The original exercise was this: Show that the Completeness Property for the real number system could equally well have been defined by the statement, “Any nonempty set of reals that has a lower bound has a greatest lower bound.”
What I have tried:
These are diagrams I made to try to understand the relationships of components:
Commutative diagram with statements
Commutative diagram with components as arrows
Commutative diagram with isomorphisms
In these diagrams it doesn't matter to what node we connect the domain, because the middle part commutes. It is as if the middle part was a single node, all the nodes and arrows are the same for the input.
I believe this is because lower bound and upper bound are symmetric notions. Flipping the ≤ sign in their definitions maps one to the other. Does this mean they are isomorphic? I think so. They are invertible, by flipping the sign we get one from the other.
The definitions of infimum and supremum have a similar (the same?) relationship.
Something that seems very interesting, is that if we have lower bounds, upper bounds, infimum and supremum as isomorphic arrows in a category there seem to be isomorphic functors mapping one to the other. There are similar functors between supremum and infimum, and there are natural transformations mapping these, and so on. Perhaps this is not the case, I'm a novice at this stuff, but my brain branches of into everywhich-way with these patterns.