The inflation functor in pluripotential homological algebra

TL;DR

This paper introduces an inflation functor from cochain complexes to bicomplexes that preserves weak equivalences and integrates into a Quillen adjunction, advancing pluripotential homological algebra.

Contribution

It defines a new inflation functor, establishes its role in a Quillen adjunction, and develops pluripotential Koszul duality theory for operads.

Findings

The inflation functor sends quasi-isomorphisms to pluripotential weak equivalences.

The functor forms part of a Quillen adjunction with a known complex geometry construction.

It enables the construction of the infinity-category of bicomplexes in the pluripotential setting.

Abstract

We introduce a functor from cochain complexes to bicomplexes, called inflation functor, which sends quasi-isomorphisms to the class of pluripotential weak equivalences. We show this functor is part of a Quillen adjunction. Its right adjoint is a well-known construction in complex geometry, which gives a sheaf-theoretic presentation of Bott-Chern and Aeppli cohomologies. The inflation functor plays a key role in pluripotential Koszul duality theory for operads and allows us to construct the infinity-category of bicomplexes in the pluripotential sense.