A universal characterization of
K K
Andrew J. Blumberg
Address: Department of Mathematics, University of Texas,
Austin, TX 78703, USA
Email address: blumberg@math.utexas.edu
,
David Gepner
Address: Fakultät für Mathematik,
Universität Regensburg, 93040 Regensburg, Germany
Email address: djgepner@gmail.com
and
Gonçalo Tabuada
Address: Gonçalo Tabuada, Department of Mathematics, MIT, Cambridge, MA 02139, USA and Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516, Caparica, Portugal
Email address: tabuada@math.mit.edu
Original source: arXiv:1001.2282v4
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Contents 1. Introduction1.1. Universal characterization1.2. Morita theory1.3. Symmetric monoidal structure and dualizable objects1.4. Trace methods1.5. Related works2. Spectral categories and stable ∞ \infty 2.1. Review of spectral categories2.2. The ∞ \infty Cat ∞ ex \Cat_{\infty}^{\ex} Cat ∞ perf \Cat_{\infty}^{\perf} 2.3. Stabilization of ∞ \infty 2.4. Compact objects and compactly-generated
∞ \infty 2.5. Localization of ∞ \infty 3. Symmetric monoidal structure and dualizable objects3.1. Tensor products of stable ∞ \infty 3.2. Smooth and proper stable ∞ \infty 3.3. Dualizability4. Morita theory4.1. Stable envelopes of spectral categories4.2. Spectral enrichment of stable ∞ \infty 4.3. The triangulated and Morita localizations5. Exact sequences5.1. The Verdier quotient as the cofiber in Cat ∞ perf \Cat_{\infty}^{\perf} 5.2. The Thomason-Neeman localization theorem5.3. Split-exact sequences5.4. Approximating split-exact sequences5.5. Strict-exact sequences6. Additivity6.1. Unstable version6.2. Universal additive invariant7. Connective K K 7.1. Algebraic K K ∞ \infty 7.2. Comparison with Waldhausen’s K K 7.3. Co-representability8. Localization8.1. Additive κ \kappa 8.2. Morita equivalences8.3. Universal localizing invariant9. Non-connective K K 9.1. Non-connective K K ∞ \infty 9.2. Co-representability9.3. Non-connective K K 9.4. Extending co-representability9.5. Non-connective K K 10. Trace maps10.1. T H H THH 10.2. The topological Dennis trace map10.3. T C TC References Read the whole chapter