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functor_category [2015/02/26 13:22] nikolaj |
functor_category [2015/02/26 13:23] nikolaj |
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| === Idea === | === Idea === | ||
| - | ==Algebraic picture of functor categories:== | + | == 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 $\omega+\lambda:=\left(v\mapsto\omega(v)+\mu(v)\right)$. | + | 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. | ||
