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| 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. | + | 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 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. 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 === | ||
