is the category whose objects are frames and whose morphisms are frame homomorphisms, that is lattice homomorphisms that preserve directed joins (and thus all joins). is a subcategory of Pos, in fact a replete subcategory of both DistLat and SupLat.
is given by a variety of algebras, or equivalently by an algebraic theory, so it is an equationally presented category; however, it requires operations of arbitrarily large arity. Nevertheless, it is a monadic category (over Set), because it has free objects. Specifically, the free frame on a set is the lattice of upper subsets of the free join-semilattice on , that is .