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'generator of a category'
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| Title of object: |
generator of a category |
| Canonical Name: |
GeneratorOfACategory |
| Type: |
Definition |
| Created on: |
2008-09-03 21:48:07 |
| Modified on: |
2008-09-03 21:48:07 |
| Classification: |
msc:18A99 |
| Defines: |
generator, generating set |
Preamble:
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Content:
Let $\mathcal{C}$ be a category. A set $\lbrace X_i\mid i\in I\rbrace$ of objects (indexed by a set $I$) is called a \emph{generating set} of $\mathcal{C}$ if for any pair of distinct morphisms $f,g:A\to B$, there is a morphism $h:G_i\to A$ 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 $\mathcal{C}$.
For example, 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 $f\circ h(y)=f(x)\ne g(x)=g\circ h(y)$. |
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