Algebraic lattices and locally finitely presentable categories

Abstract

We show that subobjects and quotients respectively of any object K in a locally finitely presentable category form an algebraic lattice. The same holds for the internal equivalence relations on K. In fact, these results turn out to be—at least in the case of subobjects—nothing but simple consequences of well known closure properties of the classes of locally finitely presentable categories and accessible categories, respectively. We thus get a completely categorical explanation of the well known fact that the subobject- and congruence lattices of algebras in finitary varieties are algebraic. Moreover we also obtain new natural examples: in particular, for any (not necessarily finitary) polynomial set-functor F, the subcoalgebras of an F-coalgebra form an algebraic lattice; the same holds for the lattices of regular congruences and quotients of these F-coalgebras.