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"Locales and toposes as spaces": points
Jun. 9th, 2008 07:36 pmI've completed the first reading of Steve Vickers' rather brilliant text, "Locales and toposes as spaces", from here
http://www.cs.bham.ac.uk/~sjv/
The problem with all these categorical games is that one can understand every neat trick separately, but there are just too many of them, and it's difficult to hold the resulting picture together. So I think I'll try to write them down, one trick at a time, and this might help me.
Trick of the day: points as arrows. If a category has a terminal object 1, then (global) points of object X are (defined as) arrows 1 → X. However, in many cases there are not enough global points, so for any object A people define "points of X at stage A" as arrows A → X. In particular, the "generic point" of X is simply the identity arrow, id : X → X.
http://www.cs.bham.ac.uk/~sjv/
The problem with all these categorical games is that one can understand every neat trick separately, but there are just too many of them, and it's difficult to hold the resulting picture together. So I think I'll try to write them down, one trick at a time, and this might help me.
Trick of the day: points as arrows. If a category has a terminal object 1, then (global) points of object X are (defined as) arrows 1 → X. However, in many cases there are not enough global points, so for any object A people define "points of X at stage A" as arrows A → X. In particular, the "generic point" of X is simply the identity arrow, id : X → X.
