An axiomatic approach to virtual chains

Abstract

We introduce a category of Kuranishi presentations, whose objects are a variant of the Kuranishi structures introduced by Fukaya and Ono, and which can be seen as a refinement of the version studied by Pardon. We then formulate the notion of virtual chains categorically as a natural transformation between two functors from this category to the category of chain complexes; we call such a datum 'a theory of virtual counts'. To show that this definition carries non-trivial content, we then construct a multicategory whose objects are Kuranishi flow categories, and show that a theory of virtual counts determines a multifunctor to the multicategory of chain complexes. We then implement this construction in the setting of Hamiltonian Floer theory, borrowing from some joint work with Groman and Varolgunes, yielding a construction of Hamiltonian Floer groups (and operations on them) as an output of this machine. We plan to provide a similar account for Lagrangian Floer theory in subsequent joint work.