### Left Adjoints Between Categories of Posets

#### Posted by John Baez

I have a bunch of related questions about things like “the free lattice on a poset”, and so on. I would hope that at least some of these questions have nice answers, at least for finite posets. But I’m not having much luck finding them!

Specifically:

1) What’s the free join-semilattice (i.e. a poset where any finite subset has a supremum) on a poset?

2) What’s the free complete join-semilattice (i.e. a poset where any subset has a supremum) on a poset?

3) What’s the free lattice) (i.e. a poset where any finite subset has a supremum and infimum) on a poset?

4) What’s the free complete lattice (i.e. a poset where any subset has a supremum and infimum) on a poset?

5) What’s the free distributive lattice (i.e. a lattice where finite infs distribute over finite sups) on a poset?

6) What’s the free locale (i.e. a poset where any finite subset has an infimum and any subset has a supremum, and finite infs distribute over arbitrary sups) on a poset?

7) What’s the free completely distributive lattice (i.e. a complete lattice where arbitrary infs distribute over arbitrary sups) on a poset?

Of course these questions require specifying the allowed *morphisms* between the various kinds of objects; in many cases there are standard choices. There are also many other questions of this sort I’d be happy to hear insights about, like “what’s the free distributive lattice on a lattice?” Anyway, I’ll take whatever theorems you can tell me.

Mike Shulman suggests, quite believably, that the free complete join-semilattice on a poset $P$ is its set of **downsets**, i.e subsets $S \subseteq P$ such that

$s \in S, s' \le s \quad \implies \quad s' \in S.$

The set of all downsets in $P$, say $\hat{P}$, is ordered by inclusion, and it’s a complete join-semilattice: any union of downsets is a downset.

This makes a lot of sense because it’s analogous to how the category of presheaves on a category is its free cocompletion. In fact we can make this analogy quite precise, using enriched categories.

We can think of posets, or more generally preorders, as $Bool$-enriched categories where $Bool$ is the monoidal category with two objects $F$, $T$ and one nontrivial morphism $F \implies T$, its monoidal structure being “and”. The downsets of a poset $P$ correspond in a one-to-one way with order-preserving maps $f \colon P^{op} \to Bool$, just as presheaves on a category $C$ are functors $f \colon C^{op} \to Set$. There’s an embedding of $P$ in $\hat{P}$ that sends each $p \in P$ to its **principal** downset $\{s \in P : \; s \le p \}$. The principal downsets are the $Bool$-enriched version of representable presheaves, and the embedding of $P$ in $\hat{P}$ is the $Bool$-enriched version of the Yoneda embedding. So, just as the category of presheaves on a category $C$ is the free cocomplete category on $C$, $\hat{P}$ should be the free cocomplete $Bool$-enriched category with on $P$. But a cocomplete $Bool$-enriched category should be just the same as a complete join-semilattice, since joins are colimits.

Mike adds that probably the free sup-semilattice on $P$ likewise consists of downsets that are the union of finitely many principal ones.

So, maybe questions 1) and 2) have nice answers. Are they in the literature somewhere? What about the rest?

## Re: Left Adjoints Between Categories of Posets

Horn and Kimura (The category of semilattices, Algebra Universalis 1 (1971) no. 1, 26–38, Theorem 4.1) describe the free meet-semilattice $P^*$ on a partially ordered set $P$ as the set of all non-empty finite discrete subsets of P, with ordering defined by letting $S_1 \leq S_2$ if for all $p \in S_2$ there is a $q \in S_1$ such that $q \leq p$ in $P$. A subset is discrete if no two distinct members are comparable.

I guess that considering order-duals this will give you a description of the free join-semilattice on $P$ as well.