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Abstract Excision and ℓ 1 -Homology
Johannes Witzig1
1 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Confluentes Mathematici, Tome 15 (2023), pp. 107-136.

We use the abstract setting of excisive functors in the language of ∞-categories to show that the best approximation to the ℓ 1 -homology functor by an excisive functor is trivial.

Then we make an effort to explain the used language on a conceptual level for those who do not feel at home with ∞-categories, prove that the singular chain complex functor is indeed excisive in the abstract sense, and show how the latter leads to classical excision statements in the form of Mayer–Vietoris sequences.

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DOI : 10.5802/cml.95
Classification : 55N35, 55U35, 18F50, 18N60
Mots clés : $\ell ^1$-homology, excision, excisive approximation, $\infty $-categories

Johannes Witzig 1

1 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Licence : CC-BY-NC-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Johannes Witzig. Abstract Excision and $\ell ^1$-Homology. Confluentes Mathematici, Tome 15 (2023), pp. 107-136. doi : 10.5802/cml.95. https://cml.centre-mersenne.org/articles/10.5802/cml.95/

[1] Jiří Adámek; Horst Herrlich; George E. Strecker Abstract and Concrete Categories. The Joy of Cats, John Wiley & Sons, 1990 (addendum: Most recent version can be found here: http://katmat.math.uni-bremen.de/acc/) | Zbl

[2] C. Barwick; D. M. Kan Relative categories: Another model for the homotopy theory of homotopy theories, Indag. Math., New Ser., Volume 23 (2012) no. 1–2, pp. 42-68 | DOI | Zbl

[3] Julia E. Bergner A survey of (∞,1)-categories, Towards Higher Categories (IMA Volumes in Mathematics and its Applications), Volume 152, Springer, 2010, pp. 69-83 | DOI | Zbl

[4] Julia E. Bergner The Homotopy Theory of (∞,1)-Categories, London Mathematical Society Student Texts, 90, Cambridge University Press, 2018 | DOI | Zbl

[5] J. M. Boardman; R. M. Vogt Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, 347, Springer, 1973 | DOI | Zbl

[6] Denis-Charles Cisinski Higher Categories and Homotopical Algebra, Cambridge Studies in Advanced Mathematics, 180, Cambridge University Press, 2019 (addendum: Last version and errata available at http://www.mathematik.uni-regensburg.de/cisinski/publikationen.html) | DOI | Zbl

[7] Daniel Dugger; David I. Spivak Mapping spaces in quasi-categories, Algebr. Geom. Topol., Volume 11 (2011) no. 1, pp. 263-325 | DOI | Zbl

[8] Daniel Dugger; David I. Spivak Rigidification of quasi-categories, Algebr. Geom. Topol., Volume 11 (2011) no. 1, pp. 225-261 | DOI | Zbl

[9] W. G. Dwyer; D. M. Kan Calculating simplicial localizations, J. Pure Appl. Algebra, Volume 18 (1980) no. 1, pp. 17-35 | DOI | Zbl

[10] W. G. Dwyer; D. M. Kan Function complexes in homotopical algebra, Topology, Volume 19 (1980) no. 4, pp. 427-440 | DOI | Zbl

[11] W. G. Dwyer; D. M. Kan Simplicial localizations of categories, J. Pure Appl. Algebra, Volume 17 (1980) no. 3, pp. 267-284 | DOI | Zbl

[12] Greg Friedman Survey article: An elementary illustrated introduction to simplicial sets, Rocky Mt. J. Math., Volume 42 (2012) no. 2, pp. 353-423 | DOI | Zbl

[13] Paul G. Goerss; John F. Jardine Simplicial Homotopy Theory, Modern Birkhäuser Classics, Birkhäuser, 2010 | Zbl

[14] Michael Gromov Volume and bounded cohomology, Publ. Math., Inst. Hautes Étud. Sci., Volume 56 (1982), pp. 5-99 | Numdam | Zbl

[15] Moritz Groth A short course on ∞-categories (2015) (https://arxiv.org/abs/1007.2925)

[16] Allen E. Hatcher Algebraic Topology, Cambridge University Press, 2002 (https://www.math.cornell.edu/~hatcher/AT/ATpage.html) | Zbl

