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functor_category [2015/02/26 13:23]
nikolaj
functor_category [2015/03/16 23:01]
nikolaj
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-=== Idea ===+=== 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.  
 + 
 +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. 
 +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. 
 == Algebraic picture if the target is structured ==  == 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 $\omega+\lambda:​=\left(v\mapsto\omega(v)+\mu(v)\right)$.
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