Related papers: On the comparison of different notions of geometri…

Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology.…

We introduce a model category of spaces based on the definable sets of an o-minimal expansion of a real closed field. As a model category, it resembles the category of topological spaces, but its underlying category is a coherent topos. We…

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…

Cluckers and Lipshitz have shown that real closed fields equipped with real analytic structure are o-minimal. This generalizes the well-known subanalytic structure $\mathbb{R}_{\mathrm{an}}$ on the real numbers. We extend this line of…

We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive…

We study a metric-like structure on categories, showing that the concept of the limit of a sequence in a metric space and the concept of the colimit of a sequence in a category have a common generalization. The main concept is a norm on a…

We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open…

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure,…

The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging generalisation of semialgebraic, subanalytic and subpfaffian geometry. In these notes we try to show that this is in…

We construct a category, $\Omega$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a "spec datum" is introduced, as a certain relation between categories, of which one has been given a…

Let X be a definable sub-set of some o-minimal structure. We study the spectrum of X, in relation with the definability of types.

We give an elementary construction of an arbitrary differentially closed field and of a universal differential extension of a differential field in terms of Nash function fields. We also give a characterization of any Archimedean ordered…

We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.

We start an analysis of geometric properties of a structure relative to a reduct. In particular, we look at definability of groups and fields in this context. In the relatively one-based case, every definable group is isogenous to a…

We develop the notion of a geometric covering of a rigid space X, which yields a much larger class of covering spaces than that studied previously by de Jong. Geometric coverings of X are closed under disjoint unions and are \'etale local…

We consider notions of metrized categories, and then approximate categorical structures defined by a function of three variables generalizing the notion of $2$-metric space. We prove an embedding theorem giving sufficient conditions for an…

We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and…

We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform…

We define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a…

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various…