Difference between revisions of "SFN:Math-test"
| Line 6: | Line 6: | ||
$ | $ | ||
{{defn|name=Definition 1|note=Definition of set-based categories|label=df:sbcat | {{defn|name=Definition 1|note=Definition of set-based categories|label=df:sbcat | ||
| − | |Let $\cc$ be a small category and let $U:\cc\to\set$ be a functor. We | + | |statement=Let $\cc$ be a small category and let $U:\cc\to\set$ be a functor. We |
say that $\cc$ is ''set-based'' ($\sbcat$ for short) with respect | say that $\cc$ is ''set-based'' ($\sbcat$ for short) with respect | ||
to $U$ if the pair $(\cc,U)$ satisfy the following: | to $U$ if the pair $(\cc,U)$ satisfy the following: | ||
Revision as of 05:48, 29 March 2014
$ \newcommand\cc{\mathcal{C}} \newcommand\set{\mathsf{Set}} \newcommand\sbcat{\mathit{Category}}
$
Definition 1 (Definition of set-based categories)
Let $\cc$ be a small category and let $U:\cc\to\set$ be a functor. We
say that $\cc$ is set-based ($\sbcat$ for short) with respect to $U$ if the pair $(\cc,U)$ satisfy the following:
(Sb:a) $U$ is an embedding.
(Sb:b) $U(\im f)=\im U(f)$ for all $f\in\cc$.
(Sb:c) The functor $U$ has the following property: for $c,c'\in\vrt
\cc$ and $x\in U(c)\cap U(c')$ there is $d\in\vrt\cc$ such thatNotice that the condition (Sb:b) implies that the functor $U$ preserves image--factorizations: