Algebraic lattices and locally finitely presentable categories

作者：Porst Hans E^*

来源：Algebra Universalis, 2011, 65(3): 285-298.

DOI：10.1007/s00012-011-0129-0

• algebraic lattice
• locally finitely presentable category
• subobject
• quotient
• internal equivalence relation
• congruence

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.

• 出版日期2011-5