In this work we study exactness in the sesqui-category of n-groupoids. Using homotopy pullbacks, we construct a six term sequence of (n-1)-groupoids from an n-functor between pointed n-groupoids. We show that the sequence is exact in a suitable sense, which generalizes the usual notions of exactness for groups and categorical groups. Moreover, iterating the process, we get a ziqqurath2 of exact sequences of increasing length and decreasing dimension. For n = 1, we recover a classical result due to R. Brown and, for n = 2, its generalizations due to Hardie, Kamps and Kieboom and to Duskin, Kieboom and Vitale.

The ziqqurath of exact sequences of $n$-groupoids / S. Kasangian, G. Metere, E.M. Vitale. - In: CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES. - ISSN 1245-530X. - 52:1(2011), pp. 2-44.

The ziqqurath of exact sequences of $n$-groupoids

S. Kasangian
Primo
;
G. Metere
Secondo
;
2011

Abstract

In this work we study exactness in the sesqui-category of n-groupoids. Using homotopy pullbacks, we construct a six term sequence of (n-1)-groupoids from an n-functor between pointed n-groupoids. We show that the sequence is exact in a suitable sense, which generalizes the usual notions of exactness for groups and categorical groups. Moreover, iterating the process, we get a ziqqurath2 of exact sequences of increasing length and decreasing dimension. For n = 1, we recover a classical result due to R. Brown and, for n = 2, its generalizations due to Hardie, Kamps and Kieboom and to Duskin, Kieboom and Vitale.
Dans ce travail nous ´etudions la notion de suite exacte dans la sesquicat ´egorie des n-groupo¨ıdes. En utilisant les produits fibr´es homotopiques, `a partir d’un n-foncteur entre n-groupo¨ıdes point´es nous construisons une suite de six (n-1)- groupo¨ıdes. Nous montrons que cette suite est exacte dans un sens qui g´en´eralise les notions usuelles d’exactitude pour les groupes et les gr-cat´egories. En r´eit´erant le processus, nous obtenons une ziggourat1 de suites exactes de longueur croissante et dimension d´ecroissante. Pour n = 1, nous retrouvons un r´esultat classic du `a R. Brown et, pour n = 2, nous retrouvons ses g´en´eralisations dues `a Hardie, Kamps et Kieboom et `a Duskin, Kieboom et Vitale.
Settore MAT/02 - Algebra
Settore MAT/03 - Geometria
http://perso.uclouvain.be/enrico.vitale/Ziqqurath.pdf
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/205914
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