nLab suplattice




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 TVTV 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:SupLatPoset U \colon SupLat \to Poset

This has a left adjoint

F:PosetSupLat F \colon Poset \to SupLat

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

sS,sssS. s \in S, s' \le s \quad \implies \quad s' \in S.

This set of all downsets in PP, say P^\hat{P}, is ordered by inclusion, and it’s a suplattice: any union of downsets is a downset. There’s an embedding of PP in P^\hat{P} that sends each pPp \in P to its principal downset {sP:sp}\{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 BoolBool-enriched categories, where BoolBool is the monoidal category with two objects FF, TT and one nontrivial morphism FTF \implies T, its monoidal structure being “and”. Using this idea, the downsets of a poset PP correspond in a one-to-one way with BoolBool-enriched functors f:P opBoolf \colon P^{op} \to Bool, just as presheaves on a category CC are functors f:C opSetf \colon C^{op} \to Set. The embedding y:PP^y \colon P \to \hat{P} that sends each pPp \in P to its principal downset is the BoolBool-enriched version of the Yoneda embedding. So, just as the category of presheaves on a category CC is the free cocomplete category on CC, P^\hat{P} is the free cocomplete BoolBool-enriched category on PP. But a cocomplete BoolBool-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 ξ:P^P\xi: \hat{P} \to P is left adjoint to the Yoneda embedding y:PP^y: P \to \hat{P}. This makes suplattices the same thing (up to equivalence) as total categories in the BoolBool-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, SupLatSupLat is a symmetric monoidal closed category where the internal hom Hom(P,Q)Hom(P, Q) between two suplattices is the suplattice of cocontinuous maps PQP \to Q, which are the same as left adjoints PQP \to Q according to the poset version of the adjoint functor theorem.

SupLatSupLat is also monadic over SetSet, where the monad P:SetSetP: 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 RelRel may be thought of as the free suplattices.

As stated above, the symmetric monoidal closed category SupLatSupLat is a star-autonomous category where the star-involution takes a suplattice PP to the opposite poset P opP^{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 PQP \otimes Q may be formed as the suplattice Hom(P,Q op) opHom(P, Q^{op})^{op}. The presence of the equivalence

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

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


Last revised on February 6, 2024 at 12:12:25. See the history of this page for a list of all contributions to it.