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terminal_morphism [2014/09/26 15:58] 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 $B$. | ||
| + | |||
| + | === 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. | ||
| + | |||
| + | 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 $Z$. | ||
| + | |||
| + | 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 == | ||
