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Notes on weak units of group-like 1- and 2-stacks

Published online by Cambridge University Press:  09 November 2016

ETTORE ALDROVANDI  and
A. EMIN TATAR
ETTORE ALDROVANDI
Affiliation:
Department of Mathematics, Florida State University, 1017 Academic Way Tallahassee, FL 32312-4510, U.S.A. e-mail: aldrovandi@math.fsu.edu
A. EMIN TATAR
Affiliation:
Department of Mathematics and Statistics, KFUPM, Dhahran, 31261, Saudi Arabia. e-mail: atatar@kfupm.edu.sa
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Abstract

The weak units of strict monoidal 1- and 2-categories are defined respectively in [15] and [14]. In this paper, we define them for group-like 1- and 2-stacks. We show that they form a contractible Picard 1- and 2-stack, respectively. We give their cohomological description which provides for these stacks a representation by complexes of sheaves of groups. Later, we extend the discussion to the monoidal case. We consider the (2-)substack of cancelable objects of a monoidal 1-(2-)stack. We observe that this (2-)substack is trivially group-like, its weak units are the same as the weak units of the monoidal 1-(2-)stack, and therefore we can recover the contractibility results in [15] and [14] by analysing it.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 205 - 220
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Aldrovandi, E. 2-gerbes bound by complexes of gr-stacks, and cohomology. J. Pure Appl. Algebra 212 (5) (2008), 994–1038.Google Scholar
[2] Aldrovandi, E. and Noohi, B. Butterflies III: higher butterflies and higher gr-stacks. In Preparation.Google Scholar
[3] Aldrovandi, E. and Noohi, B. Butterflies I: morphisms of 2-group stacks. Adv. Math. 221 (3) (2009), 687–773.Google Scholar
[4] Aldrovandi, E. and Tatar, A. E. Cohomology and coherence. In preparation.Google Scholar
[5] Bertolin, C. and Tatar, A. E. Resolution of extensions of picard 2-stacks. arXiv:1401.6685 (2014).Google Scholar
[6] Breen, L. Bitorseurs et cohomologie non abélienne. In The Grothendieck Festschrift, Vol. I, Progr. Math. vol. 86. (Birkhäuser Boston, Boston, MA, 1990), pp. 401–476.Google Scholar
[7] Breen, L. Théorie de Schreier supérieure. Ann. Sci. École Norm. Sup. (4), 25 (5) (1992), 465–514.Google Scholar
[8] Breen, L. On the classification of 2-gerbes and 2-stacks. Astérisque 225 (1994), 160.Google Scholar
[9] Conduché, D. Simplicial crossed modules and mapping cones. Georgian Math. J. 10 (4) (2003), 623–636.Google Scholar
[10] Conduché, D. Modules croisés généralisés de longueur 2. In Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983) vol. 34 (1984), pp. 155–178.Google Scholar
[11] Deligne, P. La formule de dualité globale, 1973. SGA 4 III, Exposé XVIII.Google Scholar
[12] Deligne, P. Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII (Amer. Math. Soc., Providence, R.I., 1979), pp. 247–289.Google Scholar
[13] Hakim, M. Topos Annelés et Schémas Relatifs (Springer-Verlag, Berlin, 1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64.CrossRefGoogle Scholar
[14] Joyal, A. and Kock, J. Coherence for weak units. Doc. Math. 18 (2013), 71–110.CrossRefGoogle Scholar
[15] Kock, J. Elementary remarks on units in monoidal categories. Math. Proc. Camb. Phil. Soc. 144 (1) (2008), 53–76.Google Scholar
[16] Noohi, B. On weak maps between 2-groups (2005).Google Scholar
[17] Norrie, K. Actions and automorphisms of crossed modules. Bull. Soc. Math. France 118 (2) (1990), 129–146.Google Scholar
[18] Saavedra Rivano, N. Catégories Tannakiennes. Lecture Notes in Mathematics vol. 265 (Springer-Verlag, Berlin, 1972).Google Scholar