Universal Properties and Constructions of Pullback Formalisms in Terms of Invariance and Stability

TL;DR

This paper develops a theoretical framework for pullback formalisms, introducing invariance and stability concepts, and constructs a new motivic homotopy theory for complex analytic stacks with promising future properties.

Contribution

It introduces fundamental notions of pullback formalisms, constructs a new motivic homotopy theory for complex analytic stacks, and lays groundwork for further structural properties.

Findings

Constructed a pullback formalism $ extbf{SH}^{ ext{hol}}$ for complex analytic stacks.

Established foundational properties of the pullback formalism.

Set the stage for future work on gluing, six-functor formalism, and realization maps.

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.