San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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in Category Theory |
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A category is a collection of objects and a collection of arrows between those object. The collection of arrows is closed under composition. A functor is mapping between two categories. Because any cateory involves two collections, objects and arrows, a functor has to involve two mappings; one mapping between the object collections of the two categories and another between the arrow collections.
The functor mappings have satisfy further conditions. Let the two categories for a functor be denoted as C and D and a functor between them as F:C → D. Just for this presentation think of C and D as the order pairs (C_{O}, C_{A}) and (D_{O}, D_{A}). Likewise F is an ordered pair (F_{O}, F_{A}) where
The conditions which F must satisfy are:
The remarable and beautiful thing is that the collection of categories with functors as the collection of arrows is a CATEGORY!
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