# Explicit lifting characterization of complete lattices among posets?

It's well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $$L$$ is a complete lattice if and only if $$L$$ has the right lifting property with respect to the class $$Emb$$ of all embeddings of posets. This remarkable characterization would be even more useful if there were some smaller, more explicit class of embeddings $$\mathcal E \subset Emb$$ sufficient to check injectivity. That is,

Questions:

1. Is there a good sub-class $$\mathcal E \subset Emb$$ of all poset embeddings such that a poset $$L$$ is a complete lattice if and only if $$L$$ has the right lifting property with respect to all the embeddings in $$\mathcal E$$?

2. What if we restrict attention to finite posets? That is: Is there a good subclass $$\mathcal E^{fin} \subset Emb^{fin}$$ of all embeddings of finite posets such that a finite poset $$L$$ is a lattice if and only if $$L$$ has the right lifting property with respect to all the embeddings in $$\mathcal E^{fin}$$?

What is meant by "good" is a bit subjective, but to start I'd be happy with anything non-tautological. Ultimately, it would be nice to have a class $$\mathcal E$$ or $$\mathcal E^{fin}$$ which is "explicit" in some sense, so that it actually becomes easier to check that a poset $$L$$ is complete via lifting properties. Normally, I'd hope for $$\mathcal E$$ to be small, but I'm pretty sure this is not possible.

One candidate I have in mind for $$\mathcal E$$ would be the collection of embeddings $$\{S \to S^\triangleright \mid S \text{ discrete}\} \cup \{S \to S^\triangleleft \mid S \text{ discrete}\}$$ which add a new top or bottom element to a discrete poset $$S$$. But I suspect this is too naive, as it would mean that any $$\infty$$-directed and $$\infty$$-codirected poset is a complete lattice -- this is probably false.

• Does it make sense to try and go back to Baer's criterion for modules? This might be an inspiration for defining the proper $\mathcal{E}$. In Popescu's book on abelian categories one can find a generalized version of Bear's criterion which holds for nice abelian categories with a generator. – Ivan Di Liberti Mar 23 '20 at 14:59

You are right that $$\mathcal E$$ cannot be small because complete lattices are not closed in posets under $$\lambda$$-filtered colimits for any regular cardinal $$\lambda$$. A candidate for $$\mathcal E$$ consists from embeddings of posets to their Mac-Neille completions. Since the Mac-Neille completion of a finite poset is finite, it works in the finite case too. But I think that you would not consider it to be "good".

After a bit of thought, I think I have a pretty good set for $$\mathcal E^{fin}$$ and a pretty good class for $$\mathcal E$$.

Proposition: Let $$L$$ be a finite poset. Then $$L$$ is a lattice if and only if $$L$$ has the right lifting property with respect to the following set $$\mathcal E^{fin}$$ of embeddings:

1. $$\emptyset \to 1$$

2. $$2 \to 2^\triangleleft$$ and $$2 \to 2^\triangleright$$ where $$2$$ is the discrete poset with two elements and these morphisms add a cone and cocone respectively.

3. $$\newcommand{\bbowtie}{\bowtie\mkern-17mu\bullet\mkern17mu}$$ $$\bowtie \to \bbowtie$$ where $$\bbowtie$$ is the 5-element poset $$x,y < a < p,q$$ and $$\bowtie$$ is the full sub-poset on $$x,y,p,q$$.

Proof: It suffices to show that $$L$$ is a meet-semilattice, i.e. that for every finite set $$S \subseteq L$$, the poset $$L \downarrow S$$ of elements under $$S$$ has a top element. It suffices to consider the cases (i) when $$S$$ is empty and (ii) when $$S$$ has two elements. Moreover, every finite directed poset has a top element, so it will suffice to show that (i) $$L$$ is directed and (ii) for every $$p, q \in L$$, the poset $$L \downarrow \{p,q\}$$ is directed. (i) follows from (1) and the second part of (2). (ii) follows from the first part of (2) and (3).

Analogously, using $$\infty$$-directedness in place of directedness, we have

Proposition: Let $$L$$ be a poset. Then $$L$$ is a complete lattice if and only if $$L$$ has the right lifting property with respect to the following class $$\mathcal E$$ of embeddings:

1. $$\emptyset \to 1$$

2. $$S \to S^\triangleleft$$ and $$S \to S^\triangleright$$ for each discrete poset $$S$$ (i.e. we add a top element and a bottom element, respectively, to $$S$$)

3. $$\newcommand{\bbowtie}{\bowtie\mkern-17mu\bullet\mkern17mu}$$ $$\bowtie_{S,S} \to \bbowtie_{S,S}$$ for each set $$S$$, where $$\bbowtie_{S,T}$$ is the poset whose under lying set is $$S \amalg T \amalg \{a\}$$, with $$S < a < T$$, and $$\bowtie_{S,T}$$ is the full subposet on $$S \amalg T$$.