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I am getting ready for the World Congress and School on Universal Logic in Vichy, France on June 16-26. I'm learning about different aspects of logic and trying to wrap my mind around what logic is. I'm listening in on various videos and some questions arise that it would be helpful to discuss.
I have been listening to Taylor Dupuy's videos on model theory. I like how he mapped out a hierarchy of logics, and how he compares various logics, categories and set theory concepts. I became aware of Heyting categories/algebras for intuitionistic logic. And today I watched a video, Category Theory: Topos Logic by Dr.Marni Sheppeard. She gave a nice distinction between classical logic as visualized with set theory, and topological spaces, and vector spaces. In the case of topological spaces, the "topological complement" of an open set U would be "the largest open set in the complement", or more precisely, the union of all of the sets Vi that are disjoint from U. Which is to say, the complement of an open set need not be open, and so U ∨ ¬U need not be the totality. Similarly, the vector space example shows that distribution laws may not hold, because the union of two nonparallel lines would be the plane that they span, with the consequence that a third line may intersect with that plane to yield a line, whereas it may intersect with the original lines to simply yield two points. These were helpful examples of nonclassical logics.
This reminded me of what I've been learning in this class about adjoints. I am happy to think of adjoints as approximate inverses Left:D->C, Right:D->C, of a functor f:C->D from the left or from the right, much like we can have least upper bounds from above a sequence, and greatest lower bounds from below a sequence.
Here is my question: Is there a sense in which Left:D->C and Right:D->C can be considered complements of each other?