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\begin{document}

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\section*{protocategory}

\hypertarget{protocategories}{}\section*{{Protocategories}}\label{protocategories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{protocategory} is a way to present a [[category]] that makes formal the idea that a single datum (a ``protomorphism'') can represent many different [[morphisms]] with different sources and targets. For instance, in [[material set theory]], a given set of [[ordered pairs]] can represent [[functions]] with many different codomains.

As compared to the definition of category with one set of objects and one set of morphisms, a protocategory removes the requirement that each morphism have a unique source and target. As compared to the definition of category with separate hom-sets $\hom(A,B)$ for each pair of objects, a protocategory requires all these homsets to be (not necessarily disjoint) subsets of one set of all ``protomorphisms''.

It is important to recognize, however, that unlike those two definitions of category, a protocategory is \emph{not} a definition of a category. It retains information about ``which morphisms in distinct hom-sets are `equal''', which is not relevant category-theoretic information: in a true category it is never `meaningful' to compare two morphisms for equality unless one already knows for other reasons that their domains and codomains coincide. A protocategory is rather a way of \emph{presenting} a category, analogous to a [[group presentation]], formalizing the fact that in some cases morphisms in different hom-sets can be ``equal'' until we ``forget about that fact'' in a structured way.

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

A \textbf{protocategory} consists of:

\begin{itemize}%
\item Two collections of objects and protomorphisms
\item A ternary ``source-target relation'' relating two objects and one protomorphism, written ``$f : A\to B$'' and read as ``$f$ represents a morphism from $A$ to $B$''
\item A ternary ``composition predicate'' relating three protomorphisms, written ``$h = g \circ f$'' and read as ``$h$ is a possible composite of $f$ and $g$''.

\end{itemize}
satisfying the axioms

\begin{itemize}%
\item If $f:A\to B$ and $g:B\to C$, there is exactly one $h$ satisfying $h:A\to C$ and $h = g\circ f$.
\item If we define the set of morphisms $\hom(A,B)$ to be the set of protomorphisms $f$ satisfying $f:A\to B$, then the composition operation given by the previous axiom results in a [[category]] (i.e. it is associative and has identities).

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

\begin{itemize}%
\item Let the objects be the [[sets]] in [[material set theory]] (such as [[ZFC]]), and let the protomorphisms be sets $f$ of ordered pairs such that for any $x$ there is at most one $y$ such that $(x,y)\in f$. The \emph{domain} of such an $f$ is the set of all $x$ such that there is such a $y$, and its \emph{range} is the set of all $y$ such that there exists an $x$ with $(x,y)\in f$. We say $f:A\to B$ if $A$ is the domain of $f$ and $B$ is a superset of the range of $f$. With a suitable definition of composition, this yields a protocategory that generates the category [[Set]].


\item More generally, if we take the protomorphisms to be \emph{all} sets of ordered pairs, and $f:A\to B$ to mean that $A$ is a superset of the domain of $f$ in addition to $B$ being a superset of the codomain of $f$, then we get a protocategory that generates the category [[Rel]].


\item Let the objects be [[groups]], let the protomorphisms be [[functions]], and say that $f:G\to H$ when $f$ is a function between the underlying sets of $G$ and $H$ that is a group homomorphism. This protocategory generates the category [[Grp]].


\item Any category can be presented by a protocategory in a trivial way, by taking the protomorphisms to be the [[disjoint union]] of all the sets of morphisms (i.e. the set $C_1$ in the one-set-of-morphisms definition of category), with the source-target relation being simply a function from protomorphisms to pairs of objects.



\end{itemize}
\begin{uremark}
It may seem that, at least in a [[material set theory]], we should be able to modify the previous example by taking a \emph{non-disjoint} union of the sets of morphisms (assuming given a category in the many-sets-of-morphisms definition). However, this is not true, because the composition predicate in a protocategory makes no reference to objects. For example, consider the following category:

\begin{itemize}%
\item objects $0,1,2,0',1',2'$
\item $\hom(0,1) = \hom(0',1') = \{f\}$, $\hom(1,2)=\hom(1',2')=\{g\}$, $\hom(0,2)=\hom(0',2')=\{h,h'\}$, no other nonidentity morphisms, all identity morphisms distinct
\item $g\circ_{0,1,2} f = h$ and $g\circ_{0',1',2'} f = h'$.

\end{itemize}
Then if we take unions of homsets we have $g\circ f = h$ and also $g\circ f = h'$, but then the protocategory composition axiom fails: we have $f:0\to 1$ and $g:1\to 2$, but there does not exist a \emph{unique} $k$ such that $k:0\to 2$ and $g \circ f = k$.

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

\begin{itemize}%
\item The notion is due to Freyd and Scedrov, [[Categories, Allegories]].


\item It is recalled in A1.1 of [[Sketches of an Elephant]].



\end{itemize}
[[!redirects protocategories]] [[!redirects protomorphism]] [[!redirects protomorphisms]]



\end{document}