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Matteo Capucci (he/him) (Jul 03 2020 at 16:39):Today I've come to the realization that in most of the work in my thesis I should be using a Boolean prealgebra instead of its posetal reflection.
Hence I'm very tempted to change the definitions upstream to use that prealgebra. This would solve many problems, but I'm worried the switch might just move the problems somewhere else, since a lot of the definitions I use would loosen up to 'up to posetal equivalence'.
The issue is nothing new for categorists, and actually trascends my little quid-pro-quo (which can be worked around, tbh). Many 'functors' we define are only define up to equivalence.
Is anybody aware of some pitfalls of these loose definitions? The only one I can come up with is that one has to employ a lot of (possibly global!) choice to well-define stuff at the set level. But this is a merely aestethical problem, at least to me and my goals.