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\begin{document}

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\section*{codomain fibration}

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

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

[[!include category theory - contents]]

\hypertarget{universes}{}\paragraph*{{Universes}}\label{universes}

[[!include universe - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{the_arrow_category}{The arrow category}\dotfill \pageref*{the_arrow_category} \linebreak
\noindent\hyperlink{the_functor}{The functor}\dotfill \pageref*{the_functor} \linebreak
\noindent\hyperlink{the_opcartesian_lifts}{The op-cartesian lifts}\dotfill \pageref*{the_opcartesian_lifts} \linebreak
\noindent\hyperlink{the_cartesian_lifts}{The cartesian lifts}\dotfill \pageref*{the_cartesian_lifts} \linebreak
\noindent\hyperlink{direct_image_operation}{Direct image operation}\dotfill \pageref*{direct_image_operation} \linebreak
\noindent\hyperlink{inverse_image_operation}{Inverse image operation}\dotfill \pageref*{inverse_image_operation} \linebreak
\noindent\hyperlink{adjointness_of_direct_and_inverse_image}{Adjointness of direct and inverse image}\dotfill \pageref*{adjointness_of_direct_and_inverse_image} \linebreak
\noindent\hyperlink{monadic_descent}{Monadic descent}\dotfill \pageref*{monadic_descent} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{in_2category_theory}{In 2-category theory}\dotfill \pageref*{in_2category_theory} \linebreak
\noindent\hyperlink{InInfinityCategoryTheory}{In $(\infty,1)$-category theory}\dotfill \pageref*{InInfinityCategoryTheory} \linebreak
\noindent\hyperlink{as_a_universe}{As a universe}\dotfill \pageref*{as_a_universe} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $C$ any [[category]], there is a [[functor]]

\begin{displaymath}
t : [I,C] \to C
  \,,
\end{displaymath}
from the [[arrow category]] $[I,C] = Arr(C)$ that sends each [[morphism]] $(c_1 \stackrel{f}{\to} c_2) \in [I,C]$ to its codomain $c_2$.

This functor is always an [[Grothendieck fibration|opfibration]]. It corresponds under the [[Grothendieck construction]] to the [[pseudofunctor]]

\begin{displaymath}
C/(-) : C \to Cat
\end{displaymath}
that sends each object $c \in C$ to the [[overcategory]] $C/c$.

If $C$ has all [[pullback]]s, then the functor is in addition a [[Grothendieck fibration|fibration]], hence a [[bifibration]]. Traditionally, though, its fibered aspect is emphasised (and it even motivates the notion of cartesianess for categories over categories). A right adjoint $u_*$ of $u^*$ exists for every morphism $u$ in $C$ iff C is a [[locally cartesian closed category]].

This functor $cod : [I,C] \to C$ is called the \textbf{codomain fibration} of $C$.

Some say \textbf{basic fibration} or \textbf{[[indexed category|self-indexing]]} or the \textbf{fundamental fibration} --- anything with so many names must be important!

If instead of the codomain the domain is used, one obtains the dual notion: [[domain opfibration]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

We spell out the details of the functor, of its cartesian and opcartesian morphisms and their properties.

\hypertarget{the_arrow_category}{}\subsubsection*{{The arrow category}}\label{the_arrow_category}

Recall from the discussion at [[arrow category]] that the objects in $Arr(C)$ are morphisms in $C$ and the morphisms $(f:x_1\to x_2)\to (g: y_1\to y_2)$ in $Arr(C)$ are the commutative squares in $C$ of the form

\begin{displaymath}
\itexarray{
    x_1 &\stackrel{v}\to& y_1
    \\
    \downarrow\mathrlap{f} 
    &&
   \downarrow\mathrlap{g}
   \\
   x_2 &\stackrel{u}\to& y_2
}
\end{displaymath}
with the obvious composition.

