[Paper Review] Quasicategories of Frames of Cofibration Categories

Chris Kapulkin, Karol Szumiło|arXiv (Cornell University)|Jun 29, 2015

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper establishes that the quasicategory of frames in a cofibration category is equivalent to its simplicial localization, specifically showing that the quasicategory of frames is equivalent to the classification diagram construction. The key result is a compatibility of the frame construction with diagram categories, leading to an equivalence with standard simplicial localization functors such as the hammock localization and derived homotopy coherent nerve.

ABSTRACT

We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.

Motivation & Objective

• To establish an equivalence between the quasicategory of frames and the simplicial localization of a cofibration category.
• To resolve the issue of implicit fibrant replacements in simplicial localization by leveraging the richer structure of cofibration categories.
• To show that the quasicategory of frames, introduced by Szumiło, is equivalent to the classification diagram construction.
• To provide a foundational tool for modeling Homotopy Type Theory in higher categories, particularly showing that simplicial localizations of categorical models are locally cartesian closed quasicategories.

Proposed method

• Define an enhancement of the quasicategory of frames to a complete Segal space.
• Prove compatibility of the frame construction with the formation of diagram categories, crucial for lifting localizations.
• Use Reedy model structure techniques to factor diagrams into weak equivalences and Reedy fibrations, ensuring cofibrancy and fibrancy conditions.
• Apply the hammock localization and derived homotopy coherent nerve to relate the classification diagram to the quasicategory of frames.
• Leverage Toën's results on complete Segal spaces to deduce equivalence across different models of $(∞,1)$-categories.
• Use $E[1]$-homotopy and categorical equivalences to verify the quasicategory structure and homotopical equivalence of the resulting constructions.

Experimental results

Research questions

• RQ1 Is the quasicategory of frames in a cofibration category equivalent to its simplicial localization?
• RQ2 Can the frame construction be compatibly extended to diagram categories to preserve homotopical structure?
• RQ3 Does the classification diagram of a cofibration category yield a quasicategory equivalent to the quasicategory of frames?
• RQ4 How does the quasicategory of frames relate to other models of $(∞,1)$-categories such as complete Segal spaces?
• RQ5 Can the simplicial localization of a categorical model of Homotopy Type Theory be realized as a quasicategory of frames?

Key findings

• The quasicategory of frames of a cofibration category is equivalent to its simplicial localization, specifically the classification diagram.
• The construction of the quasicategory of frames is compatible with diagram categories, enabling a natural extension to diagram categories.
• The enhanced quasicategory of frames forms a complete Segal space equivalent to the classification diagram, establishing a bridge between frame theory and standard localization methods.
• The equivalence holds even when comparing the quasicategory of frames from the cofibration structure and the fibration structure in a model category, showing their simplicial localizations are equivalent.
• The results imply that the simplicial localization of any categorical model of Homotopy Type Theory is a locally cartesian closed quasicategory, as it arises as the quasicategory of frames.
• The proof relies on Reedy factorizations and $E[1]$-homotopy classes to establish categorical equivalences and quasicategory structure.