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\begin{document}

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\section*{cograph of a functor}

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

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

[[!include category theory - contents]]

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

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{cographs_of_functors_between_0categories}{Cographs of functors between 0-categories}\dotfill \pageref*{cographs_of_functors_between_0categories} \linebreak
\noindent\hyperlink{cographs_of_functors_between_1categories}{Cographs of functors between 1-categories}\dotfill \pageref*{cographs_of_functors_between_1categories} \linebreak
\noindent\hyperlink{AdjointFunctorsInTermsOfCographs}{Adjoint functors in terms of cographs}\dotfill \pageref*{AdjointFunctorsInTermsOfCographs} \linebreak
\noindent\hyperlink{cographs_of_functors_between_categories}{Cographs of functors between $(\infty,1)$-categories}\dotfill \pageref*{cographs_of_functors_between_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


The notion of \emph{cograph of a functor} is dual to that of [[graph of a functor]]: for $f : C \to D$ a functor between [[n-categories]] is the fibration classified by the [[profunctor]] correspondence $\chi_f : C^{op}\times D \to (n-1)Cat$. But $f$ also determines a morphism $\bar f : I \to n Cat$ from the [[interval category]] $I$. The \textbf{cograph} of $f$ is the fibration classified by $\bar f$.

Recall that the graph of a function $f: A \to B$ is the [[subset]] determined by the [[monomorphism]] $\langle 1, f \rangle: A \to A \times B$. This makes sense in any category with products. Under one definition, the notion of cograph of a function $f: A \to B$ is the categorically dual notion: it is the [[quotient object|quotient]] determined by the [[epimorphism]] $(f, 1): A \sqcup B \to B$.

A more vivid presentation of cograph is given by the pictures we draw of functions $f: A \to B$: as [[directed graph]]s (in the graph-theoretic sense!) whose vertices are elements in $A \sqcup B$, with an edge drawn from $a$ to $f(a)$ for each $a$ in $A$. This can also be conceived as a [[poset]] $P_f$ with underlying set $A \sqcup B$, in which $a \leq f(a)$ and all other instances of $\leq$ are the reflexive ones. The connected components function $(P_f)_0 \to \pi_0(P_f)$ is then the cograph in the sense given above.

In this article we give a definition of cograph which generalizes this poset picture of cograph of a function, and which applies to any functor between $n$-categories.

\hypertarget{examples}{}\section*{{Examples}}\label{examples}

\hypertarget{cographs_of_functors_between_0categories}{}\subsection*{{Cographs of functors between 0-categories}}\label{cographs_of_functors_between_0categories}

In the case that $C, D$ are [[0-category|0-categories]], i.e. [[sets]], a functor $f : C \to D$ is just a [[function]] between sets. The cograph [[2-pullback]]

\begin{displaymath}
\itexarray{
    cograph(f) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    I &\stackrel{\bar f}{\to}& Set
  }
\end{displaymath}
is computed by the ordinary [[pullback]]

\begin{displaymath}
\itexarray{
    cograph(f) &\to& Set_{*}
    \\
    \downarrow && \downarrow
    \\
    I &\stackrel{\bar f}{\to}& Set
  }
\end{displaymath}
and identifies $cograph(f)$ with the [[category of elements]] of $\bar f$, as described there: the objects of $cograph(f)$ are the disjoint union of $C$ and $D$: $Obj(cograph(f)) = C \coprod D$ and the nontrivial morphisms are of the form $x \to y$ whenever $x \in C$, $y \in D$ and $f(x) = y$.

What [[Bill Lawvere]] called the [[cograph of a function]] is the connected components $\pi_0(Cograph(f))$ of this category.

\hypertarget{cographs_of_functors_between_1categories}{}\subsection*{{Cographs of functors between 1-categories}}\label{cographs_of_functors_between_1categories}

For $f : C \to D$ an ordinary [[functor]], $cograph(f)$ is the category with $Obj(cograph(f)) = Obj(C) \coprod Obj(D)$ and with

\begin{displaymath}
Hom_{cograph(f)}(x,y) = 
  \left\{
    \itexarray{
      Hom_C(x,y) & if\; x,y \in C
      \\
      Hom_D(x,y) & if\; x,y \in D
      \\
      Hom_D(f(x),y) & if\; x \in C ,y \in D
      \\
      \emptyset & if\; x \in D ,y \in C
    }
  \right.
\end{displaymath}
with composition defined as induced from $C$, from $D$, and from the action of $f$. This is a special case of the [[cograph of a profunctor]], specialized to the representable [[profunctor]] $D(f-,-)$.

This cograph is denoted $C \star^f D$ in section 2.3.1 (\_Correspondences\_) in

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

\end{itemize}
where it is understood as a generalization of the [[join of simplicial sets]] and where it serves as a motivation for the study of cographs of functors between [[(∞,1)-category|(∞,1)-categories]] discussed below.

