Magnitude homology of enriched categories and metric spaces

Probabilistic metric spaces as enriched categories

Factorization Homology of Enriched ∞-categories

For an arbitrary symmetric monoidal∞-category V, we define the factorization homology of V-enriched∞-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace ...

Convergence and quantale-enriched categories

Generalising Nachbin's theory of ``topology and order'', in this paper we   continue the study of quantale-enriched categories equipped with a compact   Hausdorff topology. We compare these $V$-categorical compact Hausdorff spaces   with ultrafilter-quantale-enriched categories, and show that the presence of a   compact Hausdorff topology guarantees Cauchy completeness and (suitably   defined) ...

Enriched categories and models for spaces of dipaths

Partially ordered sets, causets, partially ordered spaces and their local counterparts are now often used to model systems in computer science and theoretical physics. The order models ‘time’ which is often not globally given. In this setting directed paths are important objects of study as they correspond to an evolving state or particle traversing the system. Many physical problems rely on th...

Heuristic and Computer Calculations for the Magnitude of Metric Spaces

This paper is concerned with an empirical look at the magnitude of subspaces of Euclidean space. Magnitude was introduced under the name ‘cardinality’ by Leinster in [4] as a partially defined invariant of finite metric spaces; his motivation came from enriched category theory, but the invariant had already been considered in the biological diversity literature [6]. In [5] we started to extend ...