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The really nice thing would be if it's the poset reflection of the free completion or free cocompletion as a category, although I doubt that it is. But perhaps it is a suitably enriched free (co)completion; I'll think about that.
The characterisation is nice and category-theoretic already; complete lattices form a reflective subcategory of . But presumably you know that, since you linked to completion.
I think there's a general biclosed bicategory niche into which things like MacNeille completions and Isbell envelopes fit, as follows: if is a 1-morphism in a biclosed bicategory (meaning we have right adjoints to composing with an arrow on either side, i.e., right Kan lifts and right Kan extensions), then for any object $c$ there is a contravariant adjunction between hom-categories
which in one direction maps a 1-morphism to the Kan lift of through , and in the other maps a 1-morphism to the right Kan extension of $h$ along . The point is that we have equivalences
and hence units and of suitable adjunctions.
One special case is MacNeille completion, where the bicategory is and is a binary relation on a set given by a partial order . (The letter 'h'$ is meant to suggest "hom", and at any rate in many practical cases will be a monad in the bicategory, as it is in the case of a partial order seen as a monad in .) A relation is tantamount to a subset of , and the right Kan extension of along is , the set of upper bounds of . Similarly a relation is tantamount to a subset of , and the right Kan lift of through is , the set of lower bounds of .
Another example is "conjugation" or "Isbell envelope". Here the bicategory is (bimodules between categories), and we can take to be any endobimodule, and in particular the unit bimodule . A 1-morphism is tantamount to a functor , and the right Kan extension of along is a 1-morphism , tantamount to a functor , given by the formula
In the other direction, the right Kan lift of through a 1-morphism (tantamount to a functor ) is given by the formula
This gives an contravariant adjoint pair (called conjugation) whose associated monad on is the Isbell envelope construction.
I think the saturations as in Stacey's work on diffeologies can also be fit in this general framework.
The "I think" was unnecessary, I think :-). Maybe I put it there because it the shape of the comment was still being worked out in my head.
When I saw what Toby inserted about Galois connections, I recalled that the typical source of Galois connections between power sets PX and PY is provided by a relation from X to Y. ("I think" I can prove they all arise this way, but I'll get back to that.) So I asked myself: what's the relation in this case? When I saw it was the partial order, which could be interpreted as hom, I recognized it as being a special case of something more general that Lawvere talks about in connection with dualities and adjunctions of algebro-geometric type. The comment then flowed from this observation.
I'll try to think of a good name and home for this in the Lab (there might be one already), but let me see about this typical source here and then I'll check in with Galois connection after I've had some breakfast. Yes, a Galois connection is given by an adjoint pair . But the left adjoint part preserves colimits (unions), and since every element in PX is a union of singletons or atoms, it is determined by a functor (function) , which is to say a relation from . So the whole adjunction is determined by a relation, and then it's just a matter of calculating the exact formulaic relationship. Something analogous but more general can be made at the level of enriched bimodules.
I've just added some remarks to Galois connection on the case of power sets, where they all arise from relations as I indicated above.
I've now just consulted the article Isbell envelope, and I got a bit of terminology wrong two comments above, although the general spirit is right. I should not have said that the Isbell envelope is the associated monad on presheaves or co-presheaves; those monads are rather the saturations described in that article. The Isbell envelope is a category, not a monad (knowing full well here that categorical concepts interpenetrate each other to such a degree that e.g. categories are monads if looked at the right way (-: ).
What I really want to think about though is the connection between saturations and bicompleteness. It is the bicompletion aspect that is important to MacNeille completion, and this ought to fit in a more general enriched category theory context, and I dimly recall working through such things a long time ago.
The Joy of Cats uses a lot of terminology that I haven't seen elsewhere, some of which (like ‘source’) conflicts with terminology that I use a lot.
I like their ideas and wish that I knew more. But in general, I would state it explicitly whenever we draw terminology from them, in case it conflicts with others down the line.
So an apparent asymmetry, or does cocompletion come for free?
I'm not sure if this is what you're asking about, but a poset is complete if and only if it is cocomplete.
Arguably, this is not so much about posets as about smallness; a small category is complete if and only if it is cocomplete. Of course, a small category which is either is a p(r)oset! But a large poset can be complete without being cocomplete.
Of course, a small category which is either [complete or cocomplete] is a p(r)oset!
Only if you assume classical logic! The effective topos contains an internal category which is complete and cocomplete in its internal logic, yet not a preorder.
The Joy of Cats uses a lot of terminology that I haven't seen elsewhere, some of which (like ‘source’) conflicts with terminology that I use a lot.
This is getting off-topic, but "source" (and its dual "sink") are not unique to JoC; I think they are fairly common in the literature on topological categories.
I’m not sure if this is what you’re asking about, but a poset is complete if and only if it is cocomplete.
Gosh. How have I never realised this? So when it is said that an injective object on Poset is complete, it could have said cocomplete. But is it morally one rather than the other, or both at the same time?
@ Mike
Do you know if we still have a constructive theorem that a small complete category must be cocomplete? That would be neat.
@ David
There is a moral difference, I'm sure. In particular, the equivalence between complete and cocomplete posets does not apply to large posets. It also doesn't apply if you don't accept the existence of (small) power sets, that is if your foundations of mathematics are predicative.
Yes, absolutely. Use the adjoint functor theorem.
A nice example of a large poset which is cocomplete but not complete is the class of pure sets under set-inclusion. Any set of pure sets has a union, so this class is cocomplete, but it has no terminal object.
@ David again
The proof that a complete lattice is injective in the category of posets works just as well for cocomplete lattices. It relies on our asking for injectivity only within the category of small posets but not on the poset itself being small. (That is, even in the category of large posets, a smally complete or smally cocomplete poset is M-injective where M is the class of embeddings of a small poset into a small poset, or into a large one for that matter.) And the argument is predicative; even though there may not be very many complete lattices at all predicatively, the argument applies to any that do exist.
I don't understand the argument in The Joy of Cats for the converse, that an injective poset must be complete, and I haven't come up with my own. But presumably this argument only applies to small posets and is not predicative. (And if it relies on the existence of the MacNeille completion, as JoC suggests, then that is so.)
@ Mike
A simpler (from a structural perpsective) example is the class of well-ordered sets, or equivalently the class of ordinal numbers. It works the same way: suprema and inhabited infima, but no top.
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