TY - JOUR

T1 - Decomposition spaces, incidence algebras and Möbius inversion II

T2 - Completeness, length filtration, and finiteness

AU - Gálvez-Carrillo, Imma

AU - Kock, Joachim

AU - Tonks, Andrew

N1 - Funding Information:
The first author was partially supported by grants MTM2012-38122-C03-01, MTM2013-42178-P, 2014-SGR-634, MTM2015-69135-P, MTM2016-76453-C2-2-P (AEI/FEDER, UE), and 2017-SGR-932, the second author by MTM2013-42293-P, MTM2016-80439-P (AEI/FEDER, UE), and 2017-SGR-1725, and the third author by MTM2013-42178-P and MTM2016-76453-C2-2-P (AEI/FEDER, UE).
Funding Information:
The first author was partially supported by grants MTM2012-38122-C03-01 , MTM2013-42178-P , 2014-SGR-634 , MTM2015-69135-P , MTM2016-76453-C2-2-P ( AEI/FEDER, UE ), and 2017-SGR-932, the second author by MTM2013-42293-P , MTM2016-80439-P ( AEI/FEDER, UE ), and 2017-SGR-1725, and the third author by MTM2013-42178-P and MTM2016-76453-C2-2-P ( AEI/FEDER, UE ).
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/7/31

Y1 - 2018/7/31

N2 - This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition. In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Möbius inversion principle, expressed as an explicit equivalence of ∞-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of ∞-groupoids to the level of Q-vector spaces. These three conditions — completeness, locally finite length, and local finiteness — together define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier–Foata), but which also covers many coalgebra constructions which do not arise from Möbius categories, such as the Faà di Bruno and Connes–Kreimer bialgebras. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [6] who call them unital 2-Segal spaces.

AB - This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition. In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Möbius inversion principle, expressed as an explicit equivalence of ∞-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of ∞-groupoids to the level of Q-vector spaces. These three conditions — completeness, locally finite length, and local finiteness — together define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier–Foata), but which also covers many coalgebra constructions which do not arise from Möbius categories, such as the Faà di Bruno and Connes–Kreimer bialgebras. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [6] who call them unital 2-Segal spaces.

KW - 2-Segal space

KW - decomposition space

KW - homotopy cardinality

KW - incidence algebra

KW - length filtration

KW - Möbius inversion

UR - http://www.scopus.com/inward/record.url?scp=85045327012&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.03.017

DO - 10.1016/j.aim.2018.03.017

M3 - Article

AN - SCOPUS:85045327012

SN - 0001-8708

VL - 333

SP - 1242

EP - 1292

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -