Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
|
limit_._category_theory [2014/09/30 17:01] nikolaj |
limit_._category_theory [2016/03/07 13:57] nikolaj |
||
|---|---|---|---|
| Line 5: | Line 5: | ||
| | @#FFBB00: definiendum | @#FFBB00: $\mathrm{lim}\,F$ ... terminal morphism from $\Delta$ to $F$ | | | @#FFBB00: definiendum | @#FFBB00: $\mathrm{lim}\,F$ ... terminal morphism from $\Delta$ to $F$ | | ||
| - | ==== Discussion ==== | + | ----- |
| === Elaboration === | === Elaboration === | ||
| A limit of a functor with image in ${\bf C}$, if it exists, is a particular terminal morphism in the functor category ${\bf C}^{\bf D}$. That definition emphasizes the underlying universal property, but it's a little hard to understand what's really going on. The reason is that a morphisms in a functor category is a natural transformation and this means there are a couple algebraic identities which don't get explicitly mentioned in the definition above. If you don't know limits, skip the next section and read the ones using cones first. | A limit of a functor with image in ${\bf C}$, if it exists, is a particular terminal morphism in the functor category ${\bf C}^{\bf D}$. That definition emphasizes the underlying universal property, but it's a little hard to understand what's really going on. The reason is that a morphisms in a functor category is a natural transformation and this means there are a couple algebraic identities which don't get explicitly mentioned in the definition above. If you don't know limits, skip the next section and read the ones using cones first. | ||
| + | |||
| + | 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 $F$, through which loads of other naturla transformations factor. | ||
| We 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 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. | ||
| Line 22: | Line 24: | ||
| In the more abstract definition using the functor category, these several morphisms are encoded into a natural transformation (=a single morphism, but in another category). The trick is that a natural transformation is an arrow in a functor category and this way one can specify the limit as terminal morphism. | In the more abstract definition using the functor category, these several morphisms are encoded into a natural transformation (=a single morphism, but in another category). The trick is that a natural transformation is an arrow in a functor category and this way one can specify the limit as terminal morphism. | ||
| + | |||
| + | === Discussion === | ||
| + | The object of the limit is like the full solution to a problem posed by the concept captured by the functor. | ||
| === Examples === | === Examples === | ||
| Line 34: | Line 39: | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Limit_%28category_theory%29|Limit (category theory)]] | Wikipedia: [[http://en.wikipedia.org/wiki/Limit_%28category_theory%29|Limit (category theory)]] | ||
| - | ==== Parents ==== | + | ----- |
| === Context === | === Context === | ||
| [[Functor category]], [[Diagonal functor]] | [[Functor category]], [[Diagonal functor]] | ||
| === Subset of === | === Subset of === | ||
| [[Terminal morphism]] | [[Terminal morphism]] | ||