\hypertarget{AdjointFunctorsInTermsOfCographs}{}\subsection*{{Adjoint functors in terms of cographs}}\label{AdjointFunctorsInTermsOfCographs}

As emphasized in the beginning of section 5.2 there, cographs of functors may be used to characterize [[adjoint functors]]. This is just one way of stating the characterization of adjoints in terms of [[profunctors]] (which in turn makes sense in any [[2-category equipped with proarrows]]).

\begin{prop}
\label{}\hypertarget{}{}
Two functors $L : C \to D$ and $R : D \to C$ are [[adjoint functors]] precisely if $cograph(L)$ and $cograph(R^{op})^{op}$ are isomorphic under $C$ and $D$:

\begin{displaymath}
(L \dashv R) \Leftrightarrow 
  (cograph(L) \cong cograph(R^{op})^{op})
\end{displaymath}
where the isomorphism is in the [[co-slice category]] $(C\sqcup D)/Cat$.

\end{prop}
More precisely, there is a bijection between adjunctions $L\dashv R$ and isomorphisms as above.

\begin{proof}
The category $cograph(L)$ is the category with $Obj(cograph(L)) = Obj(C) \coprod Obj(D)$ and with

\begin{displaymath}
Hom_{cograph(L)}(x,y) = 
  \left\{
    \itexarray{
      Hom_C(x,y) & if\; x,y \in C
      \\
      Hom_D(x,y) & if\; x,y \in D
      \\
      Hom_D(L(x),y) & if\; x \in C ,y \in D
      \\
      \emptyset & if\; x \in D ,y \in C
    }
  \right.
\end{displaymath}
(This set is also called the set of [[heteromorphisms]] between objects in $C$ and $D$.)

The category $cograph(R^{op})^{op}$ accordingly is the category with $Obj(cograph(R)) = Obj(C) \coprod Obj(D)$ and with

\begin{displaymath}
Hom_{cograph(R^{op})^{op}}(x,y) = 
  \left\{
    \itexarray{
      Hom_C(x,y) & if x,y \in C
      \\
      Hom_D(x,y) & if x,y \in D
      \\
      Hom_C(x,R(y)) & if x \in C ,y \in D
      \\
      \emptyset & if x \in D ,y \in C
    }
  \right.
\end{displaymath}
Evidently these categories are isomorphic under $C$ and $D$ precisely if for all $x \in C, y \in D$ we have

\begin{displaymath}
Hom_D(L(x),y) \cong Hom_C(x,R(y))
  \,.
\end{displaymath}
naturally in $x$ and $y$. It is natural because the isomorphism is an isomorphism of categories, and the functoriality of $Hom_D(L(-),-)$ and $Hom_C(-,R(-))$ is encoded by composition in the cograph. Of course, such a natural isomorphism is precisely the structure of an adjunction $L\dashv R$.

\end{proof}
Note also that just as $cograph(L)$ is the cograph of the profunctor $D(L-,-)$, also $cograph(R^{op})^{op}$ is the cograph of the profunctor $C(-,R-)$. Thus, this theorem can be viewed as one way of stating the characterization of adjunctions in terms of homsets, as can be \href{http://ncatlab.org/nlab/show/2-category+equipped+with+proarrows#HomsetAdjn}{formulated} in terms of profunctors in any 2-category equipped with proarrows.

\hypertarget{cographs_of_functors_between_categories}{}\subsection*{{Cographs of functors between $(\infty,1)$-categories}}\label{cographs_of_functors_between_categories}

In the context of [[(∞,1)-category]] theory there is a good theory of [[Cartesian fibrations]] $X \to S$ and of their classification by [[(∞,1)-functors]] $S^{op} \to (\infty,1)Cat$ to the [[(∞,1)-category of (∞,1)-categories]] as described at [[universal fibration of (∞,1)-categories]].

Accordingly, the above notion of cograph of a functor has a direct generalization to [[(∞,1)-functors]]:

\begin{defn}
\label{}\hypertarget{}{}
For $f : C \to D$ an [[(∞,1)-functor]], identified with a morphism

\begin{displaymath}
\bar f : I \to (\infty,1)Cat
\end{displaymath}
in the [[(∞,1)-category of (∞,1)-categories]], it \textbf{cograph} is the [[Cartesian fibration]] $cograph(f) \to I$ classified by it. In terms of the [[universal fibration of (∞,1)-categories]] this is the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    cograph(f) &\to& S^{op}
    \\
    \downarrow && \downarrow
    \\
    I &\stackrel{\bar f}{\to}& (\infty,1)Cat
  }
  \,.
\end{displaymath}
\end{defn}
As for every [[Cartesian fibration]] the functor $f : C \to D$ is determined uniquely up to equivalence by its cograph. In general, obtaining the classifying $(\infty,1)$-functor from a given [[Cartesian fibration]] may be difficult. In the special case of cographs as Cartesian fibrations over the simple [[interval category]] it is easier. This is discussed in the following:

\ldots{}

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

\begin{itemize}%
\item [[graph of a functor]]


\item \textbf{cograph of a functor}, [[cograph of a profunctor]]



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

The notion of cographs of $(\infty,1)$-functors and the theory of how to re-obtain $(\infty,1)$-functors from their cographs is the content of section 5.2.1, \emph{Correspondences and associated functors}, of

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

\end{itemize}


\end{document}