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initial_morphism [2014/06/14 17:29] nikolaj created |
initial_morphism [2014/06/18 15:43] nikolaj |
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| - | ===== Initial isomorphism ===== | + | ===== Initial morphism ===== |
| ==== Collection ==== | ==== Collection ==== | ||
| | @#55CCEE: context | @#55CCEE: $X:\mathrm{Ob}_{\bf C}$ | | | @#55CCEE: context | @#55CCEE: $X:\mathrm{Ob}_{\bf C}$ | | ||
| Line 9: | Line 9: | ||
| | @#FFFDDD: for all | @#FFFDDD: $f:{\bf C}[X,U(B)]$ | | | @#FFFDDD: for all | @#FFFDDD: $f:{\bf C}[X,U(B)]$ | | ||
| | @#DDDDDD: range | @#DDDDDD: $g:{\bf D}[A,B]$ | | | @#DDDDDD: range | @#DDDDDD: $g:{\bf D}[A,B]$ | | ||
| - | | @#55EE55: postulate | @#55EE55: $\exists!g.\ f=\phi\circ U(g)$ | | + | | @#55EE55: postulate | @#55EE55: $\exists!g.\ f=U(g)\circ\phi$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| + | For an elaboration, see [[terminal morphism]], the dual concept. | ||
| - | |||
| - | (?? if I restrict the class of arrows in ${\bf C}$ to a set, then I guess I can make this a set, rather than a collection) | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Universal_property|Universal property]] | Wikipedia: [[http://en.wikipedia.org/wiki/Universal_property|Universal property]] | ||
| Line 21: | Line 20: | ||
| [[Functor]] | [[Functor]] | ||
| === Requirements === | === Requirements === | ||
| - | [[Natural transformation]] | + | [[Product type]] |
