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functor_category [2015/03/16 23:01] nikolaj |
functor_category [2015/12/17 19:21] nikolaj |
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| ===== Functor category ===== | ===== Functor category ===== | ||
| - | ==== Category ==== | + | ==== Collection ==== |
| | @#55CCEE: context | @#55CCEE: ${\bf C}$ ... small category | | | @#55CCEE: context | @#55CCEE: ${\bf C}$ ... small category | | ||
| | @#55CCEE: context | @#55CCEE: ${\bf D}$ ... category | | | @#55CCEE: context | @#55CCEE: ${\bf D}$ ... category | | ||
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| === 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. |
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| + | 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 == | ||
