nLab suplattice

Definition

A suplattice is a poset that has joins of arbitrary subsets (and in particular is a join-semilattice). By the adjoint functor theorem for posets, a suplattice necessarily has all meets as well and so is a complete lattice. However, a suplattice homomorphism preserves joins, but not necessarily meets. Furthermore, a large semilattice which has all small joins need not have all meets, but might still be considered a large suplattice (even though it may not even be a lattice).

Dually, an inflattice is a poset which has all meets, and an inflattice homomorphism is a monotone function that preserves all meets.

A frame (dual to a locale) is a suplattice in which finitary meets distribute over arbitrary joins. (Frame homomorphisms preserve all joins and finitary meets.)

The category SupLat of suplattices and suplattice homomorphisms admits a tensor product which represents “bilinear maps”, i.e. functions which preserve joins separately in each variable. Under this tensor product, the category of suplattices is a star-autonomous category in which the dualizing object is the suplattice dual to the object $TV$ of truth-values. A semigroup in this monoidal category is a quantale, including frames as a special case when the quantale is idempotent and unital. Modules over them are modules over quantales (quantic modules with special case of localic modules, used in the localic analogue of the Grothendieck’s descent theory in Joyal-Tierney 84).

The free suplattice on a poset

There is a forgetful functor

$$U \colon SupLat \to Poset$$

This has a left adjoint

$$F \colon Poset \to SupLat$$

where for any poset $P$, the suplattice $F(P)$ is the poset of downsets of $P$, ordered by inclusion. Here a downset of a poset $P$ is a subset $S \subseteq P$ such that

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

This set of all downsets in $P$, say $\hat{P}$, is ordered by inclusion, and it’s a suplattice: any union of downsets is a downset. There’s an embedding of $P$ in $\hat{P}$ that sends each $p \in P$ to its principal downset $\{s \in P : \; s \le p \}$. (To give a downset is to give an antichain, and so the free suplattice is sometimes described equivalently in terms of antichains.)

To understand this description of the free suplattice on a poset, some enriched category theory is useful. Preorders are the same 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”. Using this idea, the downsets of a poset $P$ correspond in a one-to-one way with $Bool$-enriched functors $f \colon P^{op} \to Bool$, just as presheaves on a category $C$ are functors $f \colon C^{op} \to Set$. The embedding $y \colon P \to \hat{P}$ that sends each $p \in P$ to its principal downset 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 $C$, $\hat{P}$ is the free cocomplete $Bool$-enriched category on $P$. But a cocomplete $Bool$-enriched category that happens to be a poset is just the same as a suplattice.

The category of suplattices

The category of suplattices is monadic over the category of posets, and each algebra structure $\xi: \hat{P} \to P$ is left adjoint to the Yoneda embedding $y: P \to \hat{P}$. This makes suplattices the same thing (up to equivalence) as total categories in the $Bool$-enriched sense. Notice that algebra structure maps, being left adjoints, are cocontinuous and therefore suplattice morphisms. This makes the monad a commutative monad, and therefore according to general theory, $SupLat$ is a symmetric monoidal closed category where the internal hom $Hom(P, Q)$ between two suplattices is the suplattice of cocontinuous maps $P \to Q$, which are the same as left adjoints $P \to Q$ according to the poset version of the adjoint functor theorem.

$SupLat$ is also monadic over $Set$, where the monad $P: Set \to Set$ is the covariant power set functor. It therefore is a complete and cocomplete Barr-exact category. The Kleisli category of this monad is equivalent to Rel, the category of sets and relations. Thus the objects of $Rel$ may be thought of as the free suplattices.

As stated above, the symmetric monoidal closed category $SupLat$ is a star-autonomous category where the star-involution takes a suplattice $P$ to the opposite poset $P^{op}$. In part this says that a suplattice is also an inflattice, a fact which holds internally in any topos (where we use an internal covariant power-object functor to form an appropriate monad). Thus the tensor product $P \otimes Q$ may be formed as the suplattice $Hom(P, Q^{op})^{op}$. The presence of the equivalence

$$\ast = (-)^{op}: SupLat^{op} \to SupLat$$

(which takes a morphism $f: P \to Q$ to $g^{op}: Q^{op} \to P^{op}$, where $g$ is right adjoint to $f$) also means that colimits may be formed as appropriate limits, which are in turn formed pointwise by monadicity over $Set$.

References

• André Joyal, Miles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp. (ISBN: 978-1-4704-0722-3)