Towards free localic algebras

Abstract

The purpose of this paper is to establish that the underlying objectfunctor from the models of a Lawvere theory to the base category creates limitsand coequalisers of all parallel pairs of homomorphisms whose underlying pairadmit a split coequaliser. We show that for a small complete category witha well behaved proper factorisation structure, the underlying functor admitsa left adjoint and the category of such models is precisely monadic over thebase category in the sense of Beck’s Theorem. In particular, this establishesthe existence of free localic algebras for any Lawvere theory, generalising theknown results for the existence of free localic groups.

Keywords

• beck's theorem
• freyd's adjoint functor theorem
• co-well-powered category
• lawvere theory
• monadicity
• nearly ptt