The edgewise subdivision criterion for $2$-Segal objects
Authors:
Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, Martina Rovelli and Claudia I. Scheimbauer
Journal:
Proc. Amer. Math. Soc. 148 (2020), 71-82
MSC (2010):
Primary 18D35, 18G30, 19D10, 55U10
DOI:
https://doi.org/10.1090/proc/14679
Published electronically:
July 9, 2019
MathSciNet review:
4042831
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We show that the edgewise subdivision of a $2$-Segal object is always a Segal object, and furthermore that this property characterizes $2$-Segal objects.
- Clark Barwick, On the Q-construction for exact $\infty$-categories, arXiv:1301.4725, 2013.
- Clark Barwick, On the algebraic $K$-theory of higher categories, J. Topol. 9 (2016), no. 1, 245–347. MR 3465850, DOI https://doi.org/10.1112/jtopol/jtv042
- Andrew J. Blumberg, David Gepner, and Gonçalo Tabuada, A universal characterization of higher algebraic $K$-theory, Geom. Topol. 17 (2013), no. 2, 733–838. MR 3070515, DOI https://doi.org/10.2140/gt.2013.17.733
- Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, Martina Rovelli, and Claudia I. Scheimbauer, 2-Segal objects and the Waldhausen construction, arXiv:1809.10924, 2018.
- Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, Martina Rovelli, and Claudia I. Scheimbauer, 2-Segal sets and the Waldhausen construction, Topology Appl. 235 (2018), 445–484. MR 3760213, DOI https://doi.org/10.1016/j.topol.2017.12.009
- Clark Barwick and John Rognes, On the Q-construction for exact $\infty$-categories, http://www.maths.ed.ac.uk/~cbarwick/papers/qconstr.pdf.
- Tobias Dyckerhoff and Mikhail Kapranov, Higher Segal spaces I, arXiv:1212.3563, 2012.
- Imma Gálvez-Carrillo, Joachim Kock, and Andrew Tonks, Cohomology of decomposition spaces, in preparation.
- Imma Gálvez-Carrillo, Joachim Kock, and Andrew Tonks, Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory, Adv. Math. 331 (2018), 952–1015. MR 3804694, DOI https://doi.org/10.1016/j.aim.2018.03.016
- Daniel R. Grayson, Exterior power operations on higher $K$-theory, $K$-Theory 3 (1989), no. 3, 247–260. MR 1040401, DOI https://doi.org/10.1007/BF00533371
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Rune Haugseng, Iterated spans and classical topological field theories, Math. Z. 289 (2018), no. 3-4, 1427–1488. MR 3830256, DOI https://doi.org/10.1007/s00209-017-2005-x
- Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659
- Jacob Lurie, Higher algebra, http://www.math.harvard.edu/~lurie/papers/HA.pdf, (2017).
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR 0338129
- Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973–1007. MR 1804411, DOI https://doi.org/10.1090/S0002-9947-00-02653-2
- Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI https://doi.org/10.1007/BF01390197
- Katerina Velcheva, Generalized edgewise subdivisions, arXiv:1404.3416, 2014.
- Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI https://doi.org/10.1007/BFb0074449
- Charles A. Weibel, The $K$-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR 3076731
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 18D35, 18G30, 19D10, 55U10
Retrieve articles in all journals with MSC (2010): 18D35, 18G30, 19D10, 55U10
Additional Information
Julia E. Bergner
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
MR Author ID:
794441
Email:
jeb2md@virginia.edu
Angélica M. Osorno
Affiliation:
Department of Mathematics, Reed College, Portland, Oregon 97202
MR Author ID:
886548
Email:
aosorno@reed.edu
Viktoriya Ozornova
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
MR Author ID:
1124392
Email:
viktoriya.ozornova@rub.de
Martina Rovelli
Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
MR Author ID:
1204481
Email:
mrovelli@math.jhu.edu
Claudia I. Scheimbauer
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
Address at time of publication:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway
MR Author ID:
1115162
Email:
claudia.scheimbauer@ntnu.no
Received by editor(s):
July 20, 2018
Received by editor(s) in revised form:
April 16, 2019
Published electronically:
July 9, 2019
Additional Notes:
The first-named author was partially supported by NSF CAREER award DMS-1659931. The second-named author was partially supported by a grant from the Simons Foundation (#359449) and NSF grant DMS-1709302. The fourth-named author and fifth-named author were partially funded by the Swiss National Science Foundation, grants P2ELP2_172086 and P300P2_164652, respectively.
Communicated by:
Mark Behrens
Article copyright:
© Copyright 2019
American Mathematical Society
