Chris Goughnor wrote:
> Since preorders don't guarantee antisymmetry, I'm a little concerned that meets and joins might not be unique.
You're right, they're not. They're unique when our preorder is a poset. (They may still not _exist_, but they're unique.)
> Is this a case in which category theory only concerns itself with "uniqueness up to isomorphism"?
Yes. (If we think of preorders as categories, we can even show off and define a poset to be a preorder where isomorphic objects are equal, but we're getting way ahead of ourselves here.)
> That seems reasonable enough at first, but it's at odds with phrases like "the meet" or "the join" and when considering the proof of Proposition 1.88, f, defined pointwise as a meet, doesn't seem like it's well-defined.
I'll have to check this out - thanks!
Category theorists often use the word "the" in a more sophisticated way where we can talk about "the" object with some property if that property determines the object up to a unique isomorphism. For example, they talk about "the" direct sum of vector spaces, or "the" 1-element set. If we talk this way, we can talk about "the" meet or join of elements in a preorder. But it would be risky for Fong and Spivak to talk this way without explaining why it's okay. Maybe they just slipped... or maybe they're talking about "the" meet or join in a situation where it's really unique, namely a poset (which they call a "skeletal poset").