The Physical Mathematics of Segal Topoi and Strings

TL;DR

This paper develops a mathematical framework using Segal topoi to describe dynamics and quantum states, connecting abstract category theory with string theory and M-theory.

Contribution

It introduces a novel approach to model string theory within Segal topoi, defining quantum states and flows in this categorical setting.

Findings

Defined dynamics in Segal topoi using stacks and hom-objects

Constructed local and global flows of quantum states

Linked the formalism to standard string theory and M-theory

Abstract

We introduce a notion of dynamics in the setting of Segal topos, by considering the Segal category of stacks $\mathcal{X} = \text{dAff}_{\mathcal{C}}^{\, \sim, \tau}$ 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.