Difference between revisions of "SFN:Math-test"
| Line 2: | Line 2: | ||
\newcommand\cc{\mathcal{C}} | \newcommand\cc{\mathcal{C}} | ||
\newcommand\set{\mathsf{Set}} | \newcommand\set{\mathsf{Set}} | ||
| − | \newcommand\sbcat | + | \newcommand\sbcat{\mathit{Category}} |
$ | $ | ||
| Line 9: | Line 9: | ||
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: | ||
| − | + | ||
(Sb:a) $U$ is an embedding. | (Sb:a) $U$ is an embedding. | ||
| Line 16: | Line 16: | ||
(Sb:c) The functor $U$ has the following property: for $c,c'\in\vrt | (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 that | \cc$ and $x\in U(c)\cap U(c')$ there is $d\in\vrt\cc$ such that | ||
| + | <!-- | ||
\[ | \[ | ||
d\subseteq c,\quad d\subseteq c'\quad\text{and}\quad x\in U(d). | d\subseteq c,\quad d\subseteq c'\quad\text{and}\quad x\in U(d). | ||
Revision as of 05:33, 29 March 2014
$ \newcommand\cc{\mathcal{C}} \newcommand\set{\mathsf{Set}} \newcommand\sbcat{\mathit{Category}}
$
Notice that the condition (Sb:b) implies that the functor $U$ preserves image--factorizations: