [[(∞,1)-category]] (of any sort), or
\item a [[derivator]].

\end{itemize}
We write $M^A$, $M^B$, etc. for the model categories, simplicial categories, $(\infty,1)$-categories, or homotopy categories of diagrams in $M$ of whatever shape. We write $f^*\colon M^B\to B^A$, $g^*\colon M^D\to M^C$, and so on for precomposition functors, which are always homotopically meaningful, and we write $u_!\colon M^A\to M^C$, $v_!\colon M^B\to M^D$ and so on for the homotopically meaningful notions of pointwise [[left Kan extension]]. Specifically:

\begin{itemize}%
\item If $M$ is a model category, then $u_!$ denotes the left [[derived functor]] of pointwise Kan extension along $u$.
\item If $M$ is a simplicially enriched category, then $u_!$ is the coherent pointwise [[homotopy left Kan extension]] along $u$, which may be defined explicitly in various ways, such as using a [[bar construction]].
\item If $M$ is an $(\infty,1)$-category, then $u_!$ denotes the pointwise [[(∞,1)-Kan extensions]] along $u$.
\item If $M$ is a derivator, then $u_!$ simply denotes the left adjoint of $u^*$ (which is assumed to exist and to ``be pointwise'' by the derivator axioms).

\end{itemize}
Assume that $M$ is such that the relevant extensions $u_!$ and $v_!$ exist. Then there is a canonical [[Beck-Chevalley transformation]]

\begin{displaymath}
u_! f^* \to g^* v_!
\end{displaymath}
defined as the composite

\begin{displaymath}
u_! f^* \to u_! f^* v^* v_! \to u_! u^* g^* v_! \to g^* v_!.
\end{displaymath}
and we say that the given square is \textbf{$M$-exact} if this transformation is an [[equivalence]]. If the square is $M$-exact for all $M$, we say it is \textbf{homotopy exact}. Note that by the general calculus of [[mates]], this is equivalent to requiring that the dual transformation

\begin{displaymath}
v^* g_* \to f_* u^*
\end{displaymath}
is an equivalence, where $f_*$ and $g_*$ denote the analogous sort of \emph{right} Kan extension.

Of course when we say ``for all $M$'' we need to specify what sorts of $M$ we consider. However, we actually get the \emph{same} definition regardless of whether we mean ``for all model categories $M$'' or ``for all simplicially enriched categories $M$'' or ``for all $(\infty,1)$-categories $M$'' or ``for all derivators $M$''. This is a nontrivial theorem, especially in the case of derivators.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

Since any 1-category is a degenerate sort of $(\infty,1)$-category, any homotopy exact square is [[exact square|exact]] in the usual 1-categorical sense, but the converse is not true. This also implies that a square can be $M$-exact for some particular $M$ without being homotopy exact. However, there exists a ``universal'' $M$ such that $M$-exactness is equivalent to homotopy exactness, namely $M=\infty Gpd$.

\hypertarget{characterization}{}\subsection*{{Characterization}}\label{characterization}

Of course, the above definition is ``functional'', while in practice we want some more combinatorial characterization which is easier to check. This can be done completely analogously to the characterization of ordinary [[exact squares]] using comma objects, except that at the last step we need to consider a more restricted notion of ``equivalence'' (i.e. a more restricted [[basic localizer]]).

The characterization is the following. Given $b\in B$ and $c\in C$ and $\varphi\colon v(b) \to g(c)$, let $(b/A/c)_\varphi$ denote the category whose objects are triples $(a,\alpha,\beta)$ with $\alpha\colon u(a)\to c$ and $\beta\colon b\to f(a)$ such that $g(\alpha) \circ v(\beta) = \varphi$, and whose morphisms are morphisms $a\to a'$ making two triangles commute.

\begin{utheorem}
A square is homotopy exact if and only if each category $(b/A/c)_\varphi$ has a [[contractible space|contractible]] [[nerve]].

\end{utheorem}
\begin{itemize}%
\item See \href{/nlab/show/exact+square#CommaObj}{exact square} for the proof that is being generalized.
\item See \href{/nlab/show/derivator#ExactSquares}{derivator} for the proof in the case of derivators, which relies on Cisinski's characterization of [[basic localizers]].
\item The proof in the (∞,1)-categorical case generalizes the characterization of \href{/nlab/show/final+(∞,1%29-functor#Properties}{final (∞,1)-functors}.

