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\begin{document}

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\section*{presentation of a category by generators and relations}

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

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{2cells}{2-cells}\dotfill \pageref*{2cells} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Similarly to the presentation of [[groups]] by [[generators and relations]], a category may be presented by a set of generating arrows subject to certain relations.

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

Let $G$ be a [[quiver|directed graph]], and let $R$ be a function that assigns to each pair $a,b$ of objects of the [[free category]] $F(G)$ a [[binary relation]] $R_{a,b}$ on the [[hom-set]] $F(G)(a,b)$. The \textbf{category with generators $G$ and relations $R$} is the [[quotient category]] (as defined in \hyperlink{MacLane}{Mac Lane} or \hyperlink{Mitchel}{Mitchell}, for example -- this is \emph{not} the nLab definition due to issues of [[evil]]) $F(G)/R$. For a category $C$, an [[isomorphism of categories|isomorphism]] $C\to F(G)/R$ is called a \textbf{presentation} of $C$.

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

Writing $can\colon F(G)\to F(G)/R$ for the canonical functor, it follows from the universal property of the quotient category that for any functor $S\colon F(G)\to D$ that respects the relation $R$ ($f
R_{a,b}g$ implies $S(f)=S(g)$), there exists a unique functor $S'\colon
F(G)/R\to D$ with $S = S'\circ can$.

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

\begin{enumerate}%
\item Every category $C$ has a presentation by generators and relations: Take $G$ as the underlying graph of $C$, and for objects $a$, $b$, let $R_{a,b}$ be the relation on $F(G)(a,b)$ consisting of all pairs of paths from $a$ to $b$ in $G$ whose arrows have the same composition in $C$. However, there are sometimes more economical presentations for a category, as the following example shows.


\item The augmented simplex category $\Delta_a$ is generated by the face maps and the degeneracy maps, subject to the simplicial relations (see [[simplex category]] for details). The existence of a functor from the quotient category to $\Delta_a$ follows from the fact that the arrows of $\Delta_a$ do satisfy the simplicial relations, and the fact that this functor is an isomorphism may be verified using the unique decomposition of an arrow of $\Delta_a$ as the composition of degeneracies of decreasing index followed by the composition of face maps of increasing index (see the lemma on p. 177 of \hyperlink{MacLane}{Mac Lane}). Similarly, the subcategory $(\Delta_a)_{inj}$ consisting of all monics (injective monotone functions in our case) is generated by the face maps subject to the single simplicial relation involving only face maps.



\end{enumerate}
\hypertarget{2cells}{}\subsection*{{2-cells}}\label{2cells}

A useful way to think of the relations is as being 2-cells between parallel pairs of arrows, thus if $a, b$ are objects, and $(u,v)\in R_{a,b}$, we think of $(u,v)$ as a 2-cell (initially \emph{from $u$ to $v$}). In this way, one can encode [[rewriting]] systems of a certain kind in terms of the embryonic data for a 2-category. This is discussed more in the entry on [[computad]], which are also called [[polygraphs]].

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

\begin{itemize}%
\item [[Saunders Mac Lane]], [[Categories Work|Categories for the Working Mathematician]], pp. 51-52

\end{itemize}
\begin{itemize}%
\item [[Barry Mitchell]], \emph{Introduction to Category Theory and Homological Algebra}, in P. Salmon (Ed.), \emph{Categories and Commutative Algebra}. Springer, 2010. pp. 108-112.


\item [[Barry Mitchell]], \emph{Rings with several objects}, Advances in Mathematics 8 (1972), 1--161.



\end{itemize}
[[!redirects presentation of a category by generators and relations]] [[!redirects presentations of a category by generators and relations]] [[!redirects presentations of categories by generators and relations]]



\end{document}