Related papers: Explicit construction of cofree precoalgebras and …
We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. We prove that the composition of two cofree coalgebras is again cofree, and we give sufficient…
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…
We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators…
We describe the category of homotopy coalgebras, concentrating on properties of relatively cofree homotopy coalgebras, morphisms and coderivations from an ordinary coalgebra to a relatively cofree homotopy coalgebra, morphisms and…
A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and the coproduct. We here give examples of cofree Com-PreLie bialgebras, including all the ones such that the…
This paper proves that homology equivalences of cogenerating complexes induce homology equivalences of the cofree coalgebras in many interesting cases. We show that the underlying chain complex of any cofree coalgebra is naturally a direct…
We define two coproducts for cycle-free oriented graphs, thus building up two commutative con- nected graded Hopf algebras, such that one is a comodule-coalgebra on the other, thus generalizing the result obtained previously for Hopf…
We first prove that a graded, connected, free and cofree Hopf algebra is always self-dual; then that two graded, connected, free and cofree Hopf algebras are isomorphic if, and only if, they have the same Poincar\'e-Hilbert formal series.…
This paper contains an elementary proof of the existence of the classical model structure on the category of unbounded DG-Lie algebras over a field of characteristic zero, with an emphasis on the properties of free and semifree extensions,…
This paper explicitely constructs cofree coalgebras over operads in the category of DG-modules. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). It is shown that the existence…
We give a systematic construction of Hopf algebra structures on braided cofree coalgebras. The relevant underlying structures are braided algebras and braided coalgebras. We provide some interesting examples of these algebras and coalgebras…
We give a direct proof of a result of Sweedler describing the cofree cocommutative coalgebra over a vector space, and use our approach to give an explicit construction of liftings of maps into this universal coalgebra. The basic ingredients…
Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors where the colocalised coalgebra turned out to be a…
We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in…
A simply colored coalgebra is a coassociative counital coalgebra $C$ over an arbitrary ring $R$, which can be decomposed into a direct sum of two $R$-modules: one generated by set-like elements and another consisting of conilpotent…
Our constructions provide a systematic way to study cohomology pre-algebraic structures via classical cohomology, simplifying computations and enabling the use of established techniques.
We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and…
In this paper, we first recall the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then we construct the corresponding nonassociative permutative coproduct, and we prove that the…
We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an…
We give a filtration of the unoriented cobordism ring using the infinite symplectic group, with polynomial generators given one at a time. The generating manifolds are also constructed using the cup construction.