Higher Galois for Segal Topos and Natural Phenomena

Abstract

Toen and Vezzosi showed that $RHom^{geom}(T,lX)$ is a Segal groupoid, for $T$ a Segal topos, $lX = Loc(X)$ the Segal category of locally constant stacks on a CW complex $X$. Taking the realization of such a groupoid defines a pro-object $H_T = |RHom^{geom}(T, -)|$ that is defined to be the homotopy shape of the topos $T$. What we do instead is fix a Segal topos $X$, we let $T$ vary, and use the fact that $RHom^*_{Lex}(X,T) = RHom^{geom}(T,X)$ is a fundamental $\infty$-groupoid. We then prove that $X$ is a localization of the Segal category of local systems on $RHom^{geom}(T,X)$, in the spirit of Hoyois' work in his "Higher Galois Theory" paper, where it is proved, morally, that local systems on $H_T$ are equivalent to $T$ itself. We provide one application of this formalism, regarding the Segal topos $X=dSt(k)$ of derived stacks, for $k$ a commutative ring, as corresponding to manifestations of natural laws, themselves modeled by simplicial algebras, objects of $sk-CAlg$.