arXiv:1512.07573 [math.CT]

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 8-groupoids. A decomposition space is a simplicial 8-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of generic and free maps in ¿. Just as the Segal condition expresses up-to-homotopy composition, the new condition expresses decomposition, and there is an abundance of examples coming from 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 1-simplices, and with coefficients in 8-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. We treat a few examples of decomposition spaces beyond Segal spaces, the most interesting being that of Hall algebras: the Waldhausen S·-construction of an abelian (or stable infinity) category is shown to be a decomposition space.