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{comma_squares}{}\subsubsection*{{Comma squares}}\label{comma_squares}

Any [[comma square]] is homotopy exact. In other words, if $A=(v/g)$ is the [[comma category]] with $f$ and $u$ the canonical projections, then the square is homotopy exact. If in addition $C=*$ is the [[terminal category]], then $g$ just picks out an object $d\in D$ and $A$ is the comma category $(v/d)$; thus this says that (pointwise) homotopy Kan extensions can be computed pointwise as homotopy limits over such comma categories.

\hypertarget{fully_faithful_functors}{}\subsubsection*{{Fully faithful functors}}\label{fully_faithful_functors}

If $u\colon A\to B$ is a [[fully faithful functor]], then the square

\begin{displaymath}
\itexarray{A & \overset{id}{\to} & A\\
  ^{id}\downarrow && \downarrow^u\\
  A & \underset{u}{\to} & B}
\end{displaymath}
is homotopy exact. This just says that the unit $Id_A \to u^* u_!$ is an isomorphism, i.e. that left (and equivalently right) homotopy Kan extensions along $u$ are ``honest'' extensions.

\hypertarget{final_functors}{}\subsubsection*{{Final functors}}\label{final_functors}

A functor $u\colon A\to B$ is a [[homotopy final functor]] if and only if the square

\begin{displaymath}
\itexarray{A & \overset{u}{\to} & B\\
  \downarrow && \downarrow\\
  *& \underset{}{\to} & *}
\end{displaymath}
is homotopy exact. Homotopy exactness of this square says that for $F\colon B\to M$, the canonical map $hocolim_A u^*F \to hocolim_B F$ is an isomorphism, which is one equivalent definition of when $u$ is homotopy final. In this case, the characterization theorem reduces to saying that $u$ is homotopy final if and only if each comma category $b/u$ has a contractible nerve, which is a known characterization of homotopy final functors.

\hypertarget{PullbacksOfFibrations}{}\subsubsection*{{Pullbacks of fibrations}}\label{PullbacksOfFibrations}

This example is due to \hyperlink{Groth}{Moritz Groth}. Let $p\colon C\to D$ be a [[Grothendieck opfibration]], and suppose that

\begin{displaymath}
\itexarray{ A & \xrightarrow{u} & C \\
  ^v\downarrow && \downarrow^p\\
  B & \xrightarrow{q} & D }
\end{displaymath}
is a [[pullback]] in [[Cat]]. Then we claim that this square is homotopy exact.

For a proof, suppose given $b\in B$ and $c\in C$ and a morphism $\phi\colon p(c) \to q(b)$, and let $X$ be the category whose contractibility we must check. By definition, since $A = B\times_D C$, an object of $X$ consists of objects $c'\in C$ and $b'\in B$ such that $p(c') = q(b')$, and morphisms $\alpha\colon c\to c'$ and $\beta\colon b' \to b$ such that $q(\beta) . p(\alpha) = \phi$.

Let $Y$ be the full subcategory of $X$ consisting of those objects with $b'=b$ and $\beta$ the identity, so that an object of $Y$ is an object $c'\in C$ such that $p(c') = q(b)$ together with a morphism $\alpha\colon c\to c'$ such that $p(\alpha) = \phi$. Then the inclusion $Y\hookrightarrow X$ has a left adjoint, which sends $(b',c',\alpha,\beta)$ to $(b, q(\beta)_!(c'), \overline{q(\beta)}.\alpha, 1_b)$, where $\overline{q(\beta)}\colon c' \to q(\beta)_!(c')$ is an opcartesian arrow over $q(\beta)$. Therefore, the nerves of $Y$ and $X$ are homotopy equivalent.

But $Y$ has an initial object, namely $(\phi_!(c), \overline{\phi})$, where $\overline{\phi}\colon c \to \phi_!(c)$ is an opcartesian arrow over $\phi$. Thus the nerve of $Y$ is contractible, and thus so is the nerve of $X$.

Groth has also proved that homotopy-exactness of such squares can be used to replace that of comma squares in the definition of a [[derivator]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item An \href{http://golem.ph.utexas.edu/category/2010/06/exact_squares.html}{nCaf\'e{} post} about (mostly ordinary) [[exact squares]]


\item [[Moritz Groth]], ``Derivators, pointed derivators, and stable derivators'' \href{http://www.math.uni-bonn.de/~mgroth/groth_derivators.pdf}{PDF}



\end{itemize}
\begin{itemize}%
\item [[Georges Maltsiniotis]], ``Carr\'e{}s exacts homotopiques, et d\'e{}rivateurs'', Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques 53 (2012), 3-63 \href{http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/carhex.pdf}{PDF}

\end{itemize}
[[!redirects homotopy exact squares]] [[!redirects homotopically exact square]] [[!redirects homotopically exact squares]]



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