Difference between revisions of "SFN:Math-test"
(Created page with "$ \newcommand\cc{\mathcal{C}} \newcommand\set{\mathsf{Set}} $ {{defn|name=Definition 1|note=Definition of set-based categories|label=df:sbcat |Let $\cc$ be a small category a...") |
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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: | ||
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(Sb:a) $U$ is an embedding. | (Sb:a) $U$ is an embedding. | ||
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\[ | \[ | ||
d\subseteq c,\quad d\subseteq c'\quad\text{and}\quad x\in U(d). | d\subseteq c,\quad d\subseteq c'\quad\text{and}\quad x\in U(d). | ||
| − | \] | + | \]--> |
}} | }} | ||
Revision as of 05:32, 29 March 2014
$ \newcommand\cc{\mathcal{C}} \newcommand\set{\mathsf{Set}}
$
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:
Notice that the condition (Sb:b) implies that the functor $U$ preserves image--factorizations: