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\begin{document}

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\section*{categories with finite products are cosifted}

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

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

[[!include infinity-limits - contents]]

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

[[!include category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A basic example of [[limits commuting with colimits]] in [[category theory]] is that [[colimits]] over [[opposite categories]] of categories with [[finite products]] preserves [[finite products]]. One says equivalently that categories with [[finite products]] are \emph{[[cosifted categories]]}.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{example}
\label{CategoriesWithFiniteProductsAreCosifted}\hypertarget{CategoriesWithFiniteProductsAreCosifted}{}
\textbf{([[categories with finite products are cosifted]]}

Let $\mathcal{C}$ be a [[small category]] which has [[finite products]]. Then $\mathcal{C}$ is a \emph{[[cosifted category]]}, equivalently its [[opposite category]] $\mathcal{C}^{op}$ is a \emph{[[sifted category]]}, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] are \emph{[[sifted colimits]]}, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] \emph{[[limits commuting with colimits|commute]] with [[finite products]]}, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to [[functors]] on the [[opposite category]] of $\mathcal{C}$ (hence two [[presheaves]] on $\mathcal{C}$) we have a [[natural isomorphism]]

\begin{displaymath}
\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
  \left(
    \mathbf{X} \times \mathbf{Y}
  \right)
  \;\simeq\;
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{X}
  \right)
  \times
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{Y}
  \right)
  \,.
\end{displaymath}
\end{example}
\begin{proof}
First observe that for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ two [[presheaves]], their [[Cartesian product]] is a [[colimit]] over [[representable presheaf|presheaves represented]] by Cartesian products in $\mathcal{C}$. Explicity, using [[coend]]-notation, we have:

\begin{equation}
\mathbf{X} \times \mathbf{Y}
  \;\simeq\;
  \int^{c_1,c_2 \in \mathcal{C}}
  y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)
  \,,
\label{OnSiteWithProductsExpandProductOfPresheaves}\end{equation}
where $y \;\colon\; \mathcal{C} \hookrightarrow [\mathcal{C}^{op}, Set]$ denotes the [[Yoneda embedding]].

This is due to the following sequence of [[natural isomorphisms]]:

\begin{displaymath}
\begin{aligned}
    (\mathbf{X} \times \mathbf{Y})(c)
    & \simeq
    \left(
      \int^{c_1 \in \mathcal{C}}
      \mathcal{C}(c,c_1) \times \mathbf{X}(c_1)
    \right)
    \times
    \left(
      \int^{c_2 \in \mathcal{C}}
      \mathcal{C}(c,c_2) \times \mathbf{Y}(c_2)
    \right)
    \\
    & \simeq
    \int^{c_1 \in \mathcal{C}}
    \int^{c_2 \in \mathcal{C}}
    \underset{
      \simeq \mathcal{c}(c, c_1 \times c_2)
    }{
    \underbrace{
      \mathcal{C}(c,c_1) \times \mathcal{C}(c,c_2)
    }}
    \times 
    \left( 
      \mathbf{X}(c_1) \times \mathbf{X}(c_2)
    \right)
    \\
    & \simeq
    \int^{c_1 \in \mathcal{C}}
    \int^{c_2 \in \mathcal{C}}
    \mathcal{C}(c,c_1 \times c_2)
    \times 
       \mathbf{X}(c_1) \times \mathbf{X}(c_2)
    \,,
  \end{aligned}
\end{displaymath}
where the first step expands out both presheaves as colimits of representables separately, via the [[co-Yoneda lemma]], the second step uses that the [[Cartesian product]] of presheaves is a two-variable [[left adjoint]] (by the [[symmetric monoidal category|symmetric]] [[closed monoidal structure on presheaves]]) and [[adjoints preserve (co-)limits|as such preserves colimits]] (in particular [[coends]]) in each [[variable]] separately, and under the brace we use the defining [[universal property]] of the [[Cartesian products]], assumed to exist in $\mathcal{C}$.

Now observe that the [[colimit]] of a [[representable presheaf]] is the [[singleton]].

\begin{equation}
\underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c)
  \;\simeq\;
  \ast
  \,.
\label{ColimitOfRepresentableIsSingleton}\end{equation}
One way to see this is to regard the colimit as the [[left Kan extension]] along the unique functor $\mathcal{C}^{op} \overset{p}{\to} \ast$ to the [[terminal category]]. By the formula \href{Kan+extension#PointwiseByCoEnds}{there}, this yields

\begin{displaymath}
\begin{aligned}
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    y(c)
    & \simeq
    \int^{c_1 \in \mathcal{C}} 
      \underset{const_\ast(c_1)}{\underbrace{\ast(-,p(c_1))}} \times y(c)(c_1)
    \\
    & \simeq
    \int^{c_1 \in \mathcal{C}} const_\ast(c_1) \times \mathcal{C}(c_1,c) 
    \\
    & \simeq const_\ast(c)
    \\
    & \simeq \ast
  \end{aligned}
\end{displaymath}
where we made explicit the [[constant functor]] $const_\ast \;\colon\; \mathcal{C} \to Set$, constant on the [[singleton]] set $\ast$, and then applied the [[co-Yoneda lemma]].

Using this, we compute for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ the following sequence of [[natural isomorphisms]]:

\begin{displaymath}
\begin{aligned}
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    \left(
      \mathbf{X} \times \mathbf{Y}
    \right)
    & 
    \simeq 
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    \int^{c_1,c_2 \in \mathcal{C}}
    y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)  
    \\
    & \simeq
    \int^{c_1,c_2 \in \mathcal{C}}
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    \left(
      y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)  
    \right)
    \\
    & \simeq
    \int^{c_1,c_2 \in \mathcal{C}}
    \left(
      \underset{
        \simeq \ast
      }{
      \underbrace{
        \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
        y(c_1 \times c_2) 
      }}
    \right)
    \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)  
    \\
    & \simeq
    \int^{c_1,c_2 \in \mathcal{C}}
    \left( 
      \mathbf{X}(c_1) \times \mathbf{Y}(c_2)  
    \right)
    \\
    & \simeq
    \left(
      \int^{c_1\in \mathcal{C}}
      \mathbf{X}(c_1) 
    \right)
    \times 
    \left(
      \int^{c_2\in \mathcal{C}}
      \mathbf{Y}(c_2)  
    \right)
    \\
    & \simeq
    \left(
      \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
      \mathbf{X}
    \right)
    \times
    \left(
      \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
      \mathbf{Y}
    \right)
  \end{aligned}
\end{displaymath}
Here the first step is \eqref{OnSiteWithProductsExpandProductOfPresheaves}, the second uses that [[colimits commute with colimits]], the third uses again that the [[Cartesian product]] respects colimits in each variable separately, the fourth is \eqref{ColimitOfRepresentableIsSingleton}, the last step is again the respect for colimits of the Cartesian product in each variable separately.

\end{proof}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item In the definiton of \emph{[[cohesive site]]} and \emph{[[infinity-cohesive site]]}, the assumption that the site has finite products (more generally: is [[cosifted category|cosifted]]) is what guarantees that the induced [[shape modality]] preserves finite products, as (usually) required for a [[cohesive topos]] or [[cohesive (infinity,1)-topos]]

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

\begin{itemize}%
\item [[commutativity of limits and colimits]]

\end{itemize}


\end{document}