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

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.