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\begin{document}

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\section*{(infinity,1)-pullback}

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

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

[[!include quasi-category theory contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{incarnations}{Incarnations}\dotfill \pageref*{incarnations} \linebreak
\noindent\hyperlink{in_quasicategories}{In quasi-categories}\dotfill \pageref*{in_quasicategories} \linebreak
\noindent\hyperlink{QuasiCatPastingLaw}{Pasting law}\dotfill \pageref*{QuasiCatPastingLaw} \linebreak
\noindent\hyperlink{in_model_categories}{In model categories}\dotfill \pageref*{in_model_categories} \linebreak
\noindent\hyperlink{in_derivators}{In derivators}\dotfill \pageref*{in_derivators} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{fiber_sequence}{Fiber sequence}\dotfill \pageref*{fiber_sequence} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{in_homotopy_type_theory}{In homotopy type theory}\dotfill \pageref*{in_homotopy_type_theory} \linebreak


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

An \textbf{$(\infty,1)$-pullback} is a [[limit in an (∞,1)-category]] $\mathcal{C}$ over a [[diagram]] of the shape

\begin{displaymath}
\{a \to c \leftarrow b\} \to \mathcal{C}
  \,.
\end{displaymath}
In other words it is a [[cone]]

\begin{displaymath}
\itexarray{
    A \times_C B &\to& B
    \\
    \downarrow &\cong\swArrow& \downarrow
    \\
    A &\to& C
  }
\end{displaymath}
which is [[universal property|universal]] among all such cones in the $(\infty,1)$-categorical sense.

This is the analog in [[(∞,1)-category theory]] of the notion of [[pullback]] in [[category theory]].

\hypertarget{incarnations}{}\subsection*{{Incarnations}}\label{incarnations}

\hypertarget{in_quasicategories}{}\subsubsection*{{In quasi-categories}}\label{in_quasicategories}

Let $\mathcal{C}$ be a [[quasi-category]]. Recall the notion of [[limit in a quasi-category]].

The non-degenerate cells of the [[simplicial set]] $\Delta[1] \times \Delta[1]$ obtained as the [[cartesian product]] of the simplicial 1-[[simplex]] with itself look like

\begin{displaymath}
\itexarray{
    (0,0) &\to& (1,0)
    \\
    \downarrow &\searrow& \downarrow
    \\
    (0,1) &\to& (1,1)
  }
\end{displaymath}
A \textbf{square} in a [[quasi-category]] $C$ is an image of this in $C$, i.e. a morphism

\begin{displaymath}
s : \Delta[1] \times \Delta[1] \to C
  \,.
\end{displaymath}
The simplicial square $\Delta[1]^{\times 2}$ is [[isomorphism|isomorphic]], as a [[simplicial set]], to the [[join of simplicial sets]] of a 2-[[horn]] with the point:

\begin{displaymath}
\Delta[1] \times \Delta[1]
  \simeq
  \{v\} \star \Lambda[2]_2
  = 
  \left(
  \itexarray{
    v &\to& 1
    \\
    \downarrow &\searrow& \downarrow
    \\
    0 &\to& 2
  }
  \right)
\end{displaymath}
and

\begin{displaymath}
\Delta[1] \times \Delta[1]
  \simeq
  \Lambda[2]_0 \star \{v\}
  = 
  \left(
  \itexarray{
    0&\to& 1
    \\
    \downarrow &\searrow& \downarrow
    \\
    2 &\to& v
  }
  \right)
  \,.
\end{displaymath}
If a square $\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C$ exhibits $\{v\} \to C$ as a [[limit in a quasi-category|quasi-categorical limit]] over $F : \Lambda[2]_0 \to C$, we say the limit

\begin{displaymath}
v := \lim_\leftarrow F := F(1) \prod_{F(0)} F(2)
\end{displaymath}
is [[generalized the|the]] \textbf{quasi-categorical pullback} of the diagram $F$.

