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 (subject to a completeness condition weaker
than the Rezk condition) 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. 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.)