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, $f:\mathfrak{C}[K]\to {\mathcal{C}}^{\text{op}}$ 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.

arXiv comment18 pages. The equivalence of Segal categories is replaced by the more precise equivalence relative to prestacks, and correspondingly changes are made throughout. Some commutative diagrams are fixed. Use of the Hammock localization is made more precise, and so is the definition of the simplicial category of marked simplicial sets. The discussion about the reference simplicial set K is revisited

References in corpus (7)

• Higher Topos Theory
• Descente pour les n-champs (Descent for n-stacks)
• Segal topoi and stacks over Segal categories
• Algebraic Geometry over model categories (a general approach to derived algebraic geometry)
• Weak enriched categories - Categories enrichies faibles
• Higher Galois for Segal Topos and Natural Phenomena
• $\infty$-topoi and Natural Phenomena: Generation