Abstract

In the introduction of Laumon-Moret-Bailly ([LMB] p. 2) they refer to a possible theory of algebraic n-stacks: Signalons au passage que Grothendieck propose d’élargir à son tour le cadre précédent en remplaçant les 1-champs par des n-champs (grosso modo, des faisceaux en n-catégories sur (Aff) ou sur un site arbitraire) et il ne fait guère de doute qu’il existe une notion utile de n-champs algébriques.... The purpose of this paper is to propose such a theory. I guess that the main reason why Laumon and Moret-Bailly didn’t want to get into this theory was for fear of getting caught up in a horribly technical discussion of n-stacks of groupoids over a general site. In this paper we simply assume that a theory of n-stacks of groupoids exists. This is not an unreasonable assumption, first of all because there is a relatively good substitute—the theory of simplicial presheaves or presheaves of spaces ([Bro] [B-G] [Jo] [Ja] [Si3] [Si2])— which should be equivalent, in an appropriate sense, to any eventual theory of n-stacks; and second of all because it seems likely that a real theory of n-stacks of n-groupoids could be developped in the near future ([Br2], [Ta]).

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