\hypertarget{QuasiCatPastingLaw}{}\paragraph*{{Pasting law}}\label{QuasiCatPastingLaw}

We have the following quasi-categorical analog of the familiar \href{http://ncatlab.org/nlab/show/pullback#Pasting}{pasting law of pullbacks} in ordinary [[category theory]]:

A [[pasting]] diagram of two squares is a morphism

\begin{displaymath}
\sigma : \Delta[2] \times \Delta[1] \to \mathcal{c}
  \,.
\end{displaymath}
Schematically this looks like

\begin{displaymath}
\itexarray{
     a &\to& b &\to& c
     \\
     \downarrow && \downarrow && \downarrow
     \\
     d &\to& e &\to& f
  }
\end{displaymath}
in $\mathcal{C}$.

\begin{uprop}
\textbf{(pasting law for quasi-categorical pullbacks)}

If the right square is a pullback diagram in $\mathcal{C}$, then the left square is precisely if the outer square is.

\end{uprop}
This is [[Higher Topos Theory|HTT, lemma 4.4.2.1]]

\begin{proof}
Consider the diagram inclusions

\begin{displaymath}
\left(
    \itexarray{
      && && c
      \\
      && && \downarrow
      \\
      d &\to& e &\to& f
    }
  \right)
  \;\;\to\;\;
  \left(
    \itexarray{
      && b &\to& c
      \\
      && \downarrow && \downarrow
      \\
      d &\to& e &\to& f
    }
  \right)
  \;\;\leftarrow\;\;
  \left(
    \itexarray{
      && b 
      \\
      && \downarrow
      \\
      d &\to& e
    }
  \right)
\end{displaymath}
and the induced diagram of [[over quasi-categories]]

\begin{displaymath}
\mathcal{C}_{/\sigma(c,d,f)}
  \stackrel{\phi}{\leftarrow} 
  \mathcal{C}_{/\sigma(b,c,d,e,f)}
  \stackrel{\psi}{\to}
  \mathcal{C}_{/\sigma(b,d,e)}
  \,.
\end{displaymath}
Notice that by definition of [[limit in a quasi-category]] the quasi-categorical pullback $\sigma(c) \times_{\sigma(f)} \sigma(d)$ is [[generalized the|the]] [[terminal object in a quasi-category|terminal object]] in $\mathcal{C}_{/\sigma(c,d,f)}$, while $\sigma(d) \times_{\sigma(e)} \sigma(b)$ is the terminal object in $\mathcal{C}_{/\sigma(b,d,e)}$.

The strategy now is to show that both these morphisms $\phi$ and $\psi$ are acyclic [[Kan fibration]]s. That will imply that these terminal objects coincide as objects of $\mathcal{C}$.

First notice that the inclusion

\begin{displaymath}
\left(
    \itexarray{
      && b 
      \\
      && \downarrow
      \\
      d &\to& e
    }
  \right)
   \;\; \to \;\;
  \left(
    \itexarray{
      && b &\to& c
      \\
      && \downarrow && \downarrow
      \\
      d &\to& e &\to& f
    }
  \right)
\end{displaymath}
is a [[left anodyne morphism]], being the composite of [[pushout]]s of left [[horn]] inclusions

\begin{displaymath}
\begin{aligned}  
  \left(
    \itexarray{
      && b 
      \\
      && \downarrow
      \\
      d &\to& e
    }
  \right)
  & \to 
  \left(
    \itexarray{
      && b &\to& c
      \\
      && \downarrow &&       
      \\
      d &\to& e
    }
  \right)
  \\
  & \to 
  \left(
    \itexarray{
      && b &\to& c
      \\
      && \downarrow && \downarrow 
      \\
      d &\to& e && f
    }
  \right)
  \\
  & \to 
  \left(
    \itexarray{
      && b &\to& c
      \\
      && \downarrow &\searrow& \downarrow 
      \\
      d &\to& e && f
    }
  \right)
  \\
  & \to 
  \left(
    \itexarray{
      && b &\to& c
      \\
      && \downarrow &\searrow& \downarrow 
      \\
      d &\to& e &\to& f
    }
  \right)
 \end{aligned}
  \,.
\end{displaymath}
We could also prove this by showing that this functor is [[homotopy final functor|homotopy initial]] using the characterization in terms of slice categories, and then invoking the theorem of [[HTT]] 4.1.1.3(4) which says (in dual form) that an inclusion of simplicial sets is homotopy initial if and only if it is left anodyne.

One of the  is that restriction of [[over quasi-categories]] along left anodyne morphisms produces an acyclic [[Kan fibration]]. This shows the desired statement for $\psi$.

To see that $\phi$ is also an acyclic fibration observe that $\phi$ can be factored as

\begin{displaymath}
\mathcal{C}_{/\sigma(c,d,f)} \leftarrow 
\mathcal{C}_{/\sigma(c,d,e,f)} \leftarrow 
\mathcal{C}_{/\sigma(b,c,d,e,f)}
\end{displaymath}
Observe that $\mathcal{C}_{/\sigma(c,d,e,f)}\leftarrow\mathcal{C}_{/\sigma(b,c,d,e,f)}$ fits into a pullback diagram

\begin{displaymath}
\itexarray{
      \mathcal{C}_{/\sigma(c,d,e,f)} 
      & \leftarrow &
      \mathcal{C}_{/\sigma(b,c,d,e,f)} \\ 
      \downarrow & & \downarrow \\ 
     \mathcal{C}_{/\sigma(c,e,f)} 
     & \leftarrow & 
     \mathcal{C}_{/\sigma(b,c,e,f)} 
    }
\end{displaymath}
and hence is an acyclic Kan fibration since $\mathcal{C}_{/\sigma(c,e,f)} \leftarrow \mathcal{C}_{/\sigma(b,c,e,f)}$ is one, on account of the fact that the square

\begin{displaymath}
\itexarray{ 
        \sigma(b) & \to & \sigma(c) \\ 
        \downarrow & & \downarrow \\ 
        \sigma(e) & \to & \sigma(f) 
   }
\end{displaymath}
is a pullback in $\mathcal{C}$. Finally, $\mathcal{C}_{/\sigma(c,d,f)} \leftarrow \mathcal{C}_{/\sigma(c,d,e,f)}$ is a trivial fibration since

\begin{displaymath}
\itexarray{ 
      \left( 
          \itexarray{ 
              & & c \\ 
              & & \downarrow \\ 
           d & \to & f 
          } 
     \right) & \to & 
     \left( 
        \itexarray{
           & & & & c \\ 
           & & & & \downarrow \\ 
         d & \to & e & \to & f 
         }
     \right) 
   }
\end{displaymath}
is left anodyne; clearly this is a pushout of $(d\to f)\to (d\to e\to f)$ and so it suffices to show that $\Delta^{\{0,2\}}\to \Delta^{\{0,1,2\}}$ is left anodyne. But this map factors as $\Delta^{\{0,2\}}\to \Lambda^2_0 \to \Delta^{\{0,1,2\}}$ and clearly $\Delta^{\{0,2\}}\to \Lambda^2_0$ is left anodyne since it is a pushout of $\Delta^{\{0\}}\to 
\Delta^{\{0,1\}}$.

\end{proof}
\hypertarget{in_model_categories}{}\subsubsection*{{In model categories}}\label{in_model_categories}

\begin{itemize}%
\item [[homotopy pullback]]

\end{itemize}
\hypertarget{in_derivators}{}\subsubsection*{{In derivators}}\label{in_derivators}

\begin{itemize}%
\item [[pullback in a derivator]]

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

\hypertarget{fiber_sequence}{}\subsubsection*{{Fiber sequence}}\label{fiber_sequence}

If $\mathcal{C}$ has a [[terminal object]] and $* \to C \in \mathcal{C}$ is a [[pointed object]], then the \textbf{fiber} or \textbf{$(\infty,1)$-kernel} of a morphisms $f : B \to C$ is the $(\infty,1)$-pullback

\begin{displaymath}
\itexarray{
    ker(f) &\to& *
    \\
    \downarrow && \downarrow
    \\
    B &\stackrel{f}{\to}& C
  }
  \,.
\end{displaymath}
For more on this see [[fiber sequence]].

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

\hypertarget{in_homotopy_type_theory}{}\subsubsection*{{In homotopy type theory}}\label{in_homotopy_type_theory}

A formalization of homotopy pullbacks in [[homotopy type theory]] is [[Coq]]-coded in

\begin{itemize}%
\item [[Guillaume Brunerie]], \emph{\href{https://github.com/guillaumebrunerie/HoTT/blob/master/Coq/Limits/Pullbacks.v}{Hott/Coq/Limits/Pullbacks.v}}

\end{itemize}
[[!redirects (∞,1)-pullback]] [[!redirects (infinity,1)-pullbacks]] [[!redirects (∞,1)-pullbacks]]

[[!redirects (∞,1)-fiber product]] [[!redirects (∞,1)-fiber products]]

[[!redirects (infinity,1)-fiber product]] [[!redirects (infinity,1)-fiber products]]



\end{document}