generator of a categoryGeneratorOfACategory2008-09-03 21:48:072008-09-22 01:56:38Definitiongeneratorgenerating setprogenerator\usepackage{amssymb,amscd}
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\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}Let $\mathcal{C}$ be a category, and $f,g:A\to B$ a pair of distinct morphisms. A morphism $h:X\to A$ is said to \emph{distinguish} or \emph{separate} $f$ and $g$ if $f\circ h\ne g\circ h$. For example, if $f\ne g:A\to B$, then $1_A$ on $A$ distinguishes $f$ and $g$.
A set $S=\lbrace X_i\mid i\in I\rbrace$ of objects (indexed by a set $I$) is called a \emph{generating set} of $\mathcal{C}$ if any pair of distinct morphisms $f,g:A\to B$ can be distinguished by a morphism with domain in $S$ and codomain $A$. In other words, there is $h:X_i\to A$ for some $i\in I$, such that $f\circ h\ne g\circ h$. If $\lbrace X\rbrace$ is a generating family of $\mathcal{C}$, then $X$ is called a \emph{generator} of $\mathcal{C}$. Any set of morphisms containing a generator is a generating set.
\textbf{Examples}
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\item
In \textbf{Set}, the category of sets, any singleton is a generator. Suppose $f,g:A\to B$ are distinct functions, so that $f(x)\ne g(x)$ for some $x\in A$. Let $\lbrace y\rbrace$ be any singleton. Then $h:\lbrace y\rbrace \to A$ defined by $h(y)=x$ is the function distinguishing $f$ and $g$: for $f\circ h(y)=f(x)\ne g(x)=g\circ h(y)$.
\item
In \textbf{Rng}, the category of rings, the ring $\mathbb{Z}$ is a generator. If $f,g:R\to S$ are distinct ring homomorphisms, say, $f(r)\ne g(r)$ for some $r\in R$. Then the ring homomorphism $h:\mathbb{Z}\to R$ given by $h(1)=r$ distinguishes $f$ and $g$.
\end{enumerate}
\textbf{Remark}. A projective object that is also a generator is called a \emph{progenerator}.
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\bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994)
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