Cofree compositions of coalgebras
نویسندگان
چکیده
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 with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes. Résumé.
منابع مشابه
Cofree 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 کاملCofree Coalgebras over Operads
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 constructio...
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 کاملA Universal Investigation of $n$-representations of $n$-quivers
noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $math...
متن کاملSome Co-Birkhoff Type Theorems
We consider the dual of Theorem 1 from [33], relating closure conditions on subcategories with projectivity classes for collections of discrete cocones. We extend these results by adding a new operator for closure under domains of epis and show that this corresponds to taking projectivity classes for cocones with vertex a subobject of the terminal object. We show that these theorems apply to ca...
متن کامل