Joyal wrote:
Many people seem to distrust the equality relation between the objects of a (large) category. Is this a philosophical conundrum or a mathematical problem?
I don't even trust it for the *-autonomous category with two objects and three morphisms. This is the basis for the classical notion of truth as normally understood when not trying to parse the sentences of constructive mathematicians. Some dark night someone could sneak into the basement where the foundations of mathematics are stored and switch true and false around, and in the morning we'd all wake up talking Doublespeak without even realizing it. Are you suggesting we should trust the equality relation between the objects of that category? Useful categories don't come much smaller. It can't be a question of size because the free Heyting algebra on one generator is infinite yet it doesn't have this structural ambiguity of classical logic: we can trust its equality relation because we can tell which element is the generator even if the arrows are accidentally flipped---it's a sort of error-detection feature of that Heyting algebra not possessed by any Boolean algebra. This is a reason to prefer intuitionistic logic over classical (not one I'd use myself mind you). Best, Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]