[Paper Review] Approximation in K-theory for Waldhausen Quasicategories

Thomas M. Fiore|arXiv (Cornell University)|Mar 17, 2013

Homotopy and Cohomology in Algebraic Topology19 references3 citations

TL;DR

This paper establishes a comprehensive set of Approximation Theorems in the context of Waldhausen quasicategories, generalizing Waldhausen’s classical K-theory approximation theorem to the ∞-categorical setting. It proves that under conditions reflecting equivalences, factorization of morphisms, and preservation of colimits of equivalences, an exact functor induces a level-wise weak homotopy equivalence on K-theory spectra, with a key result showing that such functors induce stable equivalences of K-theory spectra when the domain admits and preserves relevant colimits.

ABSTRACT

We prove a series of Approximation Theorems in the setting of Waldhausen quasicategories. These theorems, inspired by Waldhausen's 1985 Approximation Theorem, give sufficient conditions for an exact functor of Waldhausen quasicategories to induce a level-wise weak homotopy equivalence of K-theory spectra. The Pre-Approximation Theorem, which holds in the general setting of quasicategories without Waldhausen structures, provides sufficient conditions for a functor F:A->B to restrict to an equivalence of the maximal infinity-groupoids in A and B. Our Approximation Theorems follow from the Pre-Approximation Theorem. The Approximation Theorem in the quasicategorical setting most analogous to Waldhausen's is: if an exact functor F:A -> B satisfies Waldhausen's App 1 and App 2, and the domain A admits colimits of the aforementioned type and F preserves them, then K(F) is a level-wise equivalence. As a corollary, if F is an exact functor with ho(F) an equivalence of ordinary categories, and every morphism in the domain A is a cofibration, then K(F) is a level-wise equivalence. We then introduce a version of App 2 called Cofibration App 2 that only requires factorization of cofibrations Fa >-> b as (equiv) o F(cofibration) and prove an analogous Cofibration Approximation Theorem, and a corollary for certain functors that induce an equivalence of cofibration homotopy categories. We also prove that S_n^infinity is Waldhausen equivalent to \overline{\mathcal{F}_{n-1}^infinity} using the mid anodyne maps known as spine inclusions, and clarify how hypotheses and notions in Waldhausen structures are related in new ways in the context of quasicategories.

Motivation & Objective

• To extend Waldhausen’s classical Approximation Theorem to the setting of Waldhausen quasicategories, providing a framework for K-theory approximation in ∞-category theory.
• To identify sufficient conditions under which an exact functor between Waldhausen quasicategories induces a level-wise weak homotopy equivalence on their K-theory spectra.
• To introduce and analyze a Cofibration Approximation Theorem variant that weakens App 2 to only require factorization of cofibrations, broadening applicability.
• To clarify the relationship between Waldhausen structures and quasicategorical notions, particularly regarding spine inclusions and mid-anodyne maps.
• To establish that $ S_n^∞ $ is Waldhausen equivalent to $ \overline{\mathcal{F}_{n-1}^\infty} $, providing a new quasicategorical model for these K-theory spaces.

Proposed method

• Introduces a Pre-Approximation Theorem in general quasicategories, requiring F to reflect equivalences, factor morphisms as (equivalence)∘F(cofibration), and preserve colimits of equivalences indexed by connected finite posets.
• Establishes the main Approximation Theorem by verifying Waldhausen’s App 1 and App 2 conditions in the quasicategorical setting, assuming the domain admits and F preserves colimits of equivalences.
• Develops a Cofibration Approximation Theorem by introducing Cofibration App 2, which only requires factorization of cofibrations $ Fa \rightarrowtail b $ as (equivalence)∘F(cofibration), and proves the corresponding equivalence of K-theory spectra.
• Uses the $ S_\bullet^\infty $ and $ \overline{S_\bullet^\infty} $ constructions to model K-theory spectra in the quasicategorical setting, with the latter construction omitting composites.
• Applies barycentric subdivision and Kan’s Ex functor to analyze weak contractibility and homotopy groups, proving that a simplicial set is weakly contractible if all its induced maps on homotopy groups vanish.
• Employs the Quillen Theorem A for quasicategories and over-quasicategories to verify that certain functors induce equivalences on maximal ∞-groupoids.

Experimental results

Research questions

• RQ1 Under what conditions does an exact functor between Waldhausen quasicategories induce a level-wise weak homotopy equivalence on K-theory spectra?
• RQ2 How can Waldhausen’s classical Approximation Theorem be generalized to the setting of quasicategories, particularly in terms of factorization and colimit preservation?
• RQ3 What is the role of cofibration factorization in the Approximation Theorem, and can App 2 be weakened to a Cofibration App 2 condition?
• RQ4 How do spine inclusions and mid-anodyne maps relate to the Waldhausen structure in quasicategories, and how can they be used to prove equivalences between K-theory models?
• RQ5 To what extent do the homotopy categories of Waldhausen quasicategories control the K-theory spectra, and when does an equivalence on the level of homotopy categories lift to a K-theory equivalence?

Key findings

• The Pre-Approximation Theorem establishes that if a functor F reflects equivalences, factors morphisms as (equivalence)∘F(cofibration), and preserves colimits of equivalences indexed by connected finite posets, then F induces an equivalence on the maximal ∞-groupoids of the domain and codomain.
• The main Approximation Theorem shows that if F satisfies App 1 and App 2 and the domain quasicategory admits and F preserves colimits of equivalences indexed by connected finite posets, then the induced map on K-theory spectra is a level-wise weak homotopy equivalence.
• A Cofibration Approximation Theorem is proven under the weaker Cofibration App 2 condition, which only requires factorization of cofibrations $ Fa \rightarrowtail b $ as (equivalence)∘F(cofibration), yielding the same conclusion on K-theory spectra.
• A corollary establishes that if F is an exact functor with $ ho(F) $ an equivalence of ordinary categories and every morphism in the domain is a cofibration, then $ \mathbf{K}(F) $ is a level-wise equivalence.
• The paper proves that $ S_n^\infty $ is Waldhausen equivalent to $ \overline{\mathcal{F}_{n-1}^\infty} $ using spine inclusions, which are mid-anodyne maps, providing a new quasicategorical model for these K-theory spaces.
• The paper clarifies the interplay between Waldhausen structures and quasicategorical notions, showing that the conditions in App 1 and App 2 are equivalent to certain quasicategorical colimit and homotopy group conditions, particularly in the context of weak contractibility of simplicial sets.