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functor_category [2015/02/26 13:22] 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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| - | === Idea === | + | === Discussion === |
| - | ==Algebraic picture of functor categories== | + | 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. |
| - | The nice target category ${\bf{Set}}$ is like a ring (say $\mathbb R$) and the functor category ${\bf{Set}}^{\bf{C}}$ with objects $\omega,\mu,\dots$ is like a space of functionals on a space ${\bf{C}}$. The topos/functional space is richer than the base {\bf{C}}: The target (${\bf{Set}}$ resp. $\mathbb C$) has a nice algebraic structure (e.g. co-products resp. addition), which we can pull back to define one on ${\bf{Set}}^{\bf{C}}$. As in $\omega+\lambda:=\left(v\mapsto\omega(v)+\mu(v)\right)$. | + | |
| + | == Some very simple examples == | ||
| + | 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 ${\bf D}$. | ||
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| + | == Algebraic picture if the target is structured == | ||
| + | The nice target category ${\bf{Set}}$ is like a ring (say $\mathbb R$) and the functor category ${\bf{Set}}^{\bf{C}}$ with objects $\omega,\mu,\dots$ is like a space of functionals on a space ${\bf{C}}$. The topos/functional space is richer than the base ${\bf{C}}$: The target (${\bf{Set}}$ resp. $\mathbb C$) has a nice algebraic structure (e.g. co-products resp. addition), which we can pull back to define one on ${\bf{Set}}^{\bf{C}}$. As in $\omega+\lambda:=\left(v\mapsto\omega(v)+\mu(v)\right)$. | ||
| Adding more details leads to finer analogies: If $\mathbb C$ has co-products itself, then it's like a vector space and it's object should be viewed as a set of base vectors. If a functor preserves co-product, it's like a linear functional and ${\bf{Set}}^{\bf{C}}$ becomes a kind of dual vector space. This sheds light on the (co-variant) Yoneda embedding: If we view the objects of $\mathbb C$ as a set of base vectors, then the can be mapped to functionals in the dual space, but that space is bigger / also contains a lot of other functionals. | Adding more details leads to finer analogies: If $\mathbb C$ has co-products itself, then it's like a vector space and it's object should be viewed as a set of base vectors. If a functor preserves co-product, it's like a linear functional and ${\bf{Set}}^{\bf{C}}$ becomes a kind of dual vector space. This sheds light on the (co-variant) Yoneda embedding: If we view the objects of $\mathbb C$ as a set of base vectors, then the can be mapped to functionals in the dual space, but that space is bigger / also contains a lot of other functionals. | ||
