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terminal_morphism [2014/09/12 10:53] nikolaj |
terminal_morphism [2014/09/28 19:46] nikolaj |
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| | @#FFFDDD: for all | @#FFFDDD: $\psi:{\bf C}[FA,Z]$ | | | @#FFFDDD: for all | @#FFFDDD: $\psi:{\bf C}[FA,Z]$ | | ||
| | @#DDDDDD: range | @#DDDDDD: $f:{\bf D}[A,B]$ | | | @#DDDDDD: range | @#DDDDDD: $f:{\bf D}[A,B]$ | | ||
| - | | @#55EE55: postulate | @#55EE55: $\exists!f.\ \psi=\phi\circ F(f)$ | | + | | @#55EE55: postulate | @#55EE55: $\exists_!f.\ \psi=\phi\circ F(f)$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| + | === Terminology === | ||
| Note that the name "terminal //morphisms//" for $\langle B,\phi\rangle$ is slightly confusing, because the data is not just the morphism $\phi$ going from $F(B)$ to $Z$, but also the object $B$. | Note that the name "terminal //morphisms//" for $\langle B,\phi\rangle$ is slightly confusing, because the data is not just the morphism $\phi$ going from $F(B)$ to $Z$, but also the object $B$. | ||
| - | Terminal and [[initial morphism]]s (the opposite notion, where you just flip some arrows) are called universal morphisms. They are a very important concept with a ridiculously broad range of examples, because you can essentially translate all "forall-there exists such that" characterizations to a "there is a universal morphism such that" characterizations. Another remark: Under a certain condition (namely that the construction works for all objects of a cateogry), the universal morphism gives an adjoint functor pair and indeed all adjoint functor pairs can be seen to arise that way. | + | === Idea === |
| + | You want to specify a smaller world within ${\bf C}$. You do this by setting up another category ${\bf D}$ and mapping it into ${\bf C}$ with a functor $F$. The image of $F$ is your smaller world. | ||
| - | In the following I'm going to elaborate what's going on a bit: | + | For a fixed object $Z$, a universal morphism $\phi$ is then defined by demanding that every arrow from this small world which tries to connect $Z$, must pass through it. In other words (and now I just state the definition): For all objects $B$ in ${\bf D}$, if $\psi$ is an arrow in ${\bf C}$ from $FB$ to $Z$, then there must be an arrow $f$ within ${\bf D}$, so that $\phi$ is really just a combination of "small world business" $F(f)$, followed by the kind of gate function $\phi$, finally leading to $Z$. |
| - | The main category you work in is ${\bf C}$. You fix an object $Z:\mathrm{Ob}_{\bf C}$ and what you want to do is to specify a special object and morphism with respect to it. | + | For the reason that everything going on in the image of $F$ must pass to $Z$ only through $\phi$, one says $\phi$ is the terminal morphism "from the functor $F$ to the object $Z$". |
| - | Examples: | + | |
| - | * You are given an object $Z$, which stores the information about two objects $X$ and $Y$. You want to characterize the product $X\times Y$ and the associated projection maps $\pi_1$ and $\pi_2$. With sets, that's the Cartesian product, but it also works with groups, respecting concatenation, or with topological spaces, this automatically introduces the product topology. | + | |
| - | * Say you already know what products $\times$ for your category are. Given an object $Z$, you want to characterize exponential objects $Z^Y$ and evaluation map $\mathrm{eval}:Z^Y\times Y\to Z$. For example, in the category of sets, if $Y$ and $Z$ are objects, then the hom-set from $Y$ to $Z$ is of course the function space $Y\to Z$, and $Z^Y$ is exactly that space //as object of // ${\bf Set}$. | + | |
| - | + | ||
| - | What the above definition says is the following: First you want to specify a smaller world within ${\bf C}$. You do this by setting up another category ${\bf D}$ and mapping it into ${\bf C}$ with a functor $F$. The image of $F$ is your smaller world. The universal morphism $\phi$ is then defined by demanding that every arrow from this small world, which tries to connect with the object $Z$, must pass through $\phi$. In other words (and now comes the definition): For all objects $B$ in ${\bf D}$, if $\psi$ is an arrow in ${\bf C}$ from $FB$ to $Z$, then there must be an arrow $f$ within ${\bf D}$, so that $\phi$ is really just a combination of "small world business" $F(f)$, followed by the kind of gate function $\phi$, finally leading to $Z$. For the reason that everything going on in the image of $F$ must pass to $Z$ only through $\phi$, one says $\phi$ is the terminal morphism "from the functor $F$ to the object $Z$". | + | |
| == Example: The exponential object and currying == | == Example: The exponential object and currying == | ||
