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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: dimitri.ara@univ-amu.fr
DENIS-CHARLES CISINSKI
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Deutschland. e-mail: denis-charles.cisinski@mathematik.uni-regensburg.de
IEKE MOERDIJK
Affiliation:
Department of Mathematics, Utrecht University, PO BOX 80.010, 3508 TA Utrecht, The Netherlands. e-mail: i.moerdijk@uu.nl
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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 

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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