Download this article
 Download this article For screen
For printing
Recent Issues

Volume 26
Issue 4, 1229–1596
Issue 3, 825–1227
Issue 2, 411–824
Issue 1, 1–410

Volume 25, 9 issues

Volume 24, 9 issues

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
 
Author index
To appear
 
Other MSP journals
Interleaving distances, monoidal actions and $2$-categories

Patrick K. McFaddin and Tom Needham

Algebraic & Geometric Topology 26 (2026) 227–281
Bibliography
1 R Abraham, J E Marsden, T Ratiu, Manifolds, tensor analysis, and applications, 75, Springer (1988) MR960687
2 M Bauer, M Bruveris, P Harms, J Møller-Andersen, A numerical framework for Sobolev metrics on the space of curves, SIAM J. Imaging Sci. 10 (2017) 47 MR3595875
3 M Bauer, M Bruveris, P W Michor, Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision 50 (2014) 60 MR3233135
4 M Bauer, N Charon, E Klassen, S Kurtek, T Needham, T Pierron, Elastic metrics on spaces of Euclidean curves : theory and algorithms, J. Nonlinear Sci. 34 (2024) 56 MR4737374
5 U Bauer, M Lesnick, Induced matchings and the algebraic stability of persistence barcodes, J. Comput. Geom. 6 (2015) 162 MR3333456
6 M Beg, M Miller, A Trouvé, L Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, I. J. Computer Vision 61 (2005) 139
7 J Bénabou, Introduction to bicategories, from: "Reports of the Midwest Category Seminar", Lecture Notes in Math. 47, Springer (1967) 1 MR220789
8 H B Bjerkevik, M B Botnan, M Kerber, Computing the interleaving distance is NP-hard, Found. Comput. Math. 20 (2020) 1237 MR4156997
9 O Bobrowski, P Skraba, A universal null-distribution for topological data analysis, Scientific Reports 13 (2023) 12274
10 M B Botnan, V Lebovici, S Oudot, On rectangle-decomposable 2-parameter persistence modules, from: "36th International Symposium on Computational Geometry" (editors S Cabello, D Z Chen), Leibniz Int. Proc. Inform. 164, Schloss Dagstuhl. Leibniz-Zent. Inform. (2020) 22 MR4117735
11 M B Botnan, M Lesnick, An introduction to multiparameter persistence, from: "Representations of algebras and related structures" (editors A B Buan, H Krause, Ø Solberg), Eur. Math. Soc. (2023) 77 MR4693638
12 R Brown, From groups to groupoids : a brief survey, Bull. London Math. Soc. 19 (1987) 113 MR872125
13 M Bruveris, P W Michor, D Mumford, Geodesic completeness for Sobolev metrics on the space of immersed plane curves, Forum Math. Sigma 2 (2014) MR3264258
14 P Bubenik, J A Scott, Categorification of persistent homology, Discrete Comput. Geom. 51 (2014) 600 MR3201246
15 P Bubenik, V de Silva, J Scott, Metrics for generalized persistence modules, Found. Comput. Math. 15 (2015) 1501 MR3413628
16 P Bubenik, V de Silva, J Scott, Interleaving and Gromov–Hausdorff distance, preprint (2017) arXiv:1707.06288
17 D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
18 F Cagliari, M Ferri, P Pozzi, Size functions from a categorical viewpoint, Acta Appl. Math. 67 (2001) 225 MR1861130
19 G Carlsson, Topological pattern recognition for point cloud data, Acta Numer. 23 (2014) 289 MR3202240
20 W Chachólski, H Riihimäki, Metrics and stabilization in one parameter persistence, SIAM J. Appl. Algebra Geom. 4 (2020) 69 MR4057607
21 F Chazal, D Cohen-Steiner, M Glisse, L Guibas, S Oudot, Proximity of persistence modules and their diagrams, from: "Proceedings of the twenty-fifth annual symposium on computational geometry" (2009) 237
22 D Cohen-Steiner, H Edelsbrunner, J Harer, Stability of persistence diagrams, from: "Computational geometry", ACM (2005) 263 MR2460372
