Segal Topoi and Natural Phenomena: Universality of Physical Laws

Abstract

J. Lurie proved in Higher Topos Theory that for $K$ a simplicial set, $\mathcal{C}$ a simplicial category, $f: \mathfrak{C}[K] \rightarrow \mathcal{C}^{\text{op}}$ an equivalence of simplicial categories, we have a Quillen equivalence $(\text{Set}^+_{\Delta})_{/K} \rightleftarrows (\text{Set}^+_{\Delta})^{\mathcal{C}}$. We prove a partial converse to this theorem at the level of Segal categories, namely that if $L(\text{Set}^+_{\Delta})_{/K}$ is isomorphic to $L(\text{Set}^+_{\Delta})^{\mathcal{C}}$ in Ho(SePC), then $L \mathfrak{C}[K]^{\text{op}}$ and $L\mathcal{C}$ are equivalent as Segal pre-categories relative to Segal categories of pre-stacks. We interpret this as indicating that the Segal category of pre-stacks $L(\text{Set}^+_{\Delta})^{\mathcal{C}} \cong \mathbb{R} \underline{\text{Hom}} (L \mathcal{C}, L \text{Set}^+_{\Delta})$ on $L \mathcal{C}$ is equivalently given by a choice of simplicial set $K$, relative to which phenomena in $\text{Top}^+ = L \text{Set}^+_{\Delta}$ are considered, a sort of relativity principle. If we further take the Bousfield localizations of $L(\text{Set}^+_{\Delta})^{\mathfrak{C}[K]^{\text{op}}} \cong L( \text{Set}^+_{\Delta})_{/K}$ and $L(\text{Set}^+_{\Delta})^{\mathcal{C}}$ with respect to hypercovers, then regarding $L_{\text{Bous}}(L(\text{Set}^+_{\Delta})^{\mathcal{C}})$ as the Segal topos of natural phenomena on $L\mathcal{C}$, we also obtain an isomorphism $L_{\text{Bous}}(L(\text{Set}^+_{\Delta})^{\mathfrak{C}[K]^{\text{op}}}) \cong L_{\text{Bous}} (L(\text{Set}^+_{\Delta})^{\mathcal{C}})$ of Segal topoi of stacks. This provides two representations of the same natural phenomena, concurrently with the equivalence $L \mathfrak{C}[K]^{\text{op}} \simeq L\mathcal{C}$ relative to prestacks, which we interpret as a weak universality of natural laws.