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\begin{document}

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\section*{comma category}

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

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

[[!include category theory - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InComponents}{Via components: the objectwise definition}\dotfill \pageref*{InComponents} \linebreak
\noindent\hyperlink{AsAFiberProduct}{Via fiber products in the 1-category Cat}\dotfill \pageref*{AsAFiberProduct} \linebreak
\noindent\hyperlink{AsA2Limit}{Via 2-category theory: as a 2-limit}\dotfill \pageref*{AsA2Limit} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{completeness_and_cocompleteness}{Completeness and cocompleteness}\dotfill \pageref*{completeness_and_cocompleteness} \linebreak
\noindent\hyperlink{functors_and_comma_categories}{Functors and comma categories}\dotfill \pageref*{functors_and_comma_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{comma category} of two [[functors]] $f : C \to E$ and $g : D \to E$ is a [[category]] like an [[arrow category]] of $E$ where all arrows have their [[source]] in the image of $f$ and their [[target]] in the image of $g$ (and the morphisms between arrows keep track of how these sources and targets are in these images). It can also be seen a kind of [[2-limit]]: a [[directed homotopy theory|directed]] refinement of the [[homotopy pullback]] of two functors between [[groupoids]].

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

We discuss three equivalent definitions of comma categories

\begin{itemize}%
\item \hyperlink{InComponents}{Explicitly in components}


\item \hyperlink{AsAFiberProduct}{As a fiber product}


\item \hyperlink{AsA2Limit}{As a 2-limit}



\end{itemize}
$\backslash$begin\{rmk\} The terminology ``comma category'' is a holdover from the original notation $(f,g)$ for such a category, which generalises $(x,y)$ or $C(x,y)$ for a [[hom-set]]. This is rarely used any more. More common modern notations for the comma category are $(f/g)$, which we will use on this page, and $(f\downarrow g)$. $\backslash$end\{rmk\}

\hypertarget{InComponents}{}\subsubsection*{{Via components: the objectwise definition}}\label{InComponents}

$\backslash$begin\{defn\} If $f:C\to E$ and $g:D\to E$ are [[functors]], their \textbf{comma category} is the category $(f/g)$ whose

\begin{itemize}%
\item [[objects]] are triples $(c,d,\alpha)$ where $c\in C$, $d\in D$, and $\alpha:f(c)\to g(d)$ is a morphism in $E$, and whose


\item [[morphisms]] from $(c_1,d_1,\alpha_1)$ to $(c_2,d_2,\alpha_2)$ are pairs $(\beta,\gamma)$, where $\beta:c_1\to c_2$ and $\gamma:d_1\to d_2$ are morphisms in $C$ and $D$, respectively, such that $\alpha_2 . f(\beta) = g(\gamma) . \alpha_1$.



\end{itemize}
\begin{displaymath}
\itexarray{
    f(c_1)
    &\stackrel{f(\beta)}{\rightarrow}&
    f(c_2)
    \\
    \downarrow^{\alpha_1}
    &&
    \downarrow^{\alpha_2}
    \\
    g(d_1)
    &\stackrel{g(\gamma)}{\to}&
    g(d_2)    
    \\
    \\
    (c_1,d_1, \alpha_1)
    &\stackrel{(\beta,\gamma)}{\to}&
    (c_2,d_2, \alpha_2)
  }
\end{displaymath}
\begin{itemize}%
\item [[composition]] of morphisms is given on components by composition in $C$ and $D$.

\end{itemize}
$\backslash$end\{defn\}

The definition of $(f/g)$ is now complete. In addition, there are two canonical [[forgetful functors]] defined on the comma category:

\begin{itemize}%
\item there is a functor $H_C\colon (f/g)\rightarrow C$ which sends each object $(c,d,\alpha)$ to $c$, and each pair $(\beta,\gamma)$ to $\beta$.


\item there is a functor $H_D\colon (f/g)\rightarrow D$ which sends each object $(c,d,\alpha)$ to $d$, and each pair $(\beta,\gamma)$ to $\gamma$.



\end{itemize}
Furthermore:

\begin{itemize}%
\item there is a [[natural transformation]] $\theta : f \circ H_C \to g\circ H_D$ defined by $\theta_{(c,d,\alpha)} = \alpha$.

\end{itemize}
These functors and natural transformation together give the comma category a 2-categorical universal property; see \hyperlink{AsA2Limit}{this section} for more.

\hypertarget{AsAFiberProduct}{}\subsubsection*{{Via fiber products in the 1-category Cat}}\label{AsAFiberProduct}

Let $I = \{a \to b\}$ be the (directed) [[interval category]] and $E^I = Funct(I,E)$ the [[functor category]].

The comma category is the [[pullback]]

\begin{displaymath}
\itexarray{
    (f/g) 
    &\to&
    E^I
    \\
    \downarrow & (pb) & \downarrow^{\mathrlap{(F\mapsto F(a))\times(F\mapsto F(b))}}
    \\
    C \times D
    &\stackrel{f \times g}{\to}&
    E \times E
  }
\end{displaymath}
in the standard sense of pullback of morphisms in the 1-category [[Cat]] of categories.

Compare this with the construction of [[homotopy pullback]] (\href{homotopy+pullback#HomotopyPullbackByFactorizationLemma}{here}), hence with the definition of [[loop space object]] and also with [[generalized universal bundle]].

\hypertarget{AsA2Limit}{}\subsubsection*{{Via 2-category theory: as a 2-limit}}\label{AsA2Limit}

The comma category is the [[comma object]] of the [[cospan]] $C\overset{f}{\rightarrow}E\overset{g}{\leftarrow}D$ in the [[2-category]] $Cat$. This means it is an appropriate [[weighted limit|weighted]] 2-categorical [[2-limit|limit]] (in fact, a [[strict 2-limit]]) of the diagram

\begin{displaymath}
\itexarray{
    && C
    \\
    && \downarrow^f
    \\
    D &\stackrel{g}{\to}& E
  }
\end{displaymath}
Specifically, it is the universal [[span]] making the following square commute up to a specified [[natural transformation]] (such a universal square is in general called a [[comma square]]):

\begin{displaymath}
\itexarray{
    (f/g) &\overset{H_C}{\to}& C
    \\
    \mathllap{{}^{H_D}} \downarrow &\swArrow& \downarrow^{\mathrlap{f}}
    \\
    D &\stackrel{g}{\to}& E
  }
\end{displaymath}
(Sometimes this is called a ``lax pullback'', but that terminology properly refers to something else; see [[comma object]] and [[2-limit]].)

Notably, the forgetful functors $H_C$ and $H_D$ from the ``objectwise'' definition are thus recovered via a categorical construction: they are the projections from the summit of the ``appropriate'' 2-categorical limit.

In terms of the imagery of [[loop space objects]], the comma category is the category of [[interval object|directed paths]] in $E$ which start in the image of $f$ and end in the image of $g$.

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

\begin{itemize}%
\item If $f$ and $g$ are both the identity functor of a category $C$, then $(f/g)$ is the category $C ^{\mathbf{2}}$ of arrows in $C$.


\item If $f$ is the identity functor of $C$ and $g$ is the inclusion $1\to C$ of an object $c\in C$, then $(f/g)$ is the [[over category|slice category]] $C/c$.


\item Likewise if $g$ is the identity and $f$ is the inclusion of $c$, then $(f/g)$ is the [[under category|coslice category]] $c/C$.


\item A [[natural transformation]] $\tau \colon F \to G$ with $F,G :\colon C\to D$ may be regarded as a [[functor]] $T \colon C\to (F/G)$ with $T(c)=(c,c,\tau_c)$ and $T(f)=(f,f)$. Conversely, any such functor $T$ such that the two projections from $(F/G)$ back to $C$ are both [[left inverses]] for $T$ yields a corresponding natural transformation. This is an expression of the universal property of $(F/G)$ as a [[comma object]].



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

\hypertarget{completeness_and_cocompleteness}{}\subsubsection*{{Completeness and cocompleteness}}\label{completeness_and_cocompleteness}

If $C$ and $D$ are [[complete category|complete]] and $f: C \to E$ is [[continuous functor|continuous]] and $g: D \to E$ is arbitrary functor (not necessarily continuous) then the comma category $(f/g)$ is complete. Similarly, if $C$ and $D$ are [[cocomplete category|cocomplete]] and $f: C \to E$ is [[cocontinuous functor|cocontinuous]] then $(f/g)$ is cocomplete. For a proof, see

\begin{itemize}%
\item Rydeheard, David E., and Rod M. Burstall. Computational category theory. Vol. 152. Englewood Cliffs: Prentice Hall, 1988. Section 5.2: colimits in comma categories.

\end{itemize}
\hypertarget{functors_and_comma_categories}{}\subsubsection*{{Functors and comma categories}}\label{functors_and_comma_categories}

\begin{itemize}%
\item [[functors and comma categories]]

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

\begin{itemize}%
\item [[Quillen's theorem A]]
\item [[comma object]]

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

\begin{itemize}%
\item [[Toby Bartels]], \emph{\href{http://tobybartels.name/notes/#comma}{Comma categories}}

\end{itemize}
[[!redirects comma categories]]



\end{document}