Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T17:47:56.075Z Has data issue: false hasContentIssue false
  • English
  • Français

The additivity of traces in monoidal derivators

Published online by Cambridge University Press:  14 July 2014

Moritz Groth ,
Kate Ponto  and
Michael Shulman
Moritz Groth
Affiliation:
Department of Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, Netherlands, [email protected]
Kate Ponto
Affiliation:
Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY, 40506, USA, [email protected]
Michael Shulman
Affiliation:
Department of Mathematics and Computer Science, University of San Diego, 5998 Alcalá Park San Diego, CA, 92110, USA, [email protected]
Get access

Abstract

Motivated by traces of matrices and Euler characteristics of topological spaces, we expect abstract traces in a symmetric monoidal category to be “additive”. When the category is “stable” in some sense, additivity along cofiber sequences is a question about the interaction of stability and the monoidal structure.

May proved such an additivity theorem when the stable structure is a triangulation, based on new axioms for monoidal triangulated categories. in this paper we use stable derivators instead, which are a different model for “stable homotopy theories”. We define and study monoidal structures on derivators, providing a context to describe the interplay between stability and monoidal structure using only ordinary category theory and universal properties. We can then perform May's proof of the additivity of traces in a closed monoidal stable derivator without needing extra axioms, as all the needed compatibility is automatic.

Keywords

dualitytraceEuler characteristicderivatortriangulated categorymonoidal category18D1018E3018G5555U35
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 3 , December 2014 , pp. 422 - 494
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

AB04.Arkowitz, Martin and Brown, Robert F.. The Lefschetz-Hopf theorem and axioms for the Lefschetz number. Fixed Point Theory Appl. 2004(1) (2004), 1–11.Google Scholar
Bro71.Brown, Robert F.. The Lefschetz fixed point theorem. Scott, Foresman and Co., Glenview, Ill.-London, 1971.Google Scholar
CGR12.Cheng, Eugenia, Gurski, Nick, and Riehl, Emily. Cyclic multicategories, multivariable adjunctions and mates. J. K-Theory 13(2) (2014), 337–396.Google Scholar
Cis03.Cisinski, Denis-Charles. Images directes cohomologiques dans les catégories de modèles. Ann. Math. Blaise Pascal 10(2) (2003), 195–244.Google Scholar
DHKS04.Dwyer, William G., Hirschhorn, Philip S., Kan, Daniel M., and Smith, Jeffrey H.. Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113. American Mathematical Society, 2004.Google Scholar
DP80.Dold, Albrecht and Puppe, Dieter. Duality, trace, and transfer. In Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), pages 81–102, Warsaw, 1980. PWN.Google Scholar
EK66.Eilenberg, Samuel and Kelly, G. M.. A generalization of the functorial calculus. J. Algebra 3 (1966), 366–375.Google Scholar
Fer05.Ferrand, Daniel. on the non-additivity of traces in derived categories. arXiv:math/0506589, 2005.Google Scholar
Fra96.Franke, Jens. Uniqueness theorems for certain triangulated categories with an Adams spectral sequence. Available at http://www.math.uiuc.edu/K-theory/0139/, 1996.Google Scholar
GPS13.Groth, Moritz, Ponto, Kate, and Shulman, Michael. Mayer-Vietoris sequences in stable derivators. Homology, Homotopy, and Applications, to appear.Google Scholar
Gro90.Grothendieck, Alexandre. Les dérivateurs. http://people.math.jussieu.fr/~maltsin/groth/Derivateursengl.html, 1990.Google Scholar
Gro13.Groth, Moritz. Derivators, pointed derivators, and stable derivators. Algebr. Geom. Topol. 13 (2013), 313–374.Google Scholar
GS14a.Groth, Moritz and Shulman, Michael. Enriched derivators. In preparation, 2014.Google Scholar
GŠ14b.Groth, Moritz and Št'ovíček, Jan. Tilting theory via stable homotopy theory. arXiv:1401.6451, 2014.Google Scholar
Hel88.Heller, Alex. Homotopy theories. Mem. Amer. Math. Soc. 71(383) (1988), vi + 78.Google Scholar
Hov99.Hovey, Mark. Model Categories, Mathematical Surveys and Monographs 63. American Mathematical Society, Providence, RI, 1999.Google Scholar
HPS97.Hovey, Mark, Palmieri, John H., and Strickland, Neil P.. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 128(610) (1997), x + 114.Google Scholar
Joy.Joyal, André. The theory of quasi-categories and its applications. Lectures at CRM Barcelona February 2008. Available at http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf.Google Scholar
KN02.Keller, Bernhard and Neeman, Amnon. The connection between May's axioms for a triangulated tensor product and Happel's description of the derived category of the quiver D4. Doc. Math. 7 (2002), 535–560 (electronic).Google Scholar
KS74.Kelly, G. M. and Street, Ross. Review of the elements of 2-categories. In Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Mathematics 420, pages 75–103. Springer, Berlin, 1974.Google Scholar
Lau06.Lauda, Aaron. Frobenius algebras and ambidextrous adjunctions. Theory Appl. Categ. 16(4) (2006), 84–122.Google Scholar
LMSM86.Lewis, L. G. Jr., May, J. P., Steinberger, M., and McClure, J. E.. Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213. Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure.Google Scholar
Lur.Lurie, Jacob. Higher algebra. Available at http://www.math.harvard.edu/~lurie/.Google Scholar
Lur09.Lurie, Jacob. Higher topos theory, Annals of Mathematics Studies 170. Princeton University Press, Princeton, NJ, 2009.Google Scholar
Mal.Maltsiniotis, Georges. Introduction à la théorie des dérivateurs. http://www.math.jussieu.fr/~maltsin/ps/m.ps.Google Scholar
Mar83.Margolis, H. R.. Spectra and the Steenrod algebra, North-Holland Mathematical Library 29. North-Holland Publishing Co., Amsterdam, 1983. Spectra and the Steenrod algebra: Modules over the Steenrod algebra and the stable homotopy category.Google Scholar
May01.May, J. P.. The additivity of traces in triangulated categories. Adv. Math. 163(1) (2001), 34–73.Google Scholar
ML98.Lane, Saunders Mac. Categories For the Working Mathematician, Graduate Texts in Mathematics 5. Springer-Verlag, New York, second edition, 1998.Google Scholar
MS06.May, J. P. and Sigurdsson, J.. Parametrized Homotopy Theory, Mathematical Surveys and Monographs 132. Amer. Math. Soc., Providence, RI, 2006.Google Scholar
PS12.Ponto, Kate and Shulman, Michael. Duality and traces for indexed monoidal categories. Theory Appl. Categ. 26(23) (2012), 582–659 (electronic).Google Scholar
Shu06.Shulman, Michael. Homotopy limits and colimits and enriched homotopy theory. arXiv:math.CT/0610194, 2006.Google Scholar
Shu08.Shulman, Michael. Framed bicategories and monoidal fibrations. Theory Appl. Categ. 20(18) (2008), 650–738.Google Scholar