1.3. For any morphism f : A → B of O-algebras one defines in a standard way the functor Der: Der B/A : Mod(O, B) → C(k).The functor is representable by the module Ω B/A ∈ Mod(O, B). This is the module of differentials. If O is Σ-split so that Alg(O) admits a CMC structure, one defines the relative cotangent complex L B/A ∈ D(O, B) as the module of differentials of a corresponding cofibrant resolution. This defines cohomology of O-algebra A as the functorThe most interesting cohomology is the one with coefficients in M = A. No doubt, the complex Der O (A, A) admits a dg Lie algebra structure. The main result of Section 8, Theorem 8.5.3, claims that the tangent complex T A := H(A, A) = R Hom(L A , A) admits a canonical structure of Homotopy Lie algebra. This means that T A is defined uniquely up to a unique isomorphism as an object of the category Holie(k). 1.4. Let us indicate some relevant references. Spaltenstein [Sp], Avramov-Foxby-Halperin [AFH] developed homological algebra for unbounded complexes.Operads and operad algebras were invented by J.P. May in early 70-ies in a topological context; dg operads appeared explicitly in [HS] and became popular in 90-ies mainly because of their connection to quantum field theory.M. Markl in [M] studied "minimal models" for operads -similarly to Sullivan's minimal models for commutative dg algebras over Q. In our terms, these are cofibrant operads weakly equivalent to a given one.In [SS] M.. Schlessinger and J. Stasheff propose to define the tangent complex of a commutative algebra A as Der(A) where A is a "model" i.e. a commutative dg algebra quasi-isomorphic to A and free as a graded commutative algebra. This complex has an obvious Lie algebra structure which is proven to coincide sometimes (for a standard choice of A) with the one defined by the Harrison complex of A.It is clear "morally" that the homotopy type of the Lie algebra Der(A) should not depend on the choice of A. Our main result of Section 8 says (in a more general setting) that this is really so.1.5. Notations. For a ring k we denote by C(k) the category of complexes of k-modules. If X, Y ∈ C(k) we denote by Hom k (X, Y ) the complex of maps form X to Y (not necessarily commuting with the differentials).N is the set of non-negative integers; Ens is the category of sets, Cat is the 2-category of small categories. The rest of the notations is given in the main text.

1.3. Functors on Artin rings. Thus, we consider unital coalgebras as "the most general" formal (dg) stacks concentrated in a point. It is reasonable to describe them as functors on formal spaces -as one defines stacks using functors on affine schemes. One can go even further and take into account that any formal space is a filtered colimit of finite dimensional ones which now take form A * where A ∈ dgart ≤0 (k).1.3.1. Any formal stack X ∈ dgcu(k) gives rise to a deformation functorwhich is defined up to homotopy equivalence. This corresponds to the usual description of stacks as 2-functors from affine schemes to groupoids. The deformation functor enjoys nice exactness properties (see 8.1.3). Given the tangent Lie algebra g = L(X), the functor X can be described as the nerve Σ g of the dg Lie algebra g (see [H1] and 8.1.1) defined by the formula Σ g ((A, m)) n = MC(m ⊗ Ω n ⊗ g)for (A, m) ∈ dgart ≤0 (k), where MC( ) denotes the collection of Maurer-Cartan elements of a dg Lie algebra and Ω n is the algebra of polynomial differential forms on the standard n-simplex.The nerve of a dg Lie algebra is homotopy equivalent to the Deligne groupoid (cf. [GM1], [H1]) if (m ⊗ g) i = 0 for i < 0.1.3.2. If X is a formal space, the restriction of X to the category art(k) of artinian k-algebras concentrated in degree zero, is a functor to discrete simplicial sets (i.e., essentially, to Ens)see 9.3.2.This means in particular, that the restriction of X to art(k) is representable in usual sense by H 0 (X).1.4. Rational spaces. A very "non-geometric" class of unital coalgebras is the Quillen's category dgcu 2 (Q) of 2-reduced unital coalgebras which is one of the models for simply connected rational homotopy types.One can easily calculate the deformation functor defined by a simply connected rational homotopy type. Let g ∈ dglie(Q) be the Lie algebra model for it. This is the tangent Lie algebra of the corresponding unital coalgebra. One has g i = 0 for i ≥ 0.

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We compare two approaches to the homotopy theory of ∞-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and ∞-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Γ. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.