I'm trying to think of how posets simplify the conditions of the general https://ncatlab.org/nlab/show/adjoint+functor+theorem. First, it is clear that limits and colimits in posets are only meets and joins, because diagram commutation is nothing mo…
Explaining this idea in terms of posets gives a nice, detailed and intuitive view of this classic fact about adjoints. Thanks for doing these lectures so thoroughly.
I'm surprised nobody's done these!
40 : Preimages preserve set operations (see 41), and partitions are collections of subsets defined by set operations. Then {union over f(partition)} = f(union over partition) = f*(Y) = X, and similarly for empty p…
34: there is no pair shared by both P and Q, so the meet is discrete.
35: conjunction of equivalence relations preserves reflexivity, symmetry, and transitivity (x~y~z P and x~y~z Q implies x~z P and x~z Q, etc.)
36: clearly this conjunction is fi…
Hello Bob, thank you! I'm sorry I just now saw this. Scuttlebutt looks very interesting; I definitely support better forms of social network.
Well, Dr. Baez and I are currently working "Enriched Lawvere Theories for Operational Semantics," primaril…
Ah, whoops! I hadn't seen this when I posted in lecture 4: "Boolean algebras are an important kind of poset. The power set functor defines an (contravariant) equivalence \(P: Set^{op}→Bool\): any function \(f: X→Y\) corresponds to the preimage map \…
Boolean algebras are an important kind of poset. The power set functor defines an (contravariant) equivalence \(P: \text{Set}^{\text{op}} \to \text{Bool}\): any function \(f: X \to Y\) corresponds to the preimage map \(f*: PY \to PX\), which is a mo…
Introducing category theory by means of order theory is a natural way to ease into the basics - functor, co/product, opposite, adjunction - but this has always seemed a bit odd to me. Mathematically, if not categorically, an element of a relation is…