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2010 Homotopy Classification of Gerbes
J. F. Jardine
Publ. Mat. 54(1): 83-111 (2010).


Gerbes are locally connected presheaves of groupoids on a small Grothendieck site $\mathcal{C}$. They are classified up to local weak equivalence by path components of a cocycle category taking values in the big $2$-groupoid $\mathbf{Iso}(\Gr(\mathcal{C}))$ consisting of all sheaves of groups on $\mathcal{C}$, their isomorphisms and homotopies. If $\mathcal{F}$ is a full subpresheaf of $\mathbf{Iso}(\Gr(\mathcal{C}))$ then the set $[\ast,B\mathcal{F}]$ of morphisms in the homotopy category of simplicial presheaves classifies gerbes locally weakly equivalent to objects of $\mathcal{F}$. If $\St(\pi \mathcal{F})$ is the stack completion of the fundamental groupoid $\pi\mathcal{F}$ of $\mathcal{F}$, if $L$ is a global section of $\St(\pi\mathcal{F})$, and if $F_{L}$ is the homotopy fibre over $L$ of the canonical map $B\mathcal{F} \to B\St(\pi\mathcal{F})$, then $[\ast,F_{L}]$ is in bijective correspondence with Giraud's non-abelian cohomology object $H^{2}(\mathcal{C},L)$ of equivalence classes of gerbes with band $L$.


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J. F. Jardine. "Homotopy Classification of Gerbes." Publ. Mat. 54 (1) 83 - 111, 2010.


Published: 2010
First available in Project Euclid: 8 January 2010

zbMATH: 1190.18008
MathSciNet: MR2603590

Primary: 20G10
Secondary: 03C20 , 18G30

Keywords: $2$-groupoids , cocycles , gerbes , Simplicial presheaves

Rights: Copyright © 2010 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.54 • No. 1 • 2010
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