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Monday, August 15, 2005
Thesis Update
Today is a "feeling pretty good about the thesis" day. The general idea is this. My objective is to characterize CCD lattices in a topos of the form set^C^op. A CCD lattice is a specific kind of sup lattice, which is just a partially ordered set in which every subset has a sup.
The previous thing that had been done by other people was a characterization of sup lattices in set^C^op, under the assumption that C had pullbacks. A while ago, I managed to characterize CCD lattices under the assumption that C had infinite pullbacks. This seemed like a decent assumption to me, since CCD lattices are "sort of" an infinite version of sup lattices. Unfortunately (unbeknownst to me at the time), for set^C^op to be a topos, C has to be small, and being small and having infinite pullbacks is basically equivalent to C being a partially ordered set. Not good, now what I've done only works in a special case.
However, with an idea from my supervisor, I have now modified the original result to get a characterization of sup lattices in set^C^op for any C. I'm pretty confident that using the techniques of this modification can be applied to my result to remove the infinite pullback assumption. Then everything would be complete!
Unforunately, I'm not sure if I have time to finish this last step before I have to hand in my thesis to the readers. I'm pretty confident it'll all work out, it just won't all be in the actual thesis (provided I'm not missing something else.......).
Today is a "feeling pretty good about the thesis" day. The general idea is this. My objective is to characterize CCD lattices in a topos of the form set^C^op. A CCD lattice is a specific kind of sup lattice, which is just a partially ordered set in which every subset has a sup.
The previous thing that had been done by other people was a characterization of sup lattices in set^C^op, under the assumption that C had pullbacks. A while ago, I managed to characterize CCD lattices under the assumption that C had infinite pullbacks. This seemed like a decent assumption to me, since CCD lattices are "sort of" an infinite version of sup lattices. Unfortunately (unbeknownst to me at the time), for set^C^op to be a topos, C has to be small, and being small and having infinite pullbacks is basically equivalent to C being a partially ordered set. Not good, now what I've done only works in a special case.
However, with an idea from my supervisor, I have now modified the original result to get a characterization of sup lattices in set^C^op for any C. I'm pretty confident that using the techniques of this modification can be applied to my result to remove the infinite pullback assumption. Then everything would be complete!
Unforunately, I'm not sure if I have time to finish this last step before I have to hand in my thesis to the readers. I'm pretty confident it'll all work out, it just won't all be in the actual thesis (provided I'm not missing something else.......).
