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\begin{document}

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\section*{distributivity pullback}

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

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

[[!include category theory - contents]]

\hypertarget{distributivity_pullback}{}\section*{{Distributivity pullback}}\label{distributivity_pullback}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{connection_to_exponentiation}{Connection to exponentiation}\dotfill \pageref*{connection_to_exponentiation} \linebreak
\noindent\hyperlink{connection_to_distributivity}{Connection to distributivity}\dotfill \pageref*{connection_to_distributivity} \linebreak
\noindent\hyperlink{connection_to_pullback_complements}{Connection to pullback complements}\dotfill \pageref*{connection_to_pullback_complements} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{distributivity pullback} is the data which encodes a particular [[exponentiable morphism|exponentiation]] along a [[morphism]] in a [[category]]. In other words, it is a [[cofree object]] with respect to a [[pullback]] functor.

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

\begin{udefn}
For morphisms $g:Z\to A$ and $f:A\to B$ in a category, a \textbf{pullback around $(f,g)$} is a diagram

\begin{displaymath}
\itexarray{ X & \xrightarrow{p} & Z & \xrightarrow{g} & A\\
  ^q\downarrow &&&& \downarrow^f\\
  Y && \xrightarrow{r} && B}
\end{displaymath}
in which the outer rectangle is a [[pullback]].

\end{udefn}
A morphism of pullbacks around $(f,g)$ consists of $s:X\to X'$ and $t:Y\to Y'$ such that $p's=p$, $q s=t q'$, and $r = r's$.

\begin{udefn}
For $g:Z\to A$ and $f:A\to B$ as above, a \textbf{distributivity pullback} around $(f,g)$ is a terminal object of the category of pullbacks around $(f,g)$.

\end{udefn}
If the above diagram is a distributivity pullback, we say that it exhibits $r$ as a \emph{distributivity pullback of $g$ along $f$}.

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

\hypertarget{connection_to_exponentiation}{}\subsubsection*{{Connection to exponentiation}}\label{connection_to_exponentiation}

\begin{utheorem}
A morphism $f:A\to B$ is [[exponentiable morphism|exponentiable]] if and only if all distributivity pullbacks along $f$ exist.

\end{utheorem}
\begin{proof}
The universal property of a distributivity pullback says exactly that $Y\xrightarrow{r} B$ is the exponential of $g$ along $f$.

\end{proof}
\hypertarget{connection_to_distributivity}{}\subsubsection*{{Connection to distributivity}}\label{connection_to_distributivity}

In a category with pullbacks, for any pullback around $(f,g)$ with $f$ and $q$ exponentiable, we have a canonical Beck-Chevalley isomorphism

\begin{displaymath}
r^* f_* \xrightarrow{\cong} q_* p^* g^* .
\end{displaymath}
The [[mate]] of the inverse of this is a transformation

\begin{displaymath}
\delta_{p,q,r}:r_! q_* p^* \to f_* g_!
\end{displaymath}
\begin{utheorem}
A pullback around $(f,g)$ with $f$ and $q$ exponentiable is a distributivity pullback if and only if $\delta_{p,q,r}$ is an isomorphism.

\end{utheorem}
\begin{proof}
See \hyperlink{Weber}{(Weber)}.

\end{proof}
Invertibility of $\delta$ expresses that [[dependent products]] (the functors $f_*$ and $q_*$) distribute over [[dependent sums]] (the functors $g_!$ and $r_!$).

For instance, in the category of sets, if $(C_z)_{z\in Z}$ is a $Z$-indexed family of sets, then

\begin{displaymath}
(f_* g_! C)_b = \prod_{f(a)=b} \sum_{g(z)=a} C_z
\end{displaymath}
while

\begin{displaymath}
(r_! q_* p^* C)_b = \sum_{r(y)=b} \prod_{q(x)=y} C_{p(x)}
\end{displaymath}
As a very simple example, if $B=1$, $A=\{0,1\}$, and $Z=\{00,01,02,10,11\}$ with $g(i j) = i$, then $Y$ is the set of sections of $g$, $X$ the set of pairs of a section and an element of $A$, and $p$ the evaluation. Then if $C = (C_{00}, C_{01}, C_{02}, C_{10}, C_{11})$, we have

\begin{displaymath}
f_* g_! C = (C_{00} + C_{01} + C_{02}) \times (C_{10} + C_{11})
\end{displaymath}
and

\begin{displaymath}
r_! q_* p^* C = (C_{00}\times C_{10}) + (C_{00}\times C_{11}) + 
(C_{01}\times C_{10}) + (C_{01}\times C_{11}) +
(C_{02}\times C_{10}) + (C_{02}\times C_{11}).
\end{displaymath}
In the [[internal logic|internal]] [[dependent type theory]] of the ambient category, if we express $f$ and $g$ as [[dependent types]]

\begin{displaymath}
b:B \vdash A(b) : Type \qquad and \qquad b:B, a:A(b) \vdash Z(b,a) : Type
\end{displaymath}
then the exponential $r$ of $g$ along $f$ in a distributivity pullback is the [[dependent product type]]

\begin{displaymath}
b:B \vdash \prod_{a:A(b)} Z(b,a) : Type.
\end{displaymath}
and the distributivity isomorphism $\delta$ says that for any further dependent type

\begin{displaymath}
b:B, a:A(b), z:Z(b,a) \vdash W(b,a,z) : Type
\end{displaymath}
in the context of $b:B$ we have an isomorphism

\begin{displaymath}
\sum_{\phi:\prod_{a:A(b)} Z(b,a) } \prod_{a:A(b)} W(b,a,\phi(a)) \qquad \cong \qquad  \prod_{a:A(b)} \sum_{z:Z(b,a)} W(b,a,z).
\end{displaymath}
This isomorphism (or more specifically the left-to-right map) has traditionally been called the ``axiom of choice'' in [[Martin-Lof type theory]], since if $\sum$ and $\prod$ are interpreted according to [[propositions as types]] then it looks like the set-theoretic axiom of choice. However, this is not really appropriate, since this is a provable statement rather than an additional axiom, and does not have any of the usual strong consequences of the set-theoretic axiom of choice. See [[axiom of choice]] for further discussion.

\hypertarget{connection_to_pullback_complements}{}\subsubsection*{{Connection to pullback complements}}\label{connection_to_pullback_complements}

If $p$ is an isomorphism, then a distributivity pullback is also a [[final pullback complement]].

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

\begin{itemize}%
\item [[Mark Weber]], ``Polynomials in categories with pullbacks'', \href{http://arxiv.org/abs/1106.1983}{arXiv}, \href{http://tac.mta.ca/tac/volumes/30/16/30-16abs.html}{TAC}

\end{itemize}
[[!redirects distributivity pullbacks]]



\end{document}