nLab
algebraic lattice

Contents

Definition

Definition

An algebraic lattice is a lattice which is

An algebraic lattice is a complete lattice (equivalently, a suplattice, or in different words a poset with the property of having arbitrary colimits but with the structure of directed colimits/directed joins) in which every element is the supremum of the compact elements below it (an element ee is compact if, for every subset SS of the lattice, ee is less than or equal to the supremum of SS just in case ee is less than or equal to the supremum of some finite subset of SS).

Here is an alternative formulation:

Definition

An algebraic lattice is a poset which is locally finitely presentable as a category.

This formulation suggests useful a way of viewing algebraic lattices in terms of Gabriel-Ulmer duality (but with regard to enrichment in truth values, instead of in SetSet).

As this last formulation suggests, algebraic lattices typically arise as subobject lattices for objects in locally finitely presentable categories. As an example, for any (finitary) Lawvere theory TT, the subobject lattice of an object in TT-AlgAlg is an algebraic lattice (this class of examples explains the origin of the term “algebraic lattice”, which is due to Garrett Birkhoff).

Properties

The category of algebraic lattices

The morphisms most commonly considered between algebraic lattices are the finitary functors? between them, which is to say, the Scott-continuous functions between them; i.e., those functions which preserve directed joins (hence the parenthetical remarks above).

The resulting category AlgLat is cartesian closed and is dually equivalent to the category whose objects are meet semilattices (construed as categories with finite limits enriched over truth values) and whose morphisms are meet-preserving profunctors between them (using the convention that a VV-enriched profunctor from CC to DD is a functor D op×CVD^{op} \times C \rightarrow V; of course, with an opposite convention, one could similarly state a covariant equivalence).

There is a full embedding

i:AlgLatTop 0i \colon AlgLat \to Top_0

to the category of T 0T_0-spaces, taking an algebraic lattice LL to the space whose points are elements of LL, and whose open sets UU are defined by the property that their characteristic maps

χ U:L2\chi_U: L \to \mathbf{2}

(χ U(a)=1\chi_U(a) = 1 if aUa \in U, else χ U(a)=0\chi_U(a) = 0) are poset maps that preserve directed colimits. The specialization order of i(L)i(L) is LL again.

Every T 0T_0-space XX occurs as a subspace of some space i(L)i(L) associated with an algebraic lattice. Explicitly, let L(X)L(X) be the power set of the underlying set of the topology, P|𝒪(X)|P{|\mathcal{O}(X)|}, and define

X(iL)(X)X \to (i\circ L)(X)

to take xx to N(x){U𝒪(X):xU}N(x) \coloneqq \{U \in \mathcal{O}(X): x \in U\}. This gives a topological embedding of XX in i(L(X))i(L(X)).

Remark

On similar grounds, if U:AlgLatSetU \colon AlgLat \to Set is the forgetful functor, then the 2-image of the projection functor π:SetUSet\pi \colon Set\downarrow U \to Set is the category of topological spaces TopTop. In more nuts-and-bolts terms, an object (S,L,f:SU(L))(S, L, f \colon S \to U(L)) gives a space with underlying set SS and open sets those of the form f 1(O)f^{-1}(O), where OO ranges over the Scott topology on LL. Notice that if (f:SS,g:LL)(f \colon S \to S', g \colon L \to L') is a morphism in SetUSet \downarrow U, then ff is continuous with respect to these topologies. Therefore the projection π:SetUSet\pi \colon Set \downarrow U \to Set factors through the faithful forgetful functor TopSetTop \to Set. Thus, working in the factorization system (eso+full, faithful) on CatCat, we have a faithful functor 22-im(π)Topim(\pi) \to Top filling in as the diagonal

SetU Top 2-im(π) Set.\array{ Set \downarrow U & \to & Top \\ \downarrow & \nearrow & \downarrow \\ 2\text{-}im(\pi) & \to & Set. }

But notice also that SetUTopSet \downarrow U \to Top is eso and full. It is eso because any topology 𝒪(S)\mathcal{O}(S) on SS can be reconstituted from the triple (S,P|𝒪(S)|,xN(x):SP|𝒪(S)|)(S, P{|\mathcal{O}(S)|}, x \mapsto N(x) \colon S \to P{|\mathcal{O}(S)|}). We claim it is full as well. For, every continuous map XXX \to X' between topological spaces induces a continuous map between their T 0T_0 reflections X 0X 0X_0 \to X_{0}', and since algebraic lattices like P|𝒪(X)|P{|\mathcal{O}(X)|} (being continuous lattices) are injective objects in the category of T 0T_0 spaces, we are able to complete to a diagram

X X 0 P|𝒪(X)| X X 0 P|𝒪(X)|\array{ X & \to & X_0 & \to & P{|\mathcal{O}(X)|} \\ \downarrow & & \downarrow & & \downarrow \\ X' & \to & X_{0}' & \to & P{|\mathcal{O}(X')|} }

where the rightmost vertical arrow is Scott-continuous (and the horizontal composites are of the form xN(x)x \mapsto N(x)). Finally, since SetUTopSet \downarrow U \to Top is eso and full, it follows that 22-im(π)Topim(\pi) \to Top is eso, full, and faithful, and therefore an equivalence of categories.

This connection is explored in more depth with the category of equilogical spaces, which can be seen either as a category of (set-theoretic) partial equivalence relations over AlgLatAlgLat, or equivalently of (set-theoretic) total equivalence relations on T 0T_0 topological spaces.

Relation to locally finitely presentable categories

One of our definitions of algebraic lattice is: a poset LL which is locally finitely presentable when viewed as a category. The completeness of LL means that right adjoints LSetL \to Set are representable, given by L(p,):LSetL(p, -) \colon L \to Set, and we are particularly interested in those representable functors that preserve filtered colimits. These correspond precisely to finitely presentable objects pp, which in lattice theory are usually called compact elements. These compact elements are closed under finite joins.

By Gabriel-Ulmer duality, LL is determined from the join-semilattice of compact elements KK by LLex(K op,Set)L \cong Lex(K^{op}, Set). Since the elements of K opK^{op} are subterminal, we can also write LLex(K op,2)L \cong Lex(K^{op}, 2) where 2=Sub(1)2 = Sub(1).

Theorem

(Porst)

If CC is a locally finitely presentable category and XX is an object of CC, then

  • The lattice of subobjects Sub(X)Sub(X),

  • the lattice of quotient objects (equivalence classes of epis sourced at XX) Quot(X)Quot(X),

  • the lattice of congruences (internal equivalence relations) on XX

are all algebraic lattices.

This is due to Porst.

Completely distributive lattices

Proposition

The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattices.

This appears as (Caramello, remark 4.3).

The completely distributive algebraic lattices form a reflective subcategory of that of all distributive lattices. The reflector is called canonical extension.

See also compact element, compact element in a locale?.

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes\hookrightarrowalgebraic lattices\simeq Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes\hookrightarrowlocally presentable categories\simeq Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories\hookrightarrowaccessible categories
model category theorymodel toposes\hookrightarrowcombinatorial model categories\simeq Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes\hookrightarrowlocally presentable (∞,1)-categories\simeq
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories\hookrightarrowaccessible (∞,1)-categories

References

The relation to locally finitely presentable categories is discussed in

  • Hans Porst, Algebraic lattices and locally finitely presentable categories (pdf)

Revised on April 10, 2014 01:54:19 by Zoran Škoda (193.136.196.12)