Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T15:22:48.236Z Has data issue: false hasContentIssue false
  • English
  • Français

The dendroidal category is a test category

Published online by Cambridge University Press:  26 April 2018

DIMITRI ARA ,
DENIS-CHARLES CISINSKI  and
IEKE MOERDIJK
DIMITRI ARA
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France. e-mail: [email protected]
DENIS-CHARLES CISINSKI
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Deutschland. e-mail: [email protected]
IEKE MOERDIJK
Affiliation:
Department of Mathematics, Utrecht University, PO BOX 80.010, 3508 TA Utrecht, The Netherlands. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the category of trees Ω is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category Quillen-equivalent to spaces. We show that this model category structure, up to a change of cofibrations, can be obtained as an explicit left Bousfield localisation of the operadic model category structure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 107 - 121
Copyright
Copyright © Cambridge Philosophical Society 2018 

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

[1] Ara, D. The groupoidal analogue Θ̃ to Joyal's category Θ is a test category. Appl. Categ. Structures 20 (6) (2012), 603–649.Google Scholar
[2] Boardman, J. M. and Vogt, R. M. Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Math., Vol. 347 (Springer-Verlag, 1973).Google Scholar
[3] Cisinski, D.-C. Les préfaisceaux comme modèles des types d'homotopie. Astérisque. Soc. Math. France no. 308 (2006).Google Scholar
[4] Cisinski, D.-C. and Maltsiniotis, G. La catégorie Θ de Joyal est une catégorie test. J. Pure Appl. Algebra 215 (5) (2011), 962–982.Google Scholar
[5] Cisinski, D.-C. and Moerdijk, I. Dendroidal sets as models for homotopy operads. J. Topol. 4 (2) (2011), 257–299.Google Scholar
[6] Cisinski, D.-C. and Moerdijk, I. Dendroidal Segal spaces and ∞-operads. J. Topol. 6 (3) (2013), 675–704.Google Scholar
[7] Gabriel, P. and Zisman, M. Calculus of fractions and homotopy theory. Ergeb. Math. Grenzgeb. vol. 35 (Springer-Verlag, 1967).Google Scholar
[8] Grothendieck, A. Pursuing stacks. Manuscript, 1983, to be published in Documents Mathématiques.Google Scholar
[9] Heuts, G., Hinich, V. and Moerdijk, I. On the equivalence between Lurie's model and the dendroidal model for infinity-operads. Adv. Math. 302 (2016) 869–1043.Google Scholar
[10] Jardine, J. F. Categorical homotopy theory. Homology Homotopy Appl. 8 (1) (2006), 71–144.Google Scholar
[11] Maltsiniotis, G. La théorie de l'homotopie de Grothendieck. Astérisque. Soc. Math. France no. 301 (2006).Google Scholar
[12] Maltsiniotis, G. La catégorie cubique avec connexions est une catégorie test stricte. Homology Homotopy Appl. 11 (2) (2009), 309–326.Google Scholar
[13] Moerdijk, I. and Weiss, I. Dendroidal sets. Algebr. Geom. Topol. 7 (2007), 1441–1470.Google Scholar
[14] Moerdijk, I. and Weiss, I. On inner Kan complexes in the category of dendroidal sets. Adv. Math. 221 (2) (2009), 343–389.Google Scholar