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cat [2014/12/02 12:05] nikolaj |
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| @#55EE55: postulate | @#55EE55: $\mathrm{Ob}_{\bf C},\mathrm{Mor}_{\bf C} $ ... small | | | @#55EE55: postulate | @#55EE55: $\mathrm{Ob}_{\bf C},\mathrm{Mor}_{\bf C} $ ... small | | ||
| @#FFFDDD: for all | @#FFFDDD: ${\bf D}\in{\bf Cat}$ | | | @#FFFDDD: for all | @#FFFDDD: ${\bf D}\in{\bf Cat}$ | | ||
- | | @#55EE55: postulate | @#55EE55: ${\bf Cat}[{\bf C},{\bf D}]$ ... functor category | | + | | @#55EE55: postulate | @#55EE55: ${\bf Cat}[{\bf C},{\bf D}]$ ... functor category $({\bf C},{\bf D})$ | |
==== Discussion ==== | ==== Discussion ==== | ||
- | ${\bf Cat}$ is the archetypical example for what is called a 2-cateogry: Each hom-class ${\bf Cat}[{\bf C},{\bf D}]$ is again a category. (More precisely, each hom-class of a 2-category is a 1-category.) | + | === Elaboration === |
+ | ${\bf Cat}$ is the archetypical example for what is called a 2-cateogry: Each hom-class ${\bf Cat}[{\bf C},{\bf D}]$ is again a (ordinary) category. | ||
- | == Predicates == | + | Specifically, in ${\bf Cat}$, the hom-classes are functor categories and the hom-classes of those are natural transformations. |
+ | |||
+ | === Predicates === | ||
| @#EEEE55: predicate | @#EEEE55: ${\bf C}$ ... small $\equiv {\bf C}$ in ${\bf Cat}$ | | | @#EEEE55: predicate | @#EEEE55: ${\bf C}$ ... small $\equiv {\bf C}$ in ${\bf Cat}$ | | ||
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[[Locally small category]] | [[Locally small category]] | ||
=== Requirements === | === Requirements === | ||
- | [[Set universe]] | + | [[Set universe]], [[Functor category]] |