\hypertarget{the_functor}{}\subsubsection*{{The functor}}\label{the_functor}

The [[functor]] $cod : Arr(C)\to C$ is given on objects by the codomain (= target) map, and on morphisms it gives the lower arrow of the commutative square.

\begin{displaymath}
cod : 
  \;\;
  \left(
  \itexarray{
    x_1 &\stackrel{v}\to& y_1
    \\
    \downarrow\mathrlap{f} 
    &&
   \downarrow\mathrlap{g}
   \\
   x_2 &\stackrel{u}\to& y_2
  }
  \right)
  \;\;
  \mapsto
  \;\;
  (x_2 \stackrel{u}\to y_2)
  \,.
\end{displaymath}
If we write $[I,C]$ for the [[arrow category]], where $I$ is the [[interval category]] $I = \{a \to b\}$, then this functor is the [[hom-functor]] applied to the inclusion $\iota_1 : {b} \to \{a \to b\}$

\begin{displaymath}
cod = Hom_C(\iota_1, -) : [I,C] \to [{*}, C] = C
  \,.
\end{displaymath}
\hypertarget{the_opcartesian_lifts}{}\subsubsection*{{The op-cartesian lifts}}\label{the_opcartesian_lifts}

That the functor $cod : [I,C] \to C$ is an [[Grothendieck fibration|opfibration]] means that for each object $\hat c_1 \to c_1$ of $[I,C]$, each morphism $c_1 \stackrel{f}{\to} c_2$ in $C$ has a lift to a morphism

\begin{displaymath}
\itexarray{
     \hat c_1 &\to& ??
     \\
     \downarrow && \downarrow
     \\
     c_1 &\to& c_2
  }
\end{displaymath}
in $[I,C]$ that is a [[cartesian morphism|opcartesian morphism]].

Such a lift is given by

\begin{displaymath}
\itexarray{
     \hat c_1 &\stackrel{Id}{\to}& \hat c_1
     \\
     \downarrow && \downarrow
     \\
     c_1 &\to& c_2
  }
  \,.
\end{displaymath}
For given any commuting triangle

\begin{displaymath}
\itexarray{
     && c_2
     \\
     & \nearrow && \searrow
     \\
     c_1 &&\to&& c_3
  }
\end{displaymath}
in $C$, and any lift

\begin{displaymath}
\itexarray{
    \hat c_1 &\to& d 
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_3
  }
\end{displaymath}
of $c_1 \to c_3$, there is the unique lift

\begin{displaymath}
\itexarray{
    \hat c_1 &\to& d
    \\
    \downarrow && \downarrow
    \\
    c_2 &\to& c_3
  }
\end{displaymath}
such that

\begin{displaymath}
\left(
  \itexarray{
    \hat c_1 &\stackrel{Id}{\to}& \hat c_1
    &\to& d
    \\
    \downarrow && \downarrow && \downarrow
    \\
    c_1 &\to& c_2 &\to& c_3
  }
  \right)
  \;\;\;
  = 
  \;\;\;
  \itexarray{
    \hat c_1 
    &\to& d
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_3
  }  
  \,.
\end{displaymath}
\hypertarget{the_cartesian_lifts}{}\subsubsection*{{The cartesian lifts}}\label{the_cartesian_lifts}

If $C$ has [[pullback]]s, then $cod : [I,C] \to C$ is in addition a [[Grothendieck fibration|fibered category]] in the sense of Grothendieck:

for every object $\hat c_2 \to c_2$ in $[I,C]$, the cartesian lift of a morphism $c_1 \to c_2$ in $C$ is given by the morphism

\begin{displaymath}
\itexarray{
    c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_2
  }
  \,.
\end{displaymath}
Because for

\begin{displaymath}
\itexarray{
     && c_3
     \\
     & \swarrow && \searrow
     \\
     c_1 &&\to&& c_2
  }
\end{displaymath}
any commuting triangle in $C$, and for

\begin{displaymath}
\itexarray{
    d &\to& \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_3 &\to& c_2
  }
\end{displaymath}
any lift of $c_3 \to c_2$ in $[I,C]$, which by the commutativity of the triangle we may write as

\begin{displaymath}
\itexarray{
    d &\to& &\to& \hat c_2
    \\
    \downarrow && && \downarrow
    \\
    c_3 &\to& c_1 &\to& c_2
  }
\end{displaymath}
there is, precisely by the universal property of the [[pullback]], a unique morphism, $d\to c_1 \times_{c_2} \hat c_2$ in $C$ such that this factors as

\begin{displaymath}
\itexarray{
    d &\to& c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
    \\
    \downarrow && \downarrow && \downarrow
    \\
    c_3 &\to& c_1 &\to& c_2
  }
  \,.
\end{displaymath}
\hypertarget{direct_image_operation}{}\subsubsection*{{Direct image operation}}\label{direct_image_operation}

Recall that in an [[Grothendieck fibration|opfibration]] $p : E\to B$ , the \emph{direct image} $f_!$ of an object $e \in E$ along a morphism $p(e) \to d$ is the codomain $f_!(e)$ of [[generalized the|the]] opcartesian lift $\hat f : e \to f_! e$ of $f$.

By the above discussion this means that in the codomain opfibration $cod : [I,C] \to C$ the direct image of an object $\hat c_1 \to c_1$ in $[I,C]$ along some morphism $f : c_1 \to c_2$ is the composite morphism $\hat c_1 \to c_1 \to c_2$ in $C$, regarded as an object in $[I,C]$: this yields the functor

\begin{displaymath}
f_! : C/{c_1} \to C/{c_2}
\end{displaymath}
of [[overcategory|overcategories]] obained by postcomposition with $f$.

\hypertarget{inverse_image_operation}{}\subsubsection*{{Inverse image operation}}\label{inverse_image_operation}

Recall that in an [[Grothendieck fibration|fibration]] $p : E\to B$ , the \emph{inverse image} $f^*$ of an object $e \in E$ along a morphism $d \to p(e)$ is the domain $f^*(e)$ of [[generalized the|the]] cartesian lift $\hat f : f^* e \to e$ of $f$.

By the above discussion this means that in the codomain fibration $cod : [I,C] \to C$ the inverse image of an object $\hat c_2 \to c_2$ in $[I,C]$ along some morphism $f : c_1 \to c_2$ is the morphism out of the [[pullback]] $f^* c_2 = c_1 \times_{c_2} \hat c_2 \to c_1$ in $C$, regarded as an object in $[I,C]$: this yields the functor

\begin{displaymath}
C/{c_1} \leftarrow C/{c_2} : f^*
\end{displaymath}
of [[overcategory|overcategories]] obained by pullback.

\hypertarget{adjointness_of_direct_and_inverse_image}{}\subsubsection*{{Adjointness of direct and inverse image}}\label{adjointness_of_direct_and_inverse_image}

For every morphism $f : c_1 \to c_2$ in $C$, the direct and inverse image functors are a pair of [[adjoint functor]]s

\begin{displaymath}
f_! : C/{c_1} \to C/{c_2} : f^*
\end{displaymath}
with $f_!$ [[left adjoint]] and $f^*$ [[right adjoint]], $f_! \dashv f^*$.

By the above discussion, the adjunction isomorphism

\begin{displaymath}
Hom_{C_2}(f_! \hat c_1, \hat c_2)
  \simeq
  Hom_{C_1}(\hat c_1, f^*\hat c_2)
\end{displaymath}
is given by the universal property of the [[pullback]] operation, which says that morphisms

\begin{displaymath}
(f_! \hat c_1 \to \hat c_2)
  =
  \left(
  \itexarray{
    \hat c_1 &\to& \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_2
  }
  \right)
\end{displaymath}
factor uniquely through the [[pullback]]

\begin{displaymath}
\left(
  \itexarray{
    \hat c_1 &\to& c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
    \\
    &\searrow & \downarrow && \downarrow
    \\
    && c_1 &\to& c_2
  }
  \right)
\end{displaymath}
and hence uniquely correspond to morphisms

\begin{displaymath}
(\hat c_1 \to f^* \hat c_2)
  =
  \left(
  \itexarray{
    \hat c_1 &\to& c_1 \times_{c_2} \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_2
  }
  \right)
  \,.
\end{displaymath}
If $C$ is a [[model category]], and $u:c\to d$ a morphism in $C$, we can consider the induced model structure on the overcategories $C/c$, and $C/d$. The adjoint pair

\begin{displaymath}
u_! : C/c \leftrightarrows  C/d : u^*
\end{displaymath}
is then a Quillen pair.

\hypertarget{monadic_descent}{}\subsection*{{Monadic descent}}\label{monadic_descent}

Since the codomain fibration $cod : [I,C] \to C$ is a [[bifibration]] when $C$ has all [[pullback]]s, there is a notion of [[monadic descent]] in this case. Details on this are at \href{http://ncatlab.org/nlab/show/monadic+descent#ForCodomainFibs}{monadic descent for codomain fibrations}.

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

We discuss the codomain fibration in [[higher category theory]].

\hypertarget{in_2category_theory}{}\subsubsection*{{In 2-category theory}}\label{in_2category_theory}

A categorification in dimension 2 (see [[2-category theory]]) is a codomain 2-fibration, whose main example is $Cat^2$ over $Cat$.

[[Mike Shulman]]: I still don't believe that that is a 2-fibration. How do you lift the 2-cells?

[[David Roberts]]: How does one lift the 2-cells in a 2-fibration anyway? The case of $Cat^\mathbf{2}\to Cat$ (using weak 2-functors in $Cat^\mathbf{2}$) should in my opinion be an guiding example for this. Although, perhaps it would be better to consider (at least at first) the underlying (2,1)-category or even the (2,1)-category $Gpd$.

[[Mike Shulman]]: I think the guiding example of a 2-fibration should actually be $Fib \to Cat$, as in \href{http://www.cs.math.ist.utl.pt/s84.www/cs/claudio/articles/2-fib.ps}{Hermida's paper}. There, you can lift the 2-cells, because in each fibration you can lift the 1-cells.

\hypertarget{InInfinityCategoryTheory}{}\subsubsection*{{In $(\infty,1)$-category theory}}\label{InInfinityCategoryTheory}

Let $\mathcal{X}$ be an [[(∞,1)-category]].

\begin{prop}
\label{}\hypertarget{}{}
The codomain fibration

\begin{displaymath}
Cod : \mathcal{X}^I \to \mathcal{X}
\end{displaymath}
is an [[coCartesian fibration]].

It is classified under the [[(∞,1)-Grothendieck construction]] by

\begin{displaymath}
A \mapsto \mathcal{X}_{/A}
  \,,
\end{displaymath}
where on the right we have the [[over-(∞,1)-category]].

\end{prop}
This is a special case of (\hyperlink{Lurie}{Lurie, corollary 2.4.7.12}).

For $\mathcal{X}$ an [[(∞,1)-topos]], this is the canonical [[(infinity,2)-sheaf]].

\hypertarget{as_a_universe}{}\subsection*{{As a universe}}\label{as_a_universe}

(\ldots{})

$\mathbf{H}$ an [[(∞,1)-topos]] the codomain fibration is the [[dependent sum]]

\begin{displaymath}
\sum_{Type} : \mathbf{H}_{/Type} \to \mathbf{H}_{/*} \simeq \mathbf{H}
\end{displaymath}
where $Type \in \mathbf{H}$ is the [[object classifier]], of some size. This is the internal [[universe]]. Since the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ is the [[context]] given by $X$, in a precise sense $\mathbf{H}_{/Type}$ is the ``context of the universe''. And so this says that the codomain fibration is the ``context of the universe'' regarded over the base $\infty$-topos which is the ``outermost universe''.

(\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[tangent category]]


\item [[tangent (infinity,1)-category]]



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

Secton 2.4.7 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
[[!redirects codomain fibrations]]

[[!redirects codomain opfibration]] [[!redirects codomain opfibrations]] [[!redirects self-indexing]] [[!redirects self-indexing of a category]]



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