Venue   Category Theory 2023, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 8 July 2023 Abstract   Magnitude is a numerical invariant of enriched categories. Magnitude homology is a categorification of magnitude, first introduced by Hepworth and Willerton for graphs (seen as categories enriched in (N, ≥, +)) and extended to enriched categories by Shulman and myself. Magnitude homology generalizes the ordinary homology of categories (which in turn includes group homology), and is most novel in the case of metric spaces. There, it provides an R+-graded homology theory of metric spaces. Work of many authors has shown how magnitude homology conveys information about convexity and curvature in metric spaces. For example, while topological homology detects the existence of holes, magnitude homology detects the size of holes. I will give a survey, including some results from ongoing joint work with Adrián Doña Mateo. References (see the magnitude bibliography for more and for links) Asao, Y. Magnitude homology of geodesic metric spaces with an upper curvature bound. Algebraic & Geometric Topology 21 (2021), 647-664. Asao, Y. Magnitude homology and path homology. Bulletin of the London Mathematical Society 55 (2023), no. 1, 375-398. Gomi, K. Magnitude homology of geodesic space. arXiv:1902.07044, 2019. Gomi, K. Smoothness filtration of the magnitude complex. Forum Mathematicum 32 (2020), 625-639. Hepworth, R; Willerton, S. Categorifying the magnitude of a graph. Homology, Homotopy and Applications 19 (2017), no. 2, 31-60. Jubin, B. On the magnitude homology of metric spaces. arXiv:1803.05062, 2018. Kaneta, R; Yoshinaga, M. Magnitude homology of metric spaces and order complexes. Bulletin of the London Mathematical Society 53 (2021), no. 3, 893-905. Leinster, T; Shulman, M. Magnitude homology of enriched categories and metric spaces. Algebraic & Geometric Topology 21 (2021), 2175-2221. Roff, E. The size and shape of things: magnitude, diversity, homology. PhD thesis, University of Edinburgh, 2022. Tajima, Y; Yoshinaga, M. Causal order complex and magnitude homotopy type of metric spaces. arXiv:2302.09752, 2023 Slides   In this PDF file. Associated paper with Adrián Doña Mateo.   This page was last changed on 18 June 2024. Home