Generalised algebraic models

Algebraic theories and algebraic categories offer an innovative and revelatory
description of the syntax and the semantics. An _algebraic theory_ is a
concrete mathematical object -- the concept -- namely a set of variables
together with formal symbols and equalities between these terms; stated
otherwise, an algebraic theory is a small category with finite products. An
algebra or model of the theory is a set-theoretical interpretation -- a
possible meaning -- or, more categorically, a finite product-preserving
functor from the theory into the category of sets. We call the category of
models of an algebraic theory an _algebraic category_. By generalising the
theory we do generalise the models. This concept is the fascinating aspect of
the subject and the reference point of our project. We are interested in the
study of categories of models. We pursue our task by considering models of
different theories and by investigating the corresponding categories of models
they constitute. We analyse l _ocalizations_ (namely, fully faithful right
adjoint functors whose left adjoint preserves finite limits) of algebraic
categories and localizations of presheaf categories. These are still
categories of models of the corresponding theory.We provide a classification
of localizations and a classification of _geometric morphisms_ (namely,
functors together with a finite limit-preserving left adjoint), in both the
presheaf and the algebraic context.