[Paper Review] Model $\infty$-categories II: Quillen adjunctions

Aaron Mazel-Gee|arXiv (Cornell University)|Oct 15, 2015

Homotopy and Cohomology in Algebraic Topology12 references3 citations

TL;DR

This paper establishes that key categorical structures—Quillen adjunctions, two-variable adjunctions, monoidal and symmetric monoidal structures, and enriched model structures—descend canonically from model ∞-categories to their localizations. Using homotopy-theoretic techniques centered on fibrations and localization functors, it proves that Quillen adjunctions induce adjunctions on localizations, and that monoidal/enriched structures on model ∞-categories lift to their localizations, preserving essential ∞-categorical coherence.

ABSTRACT

We prove that various structures on model $\infty$-categories descend to corresponding structures on their localizations: (i) Quillen adjunctions; (ii) two-variable Quillen adjunctions; (iii) monoidal and symmetric monoidal model structures; and (iv) enriched model structures.

Motivation & Objective

• To establish that Quillen adjunctions between model ∞-categories induce canonical adjunctions on their localizations.
• To show that two-variable Quillen adjunctions induce two-variable adjunctions on localizations.
• To prove that monoidal and symmetric monoidal model ∞-categories induce closed (resp. symmetric) monoidal structures on their localizations.
• To demonstrate that enriched model ∞-categories induce enriched and bitensored structures on their localizations over the localized enriching category.

Proposed method

• Uses fiberwise localization and relative co/cartesian fibrations to analyze structure descent.
• Applies the theory of cocartesian fibrations as lax colimits (via Gepner-Haugseng-Nikolaus) to handle coherence in localizations.
• Relies on subcategories of 'nice' objects (e.g., cofibrant objects) to control homotopy-theoretic behavior under localization.
• Employs derived two-variable adjunctions and Quillen's framework for model categories in the ∞-categorical setting.
• Leverages the symmetric monoidality of the localization functor to lift algebraic structures.
• Utilizes the framework of relative ∞-categories and model ∞-categories as generalizations of classical model categories.

Experimental results

Research questions

• RQ1 Does a Quillen adjunction between model ∞-categories induce an adjunction on their localizations?
• RQ2 Can two-variable Quillen adjunctions be lifted to two-variable adjunctions on localizations?
• RQ3 Does a monoidal model ∞-category induce a closed monoidal structure on its localization?
• RQ4 Does an enriched model ∞-category induce an enriched and bitensored structure on its localization over the localized enriching category?

Key findings

• A Quillen adjunction between model ∞-categories induces a canonical adjunction on their localizations, and a Quillen equivalence induces an adjoint equivalence.
• A two-variable Quillen adjunction induces a canonical two-variable adjunction on the localizations of the respective model ∞-categories.
• The localization of a (symmetric) monoidal model ∞-category is canonically a closed (resp. symmetric) monoidal ∞-category.
• The localization of a V-enriched model ∞-category is canonically enriched and bitensored over the localization of V.
• The derived two-variable adjunction from a Quillen adjunction underlies a canonical enrichment and bitensoring on the localizations.
• The results hold invariantly and do not depend on point-set models or explicit computation of hom-spaces in localizations.