Decomposition spaces, incidence algebras, and Mšbius inversion
I'll survey recent work with Imma G‡lvez and Andy Tonks
developing a homotopy version of the theory of incidence
algebras and Mšbius inversion. The 'combinatorial objects'
playing the role of posets and Mšbius categories are
decomposition spaces, simplicial infinity-groupoids satisfying
an exactness condition weaker than the Segal condition,
expressed in terms of generic and free maps in Delta. Just as
the Segal condition expresses up-to-homotopy composition, the
new condition expresses decomposition. The role of vector
spaces is played by slices over infinity-groupoids, eventually
with homotopy finiteness conditions imposed. To any
decomposition space, there is associated an incidence
(co)algebra with coefficients in infinity-groupoids, which
satisfies an objective Mšbius inversion principle in the style
of Lawvere-Menni, provided a certain completeness condition is
satisfied, weaker than the Rezk condition. Generic examples of
decomposition spaces beyond Segal spaces are given by the
Waldhausen S-construction (yielding Hall algebras) and by
Schmitt restriction species, and many examples from classical
combinatorics admit uniform descriptions in this framework.
(The notion of decomposition space is equivalent to the notion
of unital 2-Segal space of Dyckerhoff-Kapranov.)