Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
Abstract
This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Möbius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of $\infty$groupoids. A decomposition space is a simplicial $\infty$groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in Delta. Just as the Segal condition expresses composition, the new condition expresses decomposition, and there is an abundance of examples in combinatorics. After establishing some basic properties of decomposition spaces, the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra, spanned by the space of 1simplices, and with coefficients in $\infty$groupoids. We take a functorial viewpoint throughout, emphasising conservative ULF functors; these induce coalgebra homomorphisms. Reduction procedures in the classical theory of incidence coalgebras are examples of this notion, and many are examples of decalage of decomposition spaces. An interesting class of examples of decomposition spaces beyond Segal spaces is provided by Hall algebras: the Waldhausen Sconstruction of an abelian (or stable infinity) category is shown to be a decomposition space. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call them unital 2Segal spaces. Our theory is quite orthogonal to theirs: the definitions are different in spirit and appearance, and the theories differ in terms of motivation, examples and directions.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.07573
 Bibcode:
 2015arXiv151207573G
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 18G30;
 16T10;
 06A11;
 18XX;
 55Pxx
 EPrint:
 50 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v4: minor expository improvements. Final version to appear in Adv. Math