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category_._set_theory [2014/04/07 17:19] nikolaj |
category_._set_theory [2014/04/07 19:02] nikolaj |
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| | @#FFBB00: definiendum | @#FFBB00: $ \langle \mathcal{O},M,id,* \rangle \in \mathrm{it}$ | | | @#FFBB00: definiendum | @#FFBB00: $ \langle \mathcal{O},M,id,* \rangle \in \mathrm{it}$ | | ||
| | @#FF9944: definition | @#FF9944: $\mathrm{Mor}:\mathcal{O}\times\mathcal{O}\to M$ | | | @#FF9944: definition | @#FF9944: $\mathrm{Mor}:\mathcal{O}\times\mathcal{O}\to M$ | | ||
| - | | @#FF9944: definition | @#FF9944: $\circ:{\large\prod}_{A,B,C:\mathcal{O}}\,\mathrm{Mor}(B,C)\times\mathrm{Mor}(A,B)\to\mathrm{Mor}(A,C)$ | | + | | @#FF9944: definition | @#FF9944: $\circ:{\large\prod}_{A,B,C\in\mathcal{O}}\,\mathrm{Mor}(B,C)\times\mathrm{Mor}(A,B)\to\mathrm{Mor}(A,C)$ | |
| - | | @#FF9944: definition | @#FF9944: $id:{\large\prod}_{A:\mathcal{O}}\,\mathrm{Mor}_O(A,A)$ | | + | | @#FF9944: definition | @#FF9944: $id:{\large\prod}_{A\in\mathcal{O}}\,\mathrm{Mor}_O(A,A)$ | |
| | @#55EE55: postulate | @#55EE55: $\mathrm{Mor}(A,B)\cap\mathrm{Mor}(U,V)\ne\emptyset\implies A=B\land U=V$ | | | @#55EE55: postulate | @#55EE55: $\mathrm{Mor}(A,B)\cap\mathrm{Mor}(U,V)\ne\emptyset\implies A=B\land U=V$ | | ||
| | @#55EE55: postulate | @#55EE55: $(g\circ f)\circ h=g\circ (f\circ h)$ | | | @#55EE55: postulate | @#55EE55: $(g\circ f)\circ h=g\circ (f\circ h)$ | | ||
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| ==== Discussion ==== | ==== Discussion ==== | ||
| - | Within set theory, we can define a category as quintuple given by two sets and two (polymorphic) maps into them. | + | Within set theory, we can define a category as quintuple given by two sets and two maps into them. The $\prod$-notation giving the set theoretical model for dependend/polymorphic functions, is given in [[function]]. |
| The three axioms say the following: The hom-sets are pairwise disjoint, the composition is associative and $id$ denotes the identity. | The three axioms say the following: The hom-sets are pairwise disjoint, the composition is associative and $id$ denotes the identity. | ||
