A categorical generalization of Hrushovski’s limit structure

In [4], Hrushovski refutes a conjecture of Zilber by constructing a strongly minimal structure with certain geometric properties. The first part of this construction consists of extracting a limit structure from a class of finite structures via an adaptation of Fraïssé’s amalgamation construction; we are interested in generalizations of the construction of this limit structure. In [5], Wagner presents an axiomatization of the construction of Hrushovski’s limit structure. In [1], Droste and Göbel present a very abstract category theoretic construction that generalizes that of [5]. In this paper, we construct a limit structure in an intermediate level of abstraction: we generalize the construction of [5] using concepts from category theory, but we keep our focus on structures and embeddings, rather than going into the full generality of categories. For a formal statement of our main result, see Theorem 4.8. Informally, suppose C is a category whose objects are some class of structures and whose morphisms are some class of embeddings; further suppose that all the objects of C are well-approximated by the finitely generated objects of C (see Definition 3.8 for a precise definition). We can relativize familiar model-theoretic concepts such as universality, homogeneity, JEP, and AP to the category C; see Definition 3.2 and Definition 4.1. For example, an object X of C is universal with respect to the finitely generated objects of C if for every finitely generated object A of C, there is a morphism A → X. Our main result is that C has an object that is universal and homogeneous with respect to the finitely generated objects of C if and only if the finitely generated objects of C have JEP and AP; furthermore, if such an object exists, it is unique up to an isomorphism in C. In Section 2, we give a brief overview of Wagner’s axiomatization of Hrushovski’s limit construction, as presented by Ferreira in [2, Section 4.1]. In Section 3, we present our ω-generated categories of L-structures and show that they cover the setting of Hrushovski amalgamations. In Section 4, we prove our generalization. In this paper, we draw on both category theory and model theory. Some confusion may arise when dealing with a category of L-structures and L-embeddings such that the categorical and model-theoretic notions of isomorphism and automorphism do not coincide. To avoid confusion, whenever we are discussing the model theoretic notion, we will prefix it with the language in question; whenever we are discussing the categorical notion, we will prefix it the category in question. So “L-automorphism” and “L-isomorphism” refer to the model theory notions, while “C-automorphism” and “C-isomorphism” refer to the categorical notions.