Related papers: Bases as Coalgebras
The set of natural integers is fundamental for at least two reasons: it is the free induction algebra over the empty set (and at such allows definitions of maps by primitive recursion) and it is the free monoid over a one-element set, the…
Given a Hopf algebra in a symmetric monoidal category with duals, the category of modules inherits the structure of a monoidal category with duals. If the notion of algebra is replaced with that of monad on a monoidal category with duals…
We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to the one of equational classes defined by equation arrows. Free…
A class of algebras is constructed using free fermions and the invariant antisymmetric tensors associated with irreducible holonomy groups. (This version contains minor typographical corrections and some additional references. )
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…
Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the…
After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic…
It is known that the construction of the frame of ideals from a distributive lattice induces a monad whose algebras are precisely the frames and frame homomorphisms. Using the Fakir construction of an idempotent approximation of a monad, we…
For a model category, we prove that taking the category of coalgebras over a comonad commutes with left Bousfield localization in a suitable sense. Then we prove a general existence result for the left-induced model structure on the…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…
Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence…
Let $R$ denote a 2-fir. The notions of F-independence and algebraic subsets of R are defined. The decomposition of an algebraic subset into similarity classes gives a simple way of translating the F-independence in terms of dimension of…
We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones. We see that in many cases, the pie algebras are…
The notion of {\it free} generalized vertex algebras is introduced. It is equivalent to the notion of {\it generalized principal subspaces} associated with lattices which are not necessarily integral. Combinatorial bases and the characters…
Finitary monads on $\mathsf{Pos}$ are characterized as the precