Cofree Coalgebras over Operads
نویسنده
چکیده
This paper explicitly constructs cofree coalgebras over operads in the category of DG-modules. It is shown that the existence of an operadaction on a coalgebra implies a “generalized coassociativity” that facilitates the construction. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). In the pointed irreducible case, this construction takes a particularly simple form: the coproduct is the dual of the operad composition.
منابع مشابه
Cofree compositions of coalgebras
We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra w...
متن کاملCofree coalgebras and differential linear logic
We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras to model the exponential is a model of differential linear logic. Thus, in this semantics, proof denotations have natural derivatives. We give several examples of these derivatives.
متن کاملHomotopy Theory of Coalgebras over Operads
This paper constructs model structures on the categories of coalgebras and pointed irreducible coalgebras over an operad whose components are projective, finitely generated in each dimension, and satisfy a condition that allows one to take tensor products with a unit interval. The underlying chain-complex is assumed to be unbounded and the results for bounded coalgebras over an operad are deriv...
متن کاملModel-categories of Coalgebras over Operads
This paper constructs model structures on the categories of coalgebras and pointed irreducible coalgebras over an operad whose components are projective, finitely generated in each dimension, and satisfy a condition that allows one to take tensor products with a unit interval. The underlying chain-complex is assumed to be unbounded and the results for bounded coalgebras over an operad are deriv...
متن کاملCommutators of A∞ Structures
This note will be for the most part of an expository nature; many of the results here have appeared elsewhere. We would like however, to highlight the connection between Jim Stasheff’s early work on A∞ algebras in the 1960’s and the notion of sh Lie algebras (L∞ algebras) which is currently popular in mathematical physics; see, for example, [1] and [7]. This connection involves some work of Vic...
متن کامل