CHAPTER 1
Introduction
Definable additive categories and their model theory are the topic of this paper.
We begin with background and preliminary results on additive categories. Then
definable categories, their properties and the morphisms between them are investi-
gated, as are certain associated topological spaces (“spectra”). It was in the model
theory of modules that these categories were first considered and model theory
provides some of the tools for exploring them. Some general model-theoretic back-
ground is presented, then various aspects of the model theory of definable categories
are considered.
There are new results in this paper but a substantial part is a working up,
into a unified form and in a general context, of results which are scattered across
the literature and sometimes are just ‘folklore’. Primarily, this paper is about the
model theory of additive categories but there are various category-theoretic and
algebraic results. Indeed, most of the results can be presented and proved using
either a model-theoretic or a functor-theoretic approach and, in writing this paper,
I have tried to illuminate the relation between these.
Suppose that R is a ring. A subcategory of the category, Mod-R, of right R-
modules is said to be definable if it is closed in Mod-R under direct products, direct
limits and pure submodules. It was in the model theory of modules that it was first
realised that there is a rich theory associated to such subcategories. In that context
they arise as the elementary (=axiomatisable) subclasses of modules closed under
products and direct summands and they are in bijection with the closed subsets of
the Ziegler spectrum, ZgR, of R (14.2). That is a topological space whose points are
certain indecomposable modules and whose topology was originally defined using
concepts from model theory but which may also be defined purely in terms of the
category Mod-R, alternatively in terms of a certain functor category associated to
Mod-R. That functor category, which also arises as the category of model-theoretic
pp-imaginaries, could equally be regarded as the real topic of this paper. One
purpose of the paper is to explain all this.
Another purpose is to develop everything in what is arguably the correct set-
ting: namely definable subcategories of finitely accessible additive categories. This
is what results when the ring R is replaced by any small preadditive category. In
this way the context is broadened to encompass functor categories themselves, cat-
egories of comodules, certain categories of sheaves of modules and a great variety
of particular natural examples.
Yet another purpose of the paper is to update the model theory of modules as
presented in [82], more generally the model theory of definable categories, now that
it has been reshaped through its interaction with additive category theory. There is
a book, [89], which is in a sense a successor of [82], but in that book model theory
per se has been de-emphasised. There is some overlap, especially with Part III of
1
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