FORMALITY FOR ALGEBROIDS I: NERVES OF TWO-GROUPOIDS PAUL BRESSLER, ALEXANDER GOROKHOVSKY, RYSZARD NEST, AND BORIS TSYGAN Abstract. Extension of the M. Kontsevich formality theorem to gerbes [19] necessitates extension of the notion of Deligne 2-groupoid to L∞ algebras. For a differential graded Lie algebra g which is zero below the degree -1, we show that the nerve of the Deligne 2-groupoid is homotopy equivalent to the Hinich simplicial set [13] of g-valued differential forms. Contents 1. Introduction 2 2. The homotopy type of a strict 2-groupoid 3 2.1. Nerves of simplicial groupoids 3 2.2. Strict 2-groupoids 4 3. Homotopy types associated with L ∞ -algebras 5 3.1. L ∞ -algebras 5 3.2. The functor Σ 6 3.3. Deligne groupoids 8 3.4. Properties of N MC 2 9 3.5. N MC 2 vs. Σ 10 4. Non-abelian integration 12 4.1. Piecewise polynomial forms and connections 13 4.2. Holonomy 14 4.3. Logarithmic derivatives 15 4.4. Existence and uniqueness of solutions 17 5. Multiplicative integrals 18 5.1. Coordinate systems 18 5.2. A key lemma about holonomies 19 5.3. Definition and basic properties of I x D 21 5.4. Independence of a coordinate system 23 5.5. End of the proof of Theorem 5.2 23 6. Higher holonomy and Σ = MC 2 24 6.1. The construction of higher holonomy I 24 6.2. Properties of the higher holonomy map 25 References 26 A. Gorokhovsky was partially supported by NSF grant DMS-0900968. B. Tsygan was partially supported by NSF grant DMS-0906391. R. Nest was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). 1