zbMATH — the first resource for mathematics

Decomposition spaces, incidence algebras and Möbius inversion. II: Completeness, length filtration, and finiteness. (English) Zbl 1403.18016
Adv. Math. 333, 1242-1292 (2018); corrigendum ibid. 371, Article ID 107267, 5 p. (2020).
In the first part of this trilogy [Adv. Math. 331, 952–1015 (2018; Zbl 1403.00023)] the authors have introduced the notion of decomposition space as a generalization for incidence coalgebras, which is equivalent to the notion of unital \(2\)-Segal space of [T. Dyckerhoff and M. Kapranov, Higher Segal spaces. I, Lect. Notes Math. 2244. Cham: Springer (2019; Zbl 1459.18001), see also arXiv:1212.3563]. The principal objective in this second part is to establish a Möbius inversion principle within the framework of complete decomposition spaces, and to analyze their associated finiteness conditions to ensure incidence coalgebras and Möbius inversion descent to classical level of \(\mathbb{Q}\)-vector spaces on taking the homotopy cardinality of the objects involved, which results in Möbius decomposition spaces as a generalization of Möbius categories in [P. Leroux, Cah. Topologie Géom. Différ. Catégoriques 16, 280–282 (1976; Zbl 0364.18001)]. Möbius decomposition spaces cover more coalgebra constructions than Möbius categories, comprehending the Faà di Bruno and Connes-Kremer bialgebras.

18N50 Simplicial sets, simplicial objects
16T10 Bialgebras
06A11 Algebraic aspects of posets
05A19 Combinatorial identities, bijective combinatorics
55U35 Abstract and axiomatic homotopy theory in algebraic topology
PDF BibTeX Cite
Full Text: DOI
[1] Baez, J. C.; Dolan, J., From finite sets to Feynman diagrams, (Mathematics Unlimited—2001 and Beyond, (2001), Springer Berlin), 29-50 · Zbl 1004.18001
[2] Baez, J. C.; Hoffnung, A. E.; Walker, C. D., Higher dimensional algebra VII: groupoidification, Theory Appl. Categ., 24, 489-553, (2010) · Zbl 1229.18003
[3] Berger, C.; Melliès, P.-A.; Weber, M., Monads with arities and their associated theories, J. Pure Appl. Algebra, 216, 2029-2048, (2012) · Zbl 1256.18004
[4] Content, M.; Lemay, F.; Leroux, P., Catégories de Möbius et fonctorialités: un cadre général pour l’inversion de Möbius, J. Combin. Theory Ser. A, 28, 169-190, (1980) · Zbl 0449.05004
[5] Dür, A., Möbius functions, incidence algebras and power series representations, Lecture Notes in Mathematics, vol. 1202, (1986), Springer-Verlag Berlin · Zbl 0592.05006
[6] Dyckerhoff, T.; Kapranov, M., Higher Segal spaces I, in: Springer Lecture Notes in Mathematics, to appear
[7] Fiore, T. M.; Lück, W.; Sauer, R., Finiteness obstructions and Euler characteristics of categories, Adv. Math., 226, 2371-2469, (2011) · Zbl 1242.18014
[8] Gálvez-Carrillo, I.; Kock, J.; Tonks, A., Groupoids and faà di bruno formulae for Green functions in bialgebras of trees, Adv. Math., 254, 79-117, (2014) · Zbl 1295.16022
[9] Gálvez-Carrillo, I.; Kock, J.; Tonks, A., Decomposition spaces, incidence algebras and Möbius inversion, (Old omnibus version, not intended for publication). Preprint · Zbl 1403.00023
[10] Gálvez-Carrillo, I.; Kock, J.; Tonks, A., Homotopy linear algebra, Proc. Roy. Soc. Edinburgh Sect. A, 148, 293-325, (2018) · Zbl 1455.18016
[11] Gálvez-Carrillo, I.; Kock, J.; Tonks, A., Decomposition spaces, incidence algebras and Möbius inversion I: basic theory, Adv. Math., 331, 952-1015, (2018) · Zbl 1403.00023
[12] Gálvez-Carrillo, I.; Kock, J.; Tonks, A., Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals, Adv. Math., (2018), in press · Zbl 1403.00023
[13] Gálvez-Carrillo, I.; Kock, J.; Tonks, A., Decomposition spaces and restriction species, Preprint
[14] Gálvez-Carrillo, I.; Kock, J.; Tonks, A., Decomposition spaces in combinatorics, Preprint
[15] Haigh, J., On the Möbius algebra and the Grothendieck ring of a finite category, J. Lond. Math. Soc. (2), 21, 81-92, (1980) · Zbl 0417.18005
[16] Joni, S. A.; Rota, G.-C., Coalgebras and bialgebras in combinatorics, Stud. Appl. Math., 61, 93-139, (1979) · Zbl 0471.05020
[17] Joyal, A., Foncteurs analytiques et espèces de structures, (Combinatoire énumérative, Montréal/Québec, 1985, Lecture Notes in Mathematics, vol. 1234, (1986), Springer Berlin), 126-159
[18] Joyal, A., Quasi-categories and kan complexes, J. Pure Appl. Algebra, 175, 207-222, (2002) · Zbl 1015.18008
[19] Joyal, A., The theory of quasi-categories, (Advanced Course on Simplicial Methods in Higher Categories, vol. II, Quaderns, vol. 45, (2008), CRM Barcelona), available at:
[20] Kock, J., Categorification of Hopf algebras of rooted trees, Cent. Eur. J. Math., 11, 401-422, (2013) · Zbl 1270.16030
[21] Kock, J., Perturbative renormalisation for not-quite-connected bialgebras, Lett. Math. Phys., 105, 1413-1425, (2015) · Zbl 1325.81126
[22] Kock, J., Polynomial functors and combinatorial Dyson-Schwinger equations, J. Math. Phys., 58, (2017), 36 pp. · Zbl 1431.81087
[23] Lawvere, F. W.; Menni, M., The Hopf algebra of Möbius intervals, Theory Appl. Categ., 24, 221-265, (2010) · Zbl 1236.18001
[24] Leinster, T., Notions of Möbius inversion, Bull. Belg. Math. Soc., 19, 911-935, (2012) · Zbl 1269.18002
[25] Leroux, P., LES catégories de Möbius, Cah. Topol. Géom. Différ., 16, 280-282, (1976) · Zbl 0364.18001
[26] Leroux, P., The isomorphism problem for incidence algebras of Möbius categories, Illinois J. Math., 26, 52-61, (1982) · Zbl 0487.18008
[27] Lück, W., Transformation groups and algebraic K-theory, Lecture Notes in Mathematics, vol. 1408, (1989), Springer-Verlag Berlin · Zbl 0679.57022
[28] Lurie, J., Higher topos theory, Annals of Mathematics Studies, vol. 170, (2009), Princeton University Press Princeton, NJ, available from: · Zbl 1175.18001
[29] Lurie, J., Higher algebra, (2013), Available from
[30] Rota, G.-C., On the foundations of combinatorial theory. I. theory of Möbius functions, Z. Wahrsch. Verw. Gebiete, 2, 340-368, (1964) · Zbl 0121.02406
[31] Schmitt, W. R., Hopf algebras of combinatorial structures, Canad. J. Math., 45, 412-428, (1993) · Zbl 0781.16026
[32] Stanley, R. P., Enumerative combinatorics. vol. I, (1986), Wadsworth & Brooks/Cole Advanced Books & Software Monterey, CA, with a foreword by Gian-Carlo Rota · Zbl 0608.05001
[33] Stern, M., Semimodular lattices: theory and applications, Encyclopedia of Mathematics and Its Applications, vol. 73, (1999), Cambridge University Press Cambridge · Zbl 0957.06008
[34] Weber, M., Generic morphisms, parametric representations and weakly Cartesian monads, Theory Appl. Categ., 13, 191-234, (2004) · Zbl 1062.18008
[35] Weber, M., Familial 2-functors and parametric right adjoints, Theory Appl. Categ., 18, 665-732, (2007) · Zbl 1152.18005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.