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 functor_category [2015/03/16 23:01]nikolaj functor_category [2015/03/17 09:43]nikolaj Both sides previous revision Previous revision 2015/12/17 19:21 nikolaj 2015/03/17 09:43 nikolaj 2015/03/16 23:01 nikolaj 2015/02/26 13:23 nikolaj 2015/02/26 13:22 nikolaj 2015/02/26 13:22 nikolaj 2015/02/26 13:22 nikolaj 2014/12/04 22:34 nikolaj 2014/12/04 16:31 nikolaj 2014/09/17 22:24 nikolaj 2014/08/21 10:11 nikolaj 2014/04/06 16:12 nikolaj 2014/04/06 16:11 nikolaj 2014/04/06 16:10 nikolaj 2014/04/06 16:10 nikolaj old revision restored (2014/04/06 15:38) 2015/12/17 19:21 nikolaj 2015/03/17 09:43 nikolaj 2015/03/16 23:01 nikolaj 2015/02/26 13:23 nikolaj 2015/02/26 13:22 nikolaj 2015/02/26 13:22 nikolaj 2015/02/26 13:22 nikolaj 2014/12/04 22:34 nikolaj 2014/12/04 16:31 nikolaj 2014/09/17 22:24 nikolaj 2014/08/21 10:11 nikolaj 2014/04/06 16:12 nikolaj 2014/04/06 16:11 nikolaj 2014/04/06 16:10 nikolaj 2014/04/06 16:10 nikolaj old revision restored (2014/04/06 15:38) Last revision Both sides next revision Line 9: Line 9: ----- ----- === Discussion === === Discussion === - If ${\bf 5}$ is the discrete ​category of five different ​objects, ​then ${\bf Set}^{\bf ​5}$ is the category ​of all choices ​of 5 sets. ${\bf Set}^{\bf 1}$ is just  ​${\bf Set}$ itself. If we'd consider a category ${\bf 5}'$ to be the same category with some ordering of the object expressed via arrows, then ${\bf Set}^{{\bf 5}'}$ is just the category of all choices of 5 sets, together with some ordering. + Firstly, A class of sets together with functions between them form a category. The only job of the arrows between ​objects ​here is to transfer individual elements from set to sets. Secondly, A class of categories and functors between them is a category too, but here the objects exhibit some internal structure and the arrows are required to respect that structure. Finally, A class of functors and natural transformations between them also form a category, call it ${\bf D}^{\bf ​C}$. Here, the objects can be thought ​of as copies ​of the category fixed category ​${\bf C}$ seated inside of ${\bf D}$, and the arrows must respect (only) ​the ${\bf C}$-structure. - A functor $F:{\bf C}\longrightarrow{\bf D}$ just embeds the diagram ${\bf C}$ within a category ${\bf D}$. Therefore, think of the functor category ${\bf D}^{\bf C}$ as the collection of all (possibly squeezed) copies of ${\bf C}$ in ${\bf D}$. It's really like a category of sets, except that the sets generally all share some sort of structure. + == Some very simple examples == - The arrows in this category are natural transformations. Those are just function which map object to object, but crucially: Only those functions which do respect the structure. + If ${\bf 5}$ is the discrete category of five different objects, then ${\bf Set}^{\bf 5}$ is the category of all choices of up to 5 sets. ${\bf Set}^{\bf 1}$ is just  ${\bf Set}$ itself. If we'd consider a category ${\bf 5}'$ to be the same category with some ordering of the object expressed arrows, then ${\bf Set}^{{\bf 5}'}$ is just the category of all choices of up to 5 sets, where the arrows expressing ordering are substituted by some function. + + Again, a functor $F:{\bf C}\longrightarrow{\bf D}$ just embeds the diagram ${\bf C}$ within a category ${\bf D}$. Therefore, think of the functor category ${\bf D}^{\bf C}$ as the collection of all (possibly squeezed) copies of ${\bf C}$ in ${\bf D}$. == Algebraic picture if the target is structured == == Algebraic picture if the target is structured == 