Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Pseudomonadicity and 2-stack completions Marta Bunge and Claudio Hermida Dedicated to Michael Makkai on his 70th birthday Abstract. We consider the notion of stack, originally due to Grothendieck and Giraud [15], in the intrinsic sense (of Lawvere [26, 27]), that is, relative to the class of all epimorphisms in a topos S. We extend to dimension 2 the result of [7] on the fibrational (1-)stack completion of (the externalization of) a category object C in a Grothendieck topos S. In dimension 1, the monadicity and descent theorems of [2] and [4] are employed in [11] to show that S (or 0-Stack) is a 1-stack over itself, which yields Diaconescu’s theorem[23] on the classification of G-torsors for a groupoid G as an application. Likewise, in di- mension 2, we resort to the pseudomonadicity and 2-descent theorems of [20] in order to prove that Stack (or 1-Stack) is a 2-stack over S. We then derive a classification theorem for G-2-torsors for a 2-gerbe G. An axiom of stack completions (ASC) states, for a topos S, that the fibrational stack completion of any category object in S is representable, and similarly for 2-stack com- pletions. ASC holds for any Grothendieck topos S, by a general argument in [23]. Alternatively, we could appeal to the Quillen model structure on Cat(S) of [21] (for S a Grothendieck topos) to obtain (strong) stack completions in- ternal to S. While we leave the question of giving a similar construction (via Quillen model structure) in the 2-dimensional case open, we indicate how the passages of stack theory from dimension 1 to dimension 2 pave the way for similar results in higher dimensions. Introduction Section 1 discusses the beautiful theory of stacks and non-abelian topos coho- mology. This material is considered by many to be one of the pinnacles of 20th century mathematics. The theory of stacks (or champs) was developed first by Grothendieck [16] and Giraud [15] in terms of sites, and then recast in an intrinsic fashion and clarified considerably in a second wave by the topos theory community based mostly in North America. Section 1, which is largely expository, is based on [11] and [7], though formu- lated in terms of fibrations over a given topos S, as opposed to S-indexed categories 2000 Mathematics Subject Classification. 18D05, 18D30, 18F20, 18G60, 14A20. Key words and phrases. Toposes, 2-categories, 2-descent, 2-stacks, 2-gerbes, 2-torsors, Quillen model structures, classifying structures, Morita equivalence theorems, higher dimensional stacks. c 0000 (copyright holder) 1