Model ∞-categories III: the fundamental theorem

2015/10/16 by Aaron Mazel-Gee, Mazel-Gee, Aaron
Mathematics · #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #math.AT #math.CT

paper · pdf · doi:10.48550/arxiv.1510.04777

arxiv created 2015/10/16 · arxiv updated 2015/10/19

Abstract

We prove that a model structure on a relative ∞-category (M,W) gives an efficient and computable way of accessing the hom-spaces hom_M[[W^-1]](x,y) in the localization. More precisely, we show that when the source x ∈ M is *cofibrant* and the target y ∈ M is *fibrant*, then this hom-space is a "quotient" of the hom-space hom_M(x,y) by either of a *left homotopy relation* or a *right homotopy relation*.

Related

• The 2-Localization of a Quillen's model category
• The 2-localization of a model category
• Homological Localisation of Model Categories
• Precovers, localizations and stable homotopy
• Model ∞-categories II: Quillen adjunctions