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limit_._category_theory [2016/03/07 13:58]
nikolaj
limit_._category_theory [2016/03/07 14:01] (current)
nikolaj
Line 11: Line 11:
 Firstly, keep in mind that in a functor category ${\bf C}^{\bf D}$, the main data of a terminal morphism to $F\in {\bf C}^{\bf D}$ is a natural transformation $\phi$ from some functor to $F$. Loads of other natural transformations factor through this $\phi$.  ​ Firstly, keep in mind that in a functor category ${\bf C}^{\bf D}$, the main data of a terminal morphism to $F\in {\bf C}^{\bf D}$ is a natural transformation $\phi$ from some functor to $F$. Loads of other natural transformations factor through this $\phi$.  ​
  
-We now restate the above definition using a little more prose: The [[diagonal functor]] $\Delta$ maps an object $N$ in ${\bf C}$ to a pretty degenerate functor $\Delta(N)$ in ${\bf C}^{\bf D}$, namely the constant functor $\Delta(N)$,​ which itself returns $N$ on any object. A limit is a terminal morphism from $\Delta$ to $F$, which means that it's a pair $\langle L,​\phi\rangle$,​ with $L$ an object in ${\bf C}$ and $\phi:​\Delta(L)\to F$ a natural transformation,​ so that any other natural transformation $\psi:​\Delta(N)\to F$ factors as $\psi=\phi\circ u$, with $u:​\Delta(N)\to\Delta(L)$ some other natural ​transformation. Now $\Delta(N)(X):​=N$ for all $X$, i.e. the images of the constant functors comprise only one object. Therefore the arrow $u$ is determined by a single component $u_N$ from $N$ to $L$ and similarly, $\psi$'​s components $\psi_X$ are all of type $N\to F(X)$. The fact that the domains of the components are always the same makes this definition equivalent with that in terms of the cone concept below - a cone is made up from the same data as a natural transformation where all components have the same domain.+We now restate the above definition using a little more prose: The [[diagonal functor]] $\Delta$ maps an object $N$ in ${\bf C}$ to a pretty degenerate functor $\Delta(N)$ in ${\bf C}^{\bf D}$, namely the constant functor $\Delta(N)$,​ which itself returns $N$ on any object. A limit is a terminal morphism from $\Delta$ to $F$, which means that it's a pair $\langle L,​\phi\rangle$,​ with $L$ an object in ${\bf C}$ and $\phi:​\Delta(L)\to F$ a natural transformation,​ so that any other natural transformation $\psi:​\Delta(N)\to F$ factors as $\psi=\phi\circ u$, with $u:​\Delta(N)\to\Delta(L)$ some other natural ​Transformation (as a not so relevant remark, $u$ is the arrow image of $\Delta$ for some arrow in ${\bf C}$). Now $\Delta(N)(X):​=N$ for all $X$, i.e. the images of the constant functors comprise only one object. Therefore the arrow $u$ is determined by a single component $u_N$ from $N$ to $L$ and similarly, $\psi$'​s components $\psi_X$ are all of type $N\to F(X)$. The fact that the domains of the components are always the same makes this definition equivalent with that in terms of the cone concept below - a cone is made up from the same data as a natural transformation where all components have the same domain.
  
 === Alternative definitions === === Alternative definitions ===
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