23 J Curry, H Hang, W Mio, T Needham, O B Okutan, Decorated merge trees for persistent topology, J. Appl. Comput. Topol. 6 (2022) 371 MR4468593
24 J Curry, W Mio, T Needham, O Okutan, F Russold, Topologically attributed graphs for shape discrimination, from: "Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)" (editors T Doster, T Emerson, H Kvinge, N Miolane, M Papillon, B Rieck, S Sanborn), Proceedings of Machine Learning Research 221, PMLR (2023) 87
25 T K Dey, C Xin, Computing bottleneck distance for 2-D interval decomposable modules, from: "34th International Symposium on Computational Geometry" (editors B Speckmann, C D Tóth), LIPIcs. Leibniz Int. Proc. Inform. 99, Schloss Dagstuhl. Leibniz-Zent. Inform. (2018) 32 MR3824276
26 H Edelsbrunner, D Letscher, A Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28 (2002) 511 MR1949898
27 U Grenander, Elements of pattern theory, Johns Hopkins University Press (1996) MR1405164
28 R S Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982) 65 MR656198
29 G Janelidze, G M Kelly, A note on actions of a monoidal category, Theory Appl. Categ. 9 (2001) 61 MR1897810
30 N Johnson, D Yau, 2-dimensional categories, Oxford Univ. Press (2021) MR4261588
31 G M Kelly, Doctrinal adjunction, from: "Category Seminar" (editor G M Kelly), Lecture Notes in Math. 420, Springer (1974) 257 MR360749
32 M Kohan, Answer to “Relationship between distances on homogeneous spaces and their Lie groups@, MathStackExchange (2019)
33 A Kriegl, P W Michor, The convenient setting of global analysis, 53, Amer. Math. Soc. (1997) MR1471480
34 F W Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973) 135 MR352214
35 M Lesnick, The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math. 15 (2015) 613 MR3348168
36 J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) MR2522659
37 P McCrudden, Categories of representations of coalgebroids, Adv. Math. 154 (2000) 299 MR1784678
38 F Mémoli, Some properties of Gromov–Hausdorff distances, Discrete Comput. Geom. 48 (2012) 416 MR2946454
39 P W Michor, D Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal. 23 (2007) 74 MR2333829
40 D Morozov, K Beketayev, G Weber, Interleaving distance between merge trees, preprint
41 T Needham, S Kurtek, Simplifying transforms for general elastic metrics on the space of plane curves, SIAM J. Imaging Sci. 13 (2020) 445 MR4075332
42 A Neeman, Metrics on triangulated categories, J. Pure Appl. Algebra 224 (2020) 106206 MR4021920
43 B Pareigis, Non-additive ring and module theory, II : 𝒞-categories, 𝒞-functors and 𝒞-morphisms, Publ. Math. Debrecen 24 (1977) 351 MR498792
44 F Petit, P Schapira, Thickening of the diagonal and interleaving distance, Selecta Math. 29 (2023) 70 MR4643933
45 L N Scoccola, Locally persistent categories and metric properties of interleaving distances, PhD thesis, The University of Western Ontario (Canada) (2020) MR4494960
46 V de Silva, E Munch, A Patel, Categorified Reeb graphs, Discrete Comput. Geom. 55 (2016) 854 MR3505333
47 V de Silva, E Munch, A Stefanou, Theory of interleavings on categories with a flow, Theory Appl. Categ. 33 (2018) 21 MR3812461
48 P Skraba, K Turner, Wasserstein stability for persistence diagrams, preprint (2020) arXiv:2006.16824
49 A Srivastava, E P Klassen, Functional and shape data analysis, Springer (2016) MR3821566
50 A Srivastava, E Klassen, S H Joshi, I H Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence 33 (2011) 1415
51 A Stefanou, Dynamics on categories and applications, PhD thesis, State University of New York at Albany (2018)
52 O Vipond, Multiparameter persistence landscapes, J. Mach. Learn. Res. 21 (2020) 61 MR4095340