An algebraic lattice is a lattice which is
such that every element is a join of compact elements.
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 $e$ is compact if, for every subset $S$ of the lattice, $e$ is less than or equal to the supremum of $S$ just in case $e$ is less than or equal to the supremum of some finite subset of $S$).
Here is an alternative formulation:
An algebraic lattice is a poset which is locally finitely presentable as a category.
This formulation suggests a useful way of viewing algebraic lattices in terms of Gabriel-Ulmer duality (but with regard to enrichment in truth values, instead of in $Set$).
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 $T$, the subobject lattice of an object in $T$-$Alg$ is an algebraic lattice (this class of examples explains the origin of the term “algebraic lattice”, which is due to Garrett Birkhoff). In fact, all algebraic lattices arise this way (see Theorem 2 below).
It is trivial that every finite lattice is algebraic.
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 $V$-enriched profunctor from $C$ to $D$ is a functor $D^{op} \times C \rightarrow V$; of course, with an opposite convention, one could similarly state a covariant equivalence).
There is a full embedding
to the category of $T_0$-spaces, taking an algebraic lattice $L$ to the space whose points are elements of $L$, and whose open sets $U$ are defined by the property that their characteristic maps
($\chi_U(a) = 1$ if $a \in U$, else $\chi_U(a) = 0$) are poset maps that preserve directed colimits. The specialization order of $i(L)$ is $L$ again.
Every $T_0$-space $X$ occurs as a subspace of some space $i(L)$ associated with an algebraic lattice. Explicitly, let $L(X)$ be the power set of the underlying set of the topology, $P{|\mathcal{O}(X)|}$, and define
to take $x$ to $N(x) \coloneqq \{U \in \mathcal{O}(X): x \in U\}$. This gives a topological embedding of $X$ in $i(L(X))$.
On similar grounds, if $U \colon AlgLat \to Set$ is the forgetful functor, then the 2-image of the projection functor $\pi \colon Set\downarrow U \to Set$ is the category of topological spaces $Top$. In more nuts-and-bolts terms, an object $(S, L, f \colon S \to U(L))$ gives a space with underlying set $S$ and open sets those of the form $f^{-1}(O)$, where $O$ ranges over the Scott topology on $L$. Notice that if $(f \colon S \to S', g \colon L \to L')$ is a morphism in $Set \downarrow U$, then $f$ is continuous with respect to these topologies. Therefore the projection $\pi \colon Set \downarrow U \to Set$ factors through the faithful forgetful functor $Top \to Set$. Thus, working in the factorization system (eso+full, faithful) on $Cat$, we have a faithful functor $2$-$im(\pi) \to Top$ filling in as the diagonal
But notice also that $Set \downarrow U \to Top$ is eso and full. It is eso because any topology $\mathcal{O}(S)$ on $S$ can be reconstituted from the triple $(S, P{|\mathcal{O}(S)|}, x \mapsto N(x) \colon S \to P{|\mathcal{O}(S)|})$. We claim it is full as well. For, every continuous map $X \to X'$ between topological spaces induces a continuous map between their $T_0$ reflections $X_0 \to X_{0}'$, and since algebraic lattices like $P{|\mathcal{O}(X)|}$ (being continuous lattices) are injective objects in the category of $T_0$ spaces, we are able to complete to a diagram
where the rightmost vertical arrow is Scott-continuous (and the horizontal composites are of the form $x \mapsto N(x)$). Finally, since $Set \downarrow U \to Top$ is eso and full, it follows that $2$-$im(\pi) \to Top$ 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 $AlgLat$, or equivalently of (set-theoretic) total equivalence relations on $T_0$ topological spaces.
One of our definitions of algebraic lattice is: a poset $L$ which is locally finitely presentable when viewed as a category. The completeness of $L$ means that right adjoints $L \to Set$ are representable, given by $L(p, -) \colon L \to Set$, and we are particularly interested in those representable functors that preserve filtered colimits. These correspond precisely to finitely presentable objects $p$, which in lattice theory are usually called compact elements. These compact elements are closed under finite joins.
By Gabriel-Ulmer duality, $L$ is determined from the join-semilattice of compact elements $K$ by $L \cong Lex(K^{op}, Set)$. Since the elements of $K^{op}$ are subterminal, we can also write $L \cong Lex(K^{op}, 2)$ where $2 = Sub(1)$.
(Porst)
If $C$ is a locally finitely presentable category and $X$ is an object of $C$, then
The lattice of subobjects $Sub(X)$,
the lattice of quotient objects (equivalence classes of epis sourced at $X$) $Quot(X)$,
the lattice of congruences (internal equivalence relations) on $X$
are all algebraic lattices.
This is due to Porst. Of course if $C$ is the category of algebras of an Lawvere theory, then the lattice of quotient objects of an algebra is isomorphic to its congruence lattice, as such $C$ is an exact category.
The following result is due to Grätzer and Schmidt:
Every algebraic lattice is isomorphic to the congruence lattice of some model $X$ of some finitary algebraic theory.
In particular, since every finite lattice is algebraic, every finite lattice arises this way. Remarkably, it is not known at this time whether every finite lattice arises as the congruence lattice of a finite algebra $X$. It has been conjectured that this is in fact false: see this MO discussion.
Another problem which had long remained open is the congruence lattice problem: is every distributive algebraic lattice the congruence lattice (or lattice of quotient objects) of some lattice $L$? The answer is negative, as shown by Wehrung in 2007: see this Wikipedia article.
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)-categories… | satisfying Giraud's axioms | inclusion of left exact localizations | generated under colimits from small objects | localization of free cocompletion | generated under filtered colimits from small objects | ||
---|---|---|---|---|---|---|---|
(0,1)-category theory | (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | ||
category theory | toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories | $\hookrightarrow$ | accessible categories |
model category theory | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | ||
(∞,1)-topos theory | (∞,1)-toposes | $\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories | $\hookrightarrow$ | accessible (∞,1)-categories |
Andrej Bauer, Lars Birkedal, Dana Scott, Equilogical Spaces, Theoretical Computer Science, 315(1):35-59, 2004. (web)
Olivia Caramello, A topos-theoretic approach to Stone-type dualities (arXiv:1103.3493)
The relation to locally finitely presentable categories is discussed in
That every algebraic lattice is a congruence lattice is proved in