[17] Philip S. Hirschhorn Model Categories and Their Localizations, Mathematical Surveys and Monographs, 99, American Mathematical Society, 2003 | Zbl

[18] Mark Hovey Model Categories, Mathematical Surveys and Monographs, American Mathematical Society, 1999 | DOI

[19] Nikolai V. Ivanov Notes on the bounded cohomology theory (2017) (https://arxiv.org/abs/1708.05150)

[20] André Joyal Quasi-categories and Kan complexes, J. Pure Appl. Algebra, Volume 175 (2002) no. 1, pp. 207-222 | DOI | Zbl

[21] André Joyal Notes on Quasi-Categories, 2008 (http://www.math.uchicago.edu/~may/IMA/Joyal.pdf)

[22] Clara Löh Simplicial Volume (Manifold Atlas Project, http://www.map.mpim-bonn.mpg.de/Simplicial_volume)

[23] Clara Löh ℓ 1 -Homology and Simplicial Volume, Ph. D. Thesis, WWU Münster (2007) addendum: available at http://nbn-resolving.de/urn:nbn:de:hbz:6-37549578216 or (with errata) at https://loeh.app.uni-regensburg.de/preprints/

[24] Clara Löh Isomorphisms in ℓ 1 -homology, Münster J. Math., Volume 1 (2008), pp. 237-265 | Zbl

[25] Zhen Lin Low Universes for category theory (2013) (https://arxiv.org/abs/1304.5227)

[26] Jacob Lurie Kerodon (https://kerodon.net, an online resource for homotopy-coherent mathematics)

[27] Jacob Lurie Higher Topos Theory, Annals of Mathematics Studies, 170, Princeton University Press, 2009 (addendum: Most recent version can be found here: https://www.math.ias.edu/~lurie/) | DOI | Zbl

[28] Jacob Lurie Higher Algebra, 2017 (https://www.math.ias.edu/~lurie/)

[29] Saunders Mac Lane Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer, 1998 | Zbl

[30] J. P. May; K. Ponto More Concise Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press, 2012 | Zbl

[31] Nicolas Monod An invitation to bounded cohomology, Proceedings of the International Congress of Mathematicians (ICM). Vol. II, EMS Publishing House, 2006, pp. 1183-1211 (https://egg.epfl.ch/~nmonod/articles/icm.html) | Zbl

[32] Brian A. Munson; Ismar Volić Cubical Homotopy Theory, New Mathematical Monographs, 25, Cambridge University Press, 2015 | DOI | Zbl

[33] nLab authors nLab (https://ncatlab.org/nlab/show/HomePage, A Wiki for collaborative work on Mathematics, Physics, and Philosophy)

[34] George Raptis Bounded cohomology and homotopy colimits (2021) (https://arxiv.org/abs/2103.15614)

[35] Emily Riehl Homotopical categories: from model categories to (∞,1)-categories, Stable categories and structured ring spectra (Mathematical Sciences Research Institute Publications), Volume 69, Cambridge University Press, 2022, pp. 5-74 | Zbl

[36] Emily Riehl Homotopy coherent structures, Expositions in Theory and Applications of Categories (2023) no. 1, pp. 1-31

[37] Emily Riehl; Dominic Verity The theory and practice of Reedy categories, Theory Appl. Categ., Volume 29 (2014), pp. 256-301 | Zbl

[38] Michael A. Shulman Set theory for category theory (2008) (https://arxiv.org/abs/0810.1279)

[39] Robert M. Switzer Algebraic Topology – Homotopy and Homology, Classics in Mathematics, Springer, 2002 (Reprint of the 1975 original) | Zbl

[40] Tammo tom Dieck Algebraic topology, EMS Textbooks in Mathematics, European Mathematical Society, 2008 (addendum: Errata at https://www.uni-math.gwdg.de/tammo/algtop.html) | DOI | Zbl

[41] Rainer M. Vogt Homotopy limits and colimits, Math. Z., Volume 134 (1973), pp. 11-52 | DOI | Zbl

[42] Charles A. Weibel An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1995 (addendum: Errata at https://sites.math.rutgers.edu/~weibel/Hbook-corrections.html)

[43] Johannes Witzig Universality and Examples in the Context of Functorial Semi-Norms, Ph. D. Thesis, University of Regensburg (2023) (https://epub.uni-regensburg.de/54396)

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