Universal Properties and Constructions of Pullback Formalisms in Terms of Invariance and Stability
Roy Magen
TL;DR
The article develops a comprehensive framework of pullback formalisms to encode cohomology theories with local coefficients across broad geometric contexts, including schemes, stacks, and analytic spaces. It introduces the universal pullback formalism H^univ and shows how to propagate structures along morphisms, setting the stage for stable motivic theories by stabilizing via pointed objects and tensor inverses. By integrating generalized descent via pseudotopologies and invariance conditions, the authors define localization and invariance mechanisms that preserve the pullback formalism structure, enabling constructions akin to motivic homotopy theories. They apply this machinery to algebraic, differentiable, and holomorphic contexts, producing SH-like invariants (SH^alg, SH^diff, SH^hol) with descent, excision, and stabilization properties, and outlining a path toward further Grothendieck six-operations type formalisms. The framework thus provides a unified, universal approach to building and comparing cohomology theories on stacks and complex analytic spaces, with robust functoriality and localization tools.
Abstract
In this article, we introduce fundamental notions and results about pullback formalisms, building on work of Drew-Gallauer. Our main application is producing a pullback formalism $\mathbf{SH}^{\mathrm{hol}}$ that encodes a version of motivic homotopy theory for complex analytic stacks, and establishing some of its properties. The notions introduced in this article will be used in later articles in which we also establish more properties of $\mathbf{SH}^{\mathrm{hol}}$, notably the gluing property of Morel and Voevodsky, the structure of a 6-functor formalism, and a realization map from the motivic homotopy theory of algebraic stacks defined by Khan-Ravi that is compatible with Grothendieck's six operations, generalizing Ayoub's results on Betti realization for schemes.