We present the dual to Birkhoff’s variety theorem in terms of predicates over the carrier of a cofree coalgebra (i.e., in terms of “coequations”). We then discuss the dual to Birkhoff’s completeness theorem, showing how closure under deductive rules dualizes to yield two modal operators acting on coequations. We discuss the properties of these operators and show, given a familiar condition on t...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

Because of the isomorphism (X × A) → X ∼= X → X, the transition structure α : X → X of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. Here we will use this algebra-coalgebra duality of automata as a common perspective for the study of equations and coequations. Equations are sets of pairs of words (v, w) that ar...

We give an introduction to computation and logic tailored for algebraists, and use this as a springboard to discuss geometric models of computation and the role of cut-elimination in these models, following Girard’s geometry of interaction program. We discuss how to represent programs in the λ-calculus and proofs in linear logic as linear maps between infinite-dimensional vector spaces. The int...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

This is a contribution to the project of quiver approaches to quasi-quantum groups initiated in [13]. We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations. This leads to a theoretic classification of graded pointed Majid algebras over path coalgebras, or equivalently cofree pointed coalgebras, and helps to provide a projective representation-theoretic ...

Batanin and Markl's operadic categories are in which each map is endowed with a finite collection of “abstract fibres”—also objects the same category—subject to suitable axioms. We give reconstruction data axioms terms décalage comonad D on small categories. A simple case involves unary categories—ones wherein has exactly one abstract fibre—which exhibited as are, first all, coalgebras for D, a...

Received We study three comonads derived from the comma construction. The induced coalgebras correspond to the three concepts displayed in the title of the paper. The comonad that yields the-autonomous categories is, in essence, the Chu construction, which has recently awaken much interest in computer science. We describe its couniversal property. It is right adjoint to the inclusion of-autonom...

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...