This paper describes one approach for calculating the cohomology of a particular complex which arises in knot theory and which is used to prove the existence of ‘associators’, a key ingredient in the construction of a universal knot invariant. The approach taken is to show that the relevant complex can be interpreted, essentially, as the cohomology of the functor Hom(K, C), where C is the coalg...

We describe the vector space semantics for linear logic in which the exponential modality is modelled by cofree coalgebras. There is a natural differentiable structure in this semantics arising from the primitive elements. We provide an investigation of this structure, through many examples of proofs and their derivatives. In particular, an encoding of Turing machines in linear logic is given i...

The category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the...

It is shown that for any commutative unital ring R the category HopfR of R–Hopf algebras is locally presentable and a coreflective subcategory of the category BialgR of R–bialgebras, admitting cofree Hopf algebras over arbitrary R–algebras. The proofs are based on an explicit analysis of the construction of colimits of Hopf algebras, which generalizes an observation of Takeuchi. Essentially be ...

In this paper, we consider the forgetful functor from the category LDcpo of local dcpos (respectively, Dcpo of dcpos) to the category Pos of posets (respectively, LDcpo of local dcpos), and study the existence of its left and right adjoints. Moreover, we give the concrete forms of free and cofree S-ldcpos over a local dcpo, where S is a local dcpo monoid. The main results are: (1) The forgetful...

Covarieties of coalgebras are those classes of coalgebras for an endofunctor H on the category of sets that are closed under coproducts, subcoalgebras and quotients. Equivalently, covarieties are classes of H-coalgebras that can be presented by coequations. Adámek introduced a logic of coequations and proved soundness and completeness for all polynomial functors on the category of sets. Here th...

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...

Abstract The goal of this paper is to show that valuation theory and Hopf are compatible on the class generalized permutahedra. We prove structure $\textbf {GP}^+$ these polyhedra descends, modulo inclusion-exclusion relations, an indicator monoid $\mathbb {I}(\textbf {GP}^+)$ permutahedra isomorphic weighted ordered set partitions. This quotient cofree. It terminal object in category monoids w...

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. We also describe the structure constants of the multiplicati...