The Physical Mathematics of Segal Topoi and Strings

TL;DR

This work develops a principled dynamical framework for Segal topoi by treating states as objects of $\delta \mathcal{X}=\mathbb{R}\underline{\mathrm{Hom}}(\mathcal{X},\mathcal{X})$ and constructing flows via higher states and functors $\\Psi$, culminating in a tiered, infinity-level description of dynamics. It introduces quantum states through minimal, structured perturbations and a cusp–tangent–cotangent toolkit (including tangent/ cotangent complexes and generalized categories) to analyze local and global flows. The approach is then applied to strings by modeling them as algebraic equivalences in $\text{Comm}(\mathcal{C})$, yielding a coherent algebraic analogue of string theory that naturally interfaces with stack-based descriptions of physical fields and with M-theory ideas. Overall, the paper offers a functorial, higher-categorical route to dynamics in geometric contexts, with concrete constructions for tangent/cotangent data, derivations, and mapping spaces, and a principled path toward algebraic string/M-theory perspectives.

Abstract

We introduce a notion of dynamics in the setting of Segal topos, by considering the Segal category of stacks $\mathcal{X} = \text{dAff}_{\mathcal{C}}^{\, \sim, τ}$ on a Segal category $\text{dAff}_{\mathcal{C}}=$ L(Comm($\mathcal{C})^{op})$ as our system, and by regarding objects of $\mathbb{R}\underline{\text{Hom}}(\mathcal{X}, \mathcal{X})$ as its states. We develop the notion of quantum state in this setting and construct local and global flows of such states. In this formalism, strings are given by equivalences between elements of commutative monoids of $\mathcal{C}$, a base symmetric monoidal model category. The connection with standard string theory is made, and with M-theory in particular.