Power Set - Power Object

Power Object

A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra.

Now the power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard subalgebras behave analogously to subsets.

However there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way.

Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs G and H, a homomorphism h: G → H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set HG of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of G as the multigraph ΩG, called the power object of G.

What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s,t: E → V giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a presheaf. Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Ω, called a subobject classifier. Although the term "power object" is sometimes used synonymously with exponential object YX, in topos theory Y is required to be Ω.