Waldhausen Ktheory

General Introduction
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Search results
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Online References
nLab on Waldhausen category, see also Sconstruction
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Paper References
ThomasonTrobaugh, see Thomason folder.
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Some kind of relation with Grothendieck's derivators: Tabuada
arXiv:1204.3607 On the algebraic Ktheory of higher categories, I. The universal property of Waldhausen Ktheory from arXiv Front: math.KT by C. Barwick We prove that Waldhausen Ktheory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, Ktheory spaces admit canonical (connective) deloopings, and the Ktheory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic Ktheory of symmetric monoidal higher groupoids, higher topoi, associative ring spectra, and spectral DeligneMumford stacks.
[arXiv:1212.5232] On exact infinitycategories and the Theorem of the Heart from arXiv Front: math.KT by C. Barwick We introduce a notion of exact quasicategory, and we prove an analogue of Amnon Neeman's Theorem of the Heart for Waldhausen Ktheory.
Toen: Homotopical and higher categorical structures in algebraic geometry. File Toen web unpubl hab.pdf. Treats general philosophical background, various forms of homotopy theories, Segal categories, Waldhausen Kth briefly, Hochshild cohomology of Segal categories and of model cats, Scats, Segal topoi, Tannakian duality for Segal cats, and schematic homotopy types. Also letter to May about ncats.
arXiv:1207.6613 Waldhausen Additivity: Classical and Quasicategorical from arXiv Front: math.AT by Thomas M. Fiore, Wolfgang Lück We give a short proof of classical Waldhausen Additivity, and then prove Waldhausen Additivity for an infinityversion of Waldhausen Ktheory. Namely, we prove that Waldhausen Ktheory sends a splitexact sequence of Waldausen quasicategories A > E > B to a stable equivalence of spectra K(E) > K(A) v K(B) under a few mild hypotheses. For example, each cofiber sequence in A of the form a0 > a1 > * is required to have the first map an equivalence. Model structures, presentability, and stability are not needed. In an effort to make the article selfcontained, we provide many details in our proofs, recall all the prerequisites from the theory of quasicategories, and prove some of those as well. For instance, we develop the expected facts about (weak) adjunctions between quasicategories and (weak) adjunctions between simplicial categories.
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