## Algebraic & Geometric Topology

### Localization of cofibration categories and groupoid $C^*$–algebras

#### Abstract

We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid $C∗$–algebra and thereby its topological $K$–theory spectrum.

#### Article information

Source
Algebr. Geom. Topol., Volume 17, Number 5 (2017), 3007-3020.

Dates
Revised: 2 May 2017
Accepted: 14 June 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841490

Digital Object Identifier
doi:10.2140/agt.2017.17.3007

Mathematical Reviews number (MathSciNet)
MR3704250

Zentralblatt MATH identifier
06791391

#### Citation

Land, Markus; Nikolaus, Thomas; Szumiło, Karol. Localization of cofibration categories and groupoid $C^*$–algebras. Algebr. Geom. Topol. 17 (2017), no. 5, 3007--3020. doi:10.2140/agt.2017.17.3007. https://projecteuclid.org/euclid.agt/1510841490

#### References

• K,S Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973) 419–458
• C Casacuberta, M Golasiński, A Tonks, Homotopy localization of groupoids, Forum Math. 18 (2006) 967–982
• D-C Cisinski, Does the classification diagram localize a category weak equivalences?, MathOveflow post (2012) Available at \setbox0\makeatletter\@url https://mathoverflow.net/q/92916 {\unhbox0
• J,F Davis, W Lück, Spaces over a category and assembly maps in isomorphism conjectures in $K$– and $L$–theory, $K$–Theory 15 (1998) 201–252
• I Dell'Ambrogio, The unitary symmetric monoidal model category of small $\rm C^*$–categories, Homology Homotopy Appl. 14 (2012) 101–127
• D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177–201
• P,G Goerss, J,F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
• M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc., Providence, RI (1999)
• M Joachim, $K$–homology of $C^\ast$–categories and symmetric spectra representing $K$–homology, Math. Ann. 327 (2003) 641–670
• M Land, T Nikolaus, On the relation between $K$– and $L$–theory of $C^*$–algebras, preprint (2016)
• J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
• J Lurie, Higher algebra, book in progress, Harvard (2017) Available at \setbox0\makeatletter\@url http://math.harvard.edu/~lurie/papers/HA.pdf {\unhbox0
• A Mazel-Gee, The universality of the Rezk nerve, preprint (2015)
• A Radulescu-Banu, Cofibrations in homotopy theory, preprint (2009)
• I Raeburn, D,P Williams, Morita equivalence and continuous-trace $C^*$–algebras, Mathematical Surveys and Monographs 60, Amer. Math. Soc., Providence, RI (1998)
• C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973–1007
• S Schwede, The stable homotopy category is rigid, Ann. of Math. 166 (2007) 837–863