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 limit_._category_theory [2016/03/07 13:58]nikolaj limit_._category_theory [2016/03/07 14:01] (current)nikolaj Both sides previous revision Previous revision 2016/03/07 14:01 nikolaj 2016/03/07 13:58 nikolaj 2016/03/07 13:57 nikolaj 2015/03/16 18:13 nikolaj 2014/09/30 17:01 nikolaj 2014/09/30 17:01 nikolaj 2014/09/30 17:00 nikolaj 2014/09/30 16:59 nikolaj 2014/09/28 23:59 nikolaj 2014/09/28 23:58 nikolaj 2014/09/28 20:03 nikolaj old revision restored (2014/07/17 19:49)2014/09/28 19:56 nikolaj 2014/07/17 19:49 nikolaj 2014/07/13 13:29 nikolaj 2014/07/13 13:29 nikolaj 2014/07/13 13:28 nikolaj 2014/07/13 01:47 nikolaj 2014/07/13 01:46 nikolaj 2014/07/13 01:07 nikolaj 2014/07/13 01:07 nikolaj 2014/07/13 01:04 nikolaj 2014/07/13 01:03 nikolaj 2014/07/13 01:03 nikolaj 2014/07/13 01:02 nikolaj 2014/07/12 23:24 nikolaj 2014/07/12 23:22 nikolaj 2014/07/12 21:17 nikolaj 2016/03/07 14:01 nikolaj 2016/03/07 13:58 nikolaj 2016/03/07 13:57 nikolaj 2015/03/16 18:13 nikolaj 2014/09/30 17:01 nikolaj 2014/09/30 17:01 nikolaj 2014/09/30 17:00 nikolaj 2014/09/30 16:59 nikolaj 2014/09/28 23:59 nikolaj 2014/09/28 23:58 nikolaj 2014/09/28 20:03 nikolaj old revision restored (2014/07/17 19:49)2014/09/28 19:56 nikolaj 2014/07/17 19:49 nikolaj 2014/07/13 13:29 nikolaj 2014/07/13 13:29 nikolaj 2014/07/13 13:28 nikolaj 2014/07/13 01:47 nikolaj 2014/07/13 01:46 nikolaj 2014/07/13 01:07 nikolaj 2014/07/13 01:07 nikolaj 2014/07/13 01:04 nikolaj 2014/07/13 01:03 nikolaj 2014/07/13 01:03 nikolaj 2014/07/13 01:02 nikolaj 2014/07/12 23:24 nikolaj 2014/07/12 23:22 nikolaj 2014/07/12 21:17 nikolaj 2014/07/12 21:15 nikolaj 2014/07/12 21:06 nikolaj 2014/07/12 21:05 nikolaj 2014/07/12 21:04 nikolaj old revision restored (2014/07/12 16:31)2014/07/12 16:33 nikolaj 2014/07/12 16:32 nikolaj 2014/07/12 16:31 nikolaj 2014/07/12 16:30 nikolaj 2014/07/12 16:10 nikolaj 2014/07/12 15:03 nikolaj 2014/07/03 11:50 nikolaj old revision restored (2014/07/03 11:48)2014/07/03 11:50 nikolaj created2014/07/03 11:48 nikolaj removed2014/07/01 16:32 nikolaj 2014/06/14 21:54 nikolaj 2014/06/14 21:54 nikolaj 2014/06/14 21:52 nikolaj 2014/06/14 21:52 nikolaj created 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 ===