Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires *concept mining* and *concept analysis* to extract shared semantics for different network nodes. Powerful semantic analysis systems have been the drivers of nearly all paradigm shifts on the web. In categorical terms, most of them can be described as bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style completions from posets to suitably enriched categories. Yet it has been well known for more than 40 years that ordinary categories themselves in general do not permit such completions. Armed with this new semantical view of Dedekind-MacNeille completions, and of matrix bicompletions, we take another look at this ancient mystery. It turns out that simple categorical versions of the *limit superior* and *limit inferior* operations characterize a general notion of Dedekind-MacNeille completion, that seems to be appropriate for ordinary categories, and boils down to the more familiar enriched versions when the limits inferior and superior coincide. This explains away the apparent gap among the completions of ordinary categories, and broadens the path towards categorical concept mining and analysis, opened in previous work.Comment: 22 pages, 5 figures and 9 diagram

Proto-exact categories of matroids, Hall algebras, and K-theory

This paper examines the category $\mathbf{Mat}_{\bullet}$ of pointed matroids and strong maps from the point of view of Hall algebras. We show that $\mathbf{Mat}_{\bullet}$ has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory $K_* (\mathbf{Mat}_{\bullet})$ of $\mathbf{Mat}_{\bullet}$ via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections $$\pi^s_n (\mathbb{S}) \hookrightarrow K_n (\mathbf{Mat}_{\bullet})$$ from the stable homotopy groups of spheres for all $n$. Finally, we show that the Hall algebra of $\mathbf{Mat}_{\bullet}$ is a Hopf algebra dual to Schmitt's matroid-minor Hopf algebra.Comment: 29 page

Measurement spaces

The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a process via which a finite amount of classical information is generated. This motivates a mathematical definition of space of measurements that consists of a topological stably Gelfand quantale whose open sets represent measurable physical properties. It also accounts for the distinction between quantum and classical measurements, and for the emergence of "classical observers." The latter have a relation to groupoid C*-algebras, and link naturally to Schwinger's notion of selective measurement