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In the present paper, by using the colored version of the Koszul duality, the concept of a differential module with $\infty$-simplicial faces is introduced. The homotopy invariance of the structure of a differential module with $\infty$-simplicial faces is proved. The relationships between differential modules with $\infty$-simplicial faces and $A_\infty$-algebras are established. The notion of a chain realization of a differential module with $\infty$-simplicial faces and the concept of a tensor product of differential modules with $\infty$-simplicial faces are introduced. It is proved that for an arbitrary $A_\infty$-algebra the chain realization of the tensor differential module with $\infty$-simplicial faces, which corresponds to this $A_\infty$-algebra, and the $B$-construction of this $A_\infty$-algebra are isomorphic differential coalgebras.

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The notion of differential Lie module over a curved coalgebra is introduced. The homotopy invariance of the structure of a differential Liemodule over a curved coalgebra is proved. A relationship between the homotopy theory of differential Lie modules over curved coalgebras and the theory of Koszul duality for quadratic-scalar algebras over commutative unital rings is determined.

Given a type
of algebras there is a notion of “free” algebra over a generic vector space V. Let us denote it by
. Viewed as a functor from the category Vect of vector spaces to itself,
is equipped with a monoid structure, that is a transformation of functors
, which is associative, and another one
which is a unit. The existence of this structure follows readily from the universal properties of free algebras. Such a data
is called an algebraic operad.
This notion admits another equivalent definitions: classical, partial, and combinatorial.

The notion of a differential module with homotopy simplicial faces is introduced, which is a homotopy analog of the notion of a differential module with simplicial faces. The homotopy invariance of the structure of a differential module with homotopy simplicial faces is proved. Relationships between the construction of a differential module with homotopy simplicial faces and the theories of A
∞-algebras and D
∞-differential modules are found. Applications of the method of homotopy simplicial faces to describing the homology of realizations of simplicial topological spaces are presented.

In this paper the homology theory of fibre spaces is studied by introducing additional algebraic structure in homology and cohomology. All modules are assumed to be over an arbitrary associative ring Λ with unit; by a differential algebra, coalgebra, module, or comodule we mean these objects graded by non-negative integers; â denotes (−1) dega. The category A(∞). An A(∞)-algebra in the sense of Stasheff [1] is defined to be a graded Λ-module M, endowed with a set of operations {mi: ⊗ i M → M, i = 1, 2,...} satisfying the conditions mi(( ⊗ i M)q) ⊂ Mq+i−2 and

This paper is devoted to the development of algebraic devices that are necessary for describing the homotopy groups of topological spaces. As shown in the note [1], the notion of a Lie algebra over the operad must be used for this purpose. Here we consider this notion in more detail, prove its most important properties and clarify the question about the algebraic structure on the homotopy groups of topological spaces. In particular, we show that the homotopy groups of a topological space possess the structure of a Lie -algebra which determines the homotopy type of the space in the simply-connected case.
Since the notion of an operad is analogous to that of algebra, we begin with recalling the notions of algebra and coalgebra and reviewing the main constructions over them. Then we transfer these constructions to the operad case and use them to investigate the structure of Lie algebras and coalgebras over an operad.

A method is presented for describing the homology of B-constructions and co-B-constructions; it is then applied to describing for F2 term of the Adams spectral sequence and the homotopy groups of topological spaces.

In the present paper the construction of a -differential -(co)algebra is introduced and basic homotopy properties of this construction are studied. The connection between -differential -(co)algebras and spectral sequences is established, which enables us to construct the structure of an -coalgebra on the Milnor coalgebra directly from the differentials of the Adams spectral sequence.

In the present paper the notions of a D-infinity-differential and a D-infinity-differential module are introduced, which are, respectively, homotopically invariant analogues of the differential and the chain complex. Basic homotopic properties of D-infinity-differentials and D-infinity-differential modules are established. The connection between the Gugenheim-Lambe-Stasheff theory of differential perturbations in homological algebra and the construction of a D-infinity-differential module is considered.

This paper makes a study of operads and of coalgebras over operads. Certain operads En and E are defined, constituting the algebraic analogues of the "little n-cube" operads; it is then shown that the singular chain complex C*(X;R) of a topological space X is a coalgebra over the operad E, and that this structure completely determines the weak homotopy type of the space.
Bibliography: 26 titles.

It is proved in the paper that the multiplication operation on an arbitrary E
∞-algebra can be extended to an E
∞-algebra A
∞-morphism. As a corollary, it is proved that every May algebra defined by an E
∞-algebra is a Cartan object in the category of May algebras.
KeywordsE∞-algebra–A∞-morphism–Cartan object–May algebra–differential module–Steenrod operations–cohomology of a topological space