On the algebraic<i>K</i>-theory of higher categories

Barwick, Clark

doi:10.1112/jtopol/jtv042

Cited by 52 publications

(138 citation statements)

References 59 publications

(60 reference statements)

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“…A topic of much recent work has been the generalization of the classical S • -construction to more general contexts, such as for stable (∞, 1)categories, for example in [5], [8], and [14]. In a companion paper [6] we show that our construction recovers and generalizes these results for exact (∞, 1)-categories; we give a summary in Section 3.2.…”

Section: Question Is This Source Of Examples Exhaustive? In Other Womentioning

confidence: 61%

2-Segal sets and the Waldhausen construction

Bergner

Osorno

Ozornova

et al. 2018

Topology and its Applications

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In a previous paper, we showed that a discrete version of the S•-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an S•-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known S•-constructions.

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Section: Question Is This Source Of Examples Exhaustive? In Other Womentioning

confidence: 61%

2-Segal sets and the Waldhausen construction

Bergner

Osorno

Ozornova

et al. 2018

Topology and its Applications

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“…One final remark on the connection between algebraic K-theory and Goodwillie calculus is worth making here. Barwick [22] identifies the process of forming algebraic K-theory itself as the first layer of a Taylor tower. He defines an ∞-category Wald ∞ of Waldhausen ∞-categories that are ∞-category-theoretic analogues of Waldhausen's categories with cofibrations.…”

Section: Applications and Calculations In Algebraic K-theorymentioning

confidence: 99%

Goodwillie calculus

Arone¹,

Ching²

2020

Handbook of Homotopy Theory

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Contents 1. Polynomial Approximation and the Taylor Tower 3 2. The Classification of Homogeneous Functors 8 3. The Taylor Tower of the Identity Functor for Based Spaces 10 4. Operads and Tate Data: the Classification of Taylor Towers 17 5. Applications and Calculations in Algebraic K-Theory 23 6. Taylor Towers of Infinity-Categories 26 7. The Manifold and Orthogonal Calculi 28 8. Further Directions 32 References 33Goodwillie calculus is a method for analyzing functors that arise in topology. One may think of this theory as a categorification of the classical differential calculus of Newton and Leibnitz, and it was introduced by Tom Goodwillie in a series of foundational papers [45, 46, 47].The starting point for the theory is the concept of an n-excisive functor, which is a categorification of the notion of a polynomial function of degree n. One of Goodwillie's key results says that every homotopy functor F has a universal approximation by an n-excisive functor P n F , which plays the role of the n-th Taylor approximation of F . Together, the functors P n F fit into a tower of approximations of F : the Taylor tower

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“…Such an ∞-category with cofibrations is referred to as a Waldhausen ∞-category in [4,17]. Specifying a subcategory of cofibrations is a way of specifying which maps have homotopy cofibers.…”

Section: 5mentioning

confidence: 99%

𝐾-theory of endomorphisms via noncommutative motives

Blumberg

Gepner

Tabuada

2015

Trans. Amer. Math. Soc.

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Abstract. In this article we study the K-theory of endomorphisms using noncommutative motives. We start by extending the K-theory of endomorphisms functor from ordinary rings to (stable) ∞-categories. We then prove that this extended functor KEnd(−) not only descends to the category of noncommutative motives but moreover becomes co-represented by the noncommutative motive associated to the tensor algebra S[t] of the sphere spectrum S. Using this co-representability result, we then classify all the natural transformations of KEnd(−) in terms of an integer plus a fraction between polynomials with constant term 1; this solves a problem raised by Almkvist in the seventies. Finally, making use of the multiplicative co-algebra structure of S[t], we explain how the (rational) Witt vectors can also be recovered from the symmetric monoidal category of noncommutative motives. Along the way we show that the K 0 -theory of endomorphisms of a connective ring spectrum R equals the K 0 -theory of endomorphisms of the underlying ordinary ring π 0 R.

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On the algebraicK-theory of higher categories

Cited by 52 publications

References 59 publications

2-Segal sets and the Waldhausen construction

2-Segal sets and the Waldhausen construction

Goodwillie calculus

𝐾-theory of endomorphisms via noncommutative motives

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