Homotopy combinatorics and homotopy linear algebra

Algebraic methods are very powerful in combinatorics. At the same
time it is well appreciated that bijective proofs represent
deeper insight, and often form a better foundation for further
developments. The objective method, advocated by Lawvere, aims to
turn algebraic proofs into bijective proofs in a systematic way,
by exploiting the 'algebra of finite sets' directly. The best
example is the theory of species, which constitutes an objective
analogue of the machinery of generating functions: all the
operations on formal series become bijective at the level of
species. Another important algebraic toolbox is that of incidence
algebras and Möbius inversion. The goal of this talk is to
explain ongoing efforts to objectify (the basic aspects of) this
theory through the notion of decomposition spaces, also called
2-Segal spaces, which naturally work not just with coefficients
in sets, but rather coefficients in groupoids or
infinity-groupoids. I will not talk much about decomposition
spaces though, but focus instead on the underlying linear
algebra, which is linear algebra with coefficients in
infinity-groupoids. The role of vector spaces is played by slice
categories, and the role of linear maps is played by
coproduct-preserving functors. I will pay special attention to
the duality between vector spaces and profinite-dimensional
vector spaces, which looks quite nice at the objective level.
This is joint work with Imma Gálvez and Andy Tonks.