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\begin{document}

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\section*{pullback in a derivator}

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

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

[[!include homotopy - contents]]

\hypertarget{pullbacks_in_derivators}{}\section*{{Pullbacks in derivators}}\label{pullbacks_in_derivators}

\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{Pasting}{Pasting law}\dotfill \pageref*{Pasting} \linebreak
\noindent\hyperlink{Detection}{Detection Lemma}\dotfill \pageref*{Detection} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{pullback in a derivator} is the generalization to the context of a [[derivator]] of the notion of [[pullback]] in ordinary category theory. Viewing a derivator as the ``shadow'' of an [[(∞,1)-category]], the notion of pullback therein coincides with the notion of [[homotopy pullback]] in an $(\infty,1)$-category.

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

Let $\square$ denote the category

\begin{displaymath}
\itexarray{a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d}
\end{displaymath}
that is the ``free-living [[commutative square]]'', and let $L$ be the full subcategory of $\square$ on $b,c,d$, with inclusion $u\colon L\to \square$.

Let $D$ be a [[derivator]] and let $X\in D(\square)$ be a commutative square in $D$. We say that $X$ is a \textbf{pullback square}, or is \textbf{cartesian}, if the unit $X\to u_* u^* X$ of the [[adjunction]] $u^*\dashv u_*$ is an isomorphism. Since $u$ is [[fully faithful]], so is $u_*$, so this is equivalent to saying that there exists some $Y\in D(L)$ such that $X\cong u_* Y$.

If $D$ is merely a prederivator, then we can phrase the same definition by saying that $X$ has the [[free object|universal property]] that $u_* u^* X$ would, if the whole functor $u_*$ existed.

The dual notion, of course, is a \textbf{pushout} or \textbf{cocartesian} square.

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

\hypertarget{Pasting}{}\subsubsection*{{Pasting law}}\label{Pasting}

Using properties of [[homotopy exact squares]], we can prove the ``pasting law'' for pullback squares in a derivator:

\begin{ulemma}
Given a diagram

\begin{displaymath}
\itexarray{a & \to & b & \to & c \\
\downarrow & & \downarrow & & \downarrow \\
d & \to & e & \to & f }
\end{displaymath}
in which the right-hand square $bcef$ is a pullback, then the left-hand square $abde$ is a pullback if and only if the outer rectangle $acdf$ is a pullback.

\end{ulemma}
The following proof should be compared and contrasted with the \href{http://ncatlab.org/nlab/show/pullback#Pasting}{standard proof} for pullbacks in 1-categories, and the \href{http://ncatlab.org/nlab/show/%28infinity%2C1%29-pullback#QuasiCatPastingLaw}{quasi-categorical proof} for pullbacks in $(\infty,1)$-categories. In particular, note that the statement for derivators is a generalization of both, since both 1-categories and $(\infty,1)$-categories give rise to derivators.

\begin{proof}
First of all, by ``a diagram'' in a derivator, we mean an object of $D(X)$ for some suitable category $X$. In the above case, $X$ is the category consisting of two commutative squares, as pictured above. We'll write $abcdef$ for this $X$, and similarly $cdef$ for its lower-right L-shaped subcategory, and so on. We leave the verification of [[homotopy exact square|homotopy exactness]] of all squares to the reader.

Firstly, since the squares

\begin{displaymath}
\itexarray{cef & \overset{}{\to} & bcef\\
  \downarrow && \downarrow\\
  cdef & \underset{}{\to} & abcdef} \qquad and \qquad
\itexarray{cdf & \overset{}{\to} & acdf\\
  \downarrow && \downarrow\\
  cdef& \underset{}{\to} & abcdef}
\end{displaymath}
are exact, if we start from a $cdef$-diagram and right Kan extend it to a full $abcdef$-diagram, then the right-hand square and outer rectangle must be pullback squares. Moreover, by composition of adjoints, right Kan extension from $cdef$ to $abcdef$ is equivalent to first extending to $bcdef$ and then to $abcdef$, and since the square

\begin{displaymath}
\itexarray{bde & \overset{}{\to} & abde\\
  \downarrow && \downarrow\\
  bcdef & \underset{}{\to} & abcdef}
\end{displaymath}
is exact, the left-hand square in such an extension must also be a pullback.

Now if we start with an $abcdef$-diagram, say $F$, we can restrict it to a $cdef$-diagram and then right Kan-extend it to a new $abcdef$-diagram. If $u\colon cdef \to abcdef$ is the inclusion, then this results in $u_\ast u^\ast F$, and we have a canonical natural transformation $\eta \colon F \to u_\ast u^\ast F$ (the unit of the adjunction $u^\ast \dashv u_\ast$). Since the square

\begin{displaymath}
\itexarray{cdef & \overset{}{\to} & cdef \\
  \downarrow && \downarrow\\
  cdef & \underset{}{\to} & abcdef}
\end{displaymath}
is exact, the counit $u^\ast u_\ast G \to G$ is an isomorphism for any $G$, and in particular for $G=u^\ast F$, from which it follows by the triangle identities that $u^\ast F \to u^\ast u_\ast u^\ast F$ is also an isomorphism --- i.e. the components of $\eta\colon F \to u_\ast u^\ast F$ at $c$, $d$, $e$, and $f$ are isomorphisms. Now if the right-hand square of $F$ is a pullback, then the restrictions of $F$ and $u_\ast u^\ast F$ to $bcef$ are both pullback squares; hence since the $cef$-components of $\eta$ are isomorphisms, so is the $b$-component. And if the left-hand square of $F$ is a pullback, then we can play the same game with $abde$ to show that the $a$-component of $\eta$ is an isomorphism, while if the outer rectangle is a pullback, we can play it with $acdf$. Hence in both of these cases, $\eta$ itself is an isomorphism, since all of its components are --- and thus the remaining square in $F$ is also a pullback, since we have shown that it is so in $u_\ast u^\ast F$.

\end{proof}
\hypertarget{Detection}{}\subsubsection*{{Detection Lemma}}\label{Detection}

The following lemma, which detects when squares occurring in a Kan extension are pullbacks or pushouts, is due to \hyperlink{Franke}{Jens Franke}; see also \hyperlink{Groth}{Groth}. We state it in terms of pushouts.

\begin{lemma}
\label{CartesianSquareDetection}\hypertarget{CartesianSquareDetection}{}
Let $f\colon K\to J$ be any functor and let $i\colon \square \to J$ be injective on objects, with lower vertex $i(1,1) = z$. Suppose that $z$ is not in the image of $f$, and that the induced functor $\Gamma \to (J \setminus z)/z$ is a nerve equivalence (such as if it has an adjoint). Then for any derivator $D$ and any $Y\in D(K)$, the square $i^* f_! Y$ is cocartesian.

\end{lemma}
\begin{proof}
Since $f_! = j_! \bar{f}_!$ where $\bar{f}\colon K\to (J\setminus z)$ is induced by $f$ and $j\colon (J\setminus z) \to J$ is the inclusion, it suffices to suppose that $K = (J\setminus z)$. Now what we want is to prove that the following square is [[homotopy exact square|homotopy exact]]:

\begin{displaymath}
\itexarray{ \Gamma &\to & (J\setminus z) \\
  \downarrow && \downarrow\\
  \square &\to & J}
\end{displaymath}
Exactness is trivial at all objects of $\square$ except $(1,1)$. In that case, we paste with another square:

\begin{displaymath}
\itexarray{ \Gamma &\to& \Gamma &\to & (J\setminus z) \\
  \downarrow & \swArrow & \downarrow && \downarrow\\
  \ast &\underset{(1,1)}{\to} &\square &\to & J}
\end{displaymath}
The left-hand square is a [[comma square]], hence homotopy exact, so it suffices to show that the composite square is homotopy exact. But the comma object associated to the cospan $\ast \to J \leftarrow (J\setminus z)$ is $(J \setminus z)/z$, and of course this comma square is also exact. And the composite square factors through this comma square by the functor $\Gamma \to (J \setminus z)/z$ which is assumed a nerve equivalence; hence it is also homotopy exact.

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

See all references at [[derivator]]. Referred to particularly above are:

\begin{itemize}%
\item Jens Franke, \emph{Uniqueness theorems for certain triangulated categories with an Adams spectral sequence}, \href{http://www.math.uiuc.edu/K-theory/0139/}{K-theory archive}

\end{itemize}
\begin{itemize}%
\item Moritz Groth, \emph{Derivators, pointed derivators, and stable derivators} \href{http://www.math.uni-bonn.de/~mgroth/groth_derivators.pdf}{pdf}

\end{itemize}
[[!redirects pullbacks in a derivator]] [[!redirects pullbacks in derivators]] [[!redirects cartesian square in a derivator]] [[!redirects cartesian squares in a derivator]] [[!redirects cartesian squares in derivators]] [[!redirects Cartesian square in a derivator]] [[!redirects Cartesian squares in a derivator]] [[!redirects Cartesian squares in derivators]] [[!redirects fiber product in a derivator]] [[!redirects fiber products in a derivator]] [[!redirects fiber products in derivators]] [[!redirects pushout in a derivator]] [[!redirects pushouts in a derivator]] [[!redirects pushoutss in derivators]] [[!redirects cocartesian square in a derivator]] [[!redirects cocartesian squares in a derivator]] [[!redirects cocartesian squares in derivators]] [[!redirects coCartesian square in a derivator]] [[!redirects coCartesian squares in a derivator]] [[!redirects coCartesian squares in derivators]]



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