2018/03/14 by Renaud Gauthier, Gauthier, Renaud · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Topology (math.AT) #Black Holes and Theoretical Physics #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1803.09611
openalex publication_date 2018/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
J. Lurie proved in Higher Topos Theory that for K a simplicial set, C a simplicial category, f: \mathfrakC[K] → Cop an equivalence of simplicial categories, we have a Quillen equivalence (Set+Δ)/K \rightleftarrows (Set+Δ)C. We prove a partial converse to this theorem at the level of Segal categories, namely that if L(Set+Δ)/K is isomorphic to L(Set+Δ)C in Ho(SePC), then L \mathfrakC[K]op and LC 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(Set+Δ)C ≅ ℝ \underlineHom (L C, L Set+Δ) on L C is equivalently given by a choice of simplicial set K, relative to which phenomena in Top+ = L Set+Δ are considered, a sort of relativity principle. If we further take the Bousfield localizations of L(Set+Δ)^\mathfrakC[K]op ≅ L( Set+Δ)/K and L(Set+Δ)C with respect to hypercovers, then regarding LBous(L(Set+Δ)C) as the Segal topos of natural phenomena on LC, we also obtain an isomorphism LBous(L(Set+Δ)^\mathfrakC[K]op) ≅ LBous (L(Set+Δ)C) of Segal topoi of stacks. This provides two representations of the same natural phenomena, concurrently with the equivalence L \mathfrakC[K]op ≃ LC relative to prestacks, which we interpret as a weak universality of natural laws.