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\begin{document}

%-------------------------------------------------------------------


\section*{precategory}

[[!redirects precategories]]

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

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{lemma_914_idtoiso}{Lemma 9.1.4 (idtoiso)}\dotfill \pageref*{lemma_914_idtoiso} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A precategory $A$ consists of the following.

\begin{itemize}%
\item A type $A_0$, whose elements are called objects. Typically $A$ is coerced to $A_0$ in order to write $x:A$ for $x:A_0$.


\item For each $a,b:A$, a set $hom_A(a,b)$, whose elements are called \textbf{arrows} or \textbf{morphisms}.


\item For each $a:A$, a morphism $1_a:hom_A(a,a)$, called the \textbf{identity morphism}.


\item For each $a,b,c:A$, a function

\begin{displaymath}
hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)
\end{displaymath}
called composition, and denoted infix by $g \mapsto f \mapsto g \circ f$, or sometimes $gf$.


\item For each $a,b:A$ and $f:hom_A(a,b)$, we have $f=1_b \circ f$ and $f=f\circ 1_a$.


\item For eagh $a,b,c,d:A$,

\begin{displaymath}
f:hom_A(a,b),\ g:hom_A(b,c),\ h:hom_A(c,d)
\end{displaymath}
we have $h\circ (g\circ f)=(h\circ g)\circ f$.



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

There are two notions of ``sameness'' for objects of a precategory. On one hand we have an [[isomorphism]] between objects $a$ and $b$ on the other hand we have equality of objects $a$ and $b$.

There is a special kind of precategory called a [[category]] where these two notions of equality coincide and some very nice properties arise.

\hypertarget{lemma_914_idtoiso}{}\subsubsection*{{Lemma 9.1.4 (idtoiso)}}\label{lemma_914_idtoiso}

if $A$ is a precategory and $a,b:A$, then there is a map

\begin{displaymath}
idtoiso : (a=b) \to (a \cong b)
\end{displaymath}
\emph{Proof.} By induction on identity, we may assume $a$ and $b$ are the same. But then we have $1_a:hom_A(a,a)$, which is clearly an isomorphism. $\square$

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

See [[category]]

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

[[Category theory]] [[category]] [[Rezk completion]]

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

[[HoTT Book]]

category: category theory



\end{document}