Some memos on Stable Symplectic Structured Space

Abstract

In these memos, we define a pregeometry ${\mathcal{T}}_{\mathbb{S}}^{alg}$ and a geometry ${\mathcal{G}}_{\mathbb{S}}^{alg}$ which integrate symplectic manifolds with $E_{\infty }$-ring sheaves, enabling the construction of ${\mathcal{G}}_{\mathbb{S}}^{alg}$-schemes as structured $\infty$-topoi. Our framework and results establish a profound connection between algebraic invariants and homological properties, opening new pathways for exploring symplectic phenomena through the lens of higher category theory and derived geometry.

9 pages, We corrected the error in definition 2.1.1 and definition 2.2.1 example 2.3.1 and definition 3.2.1 !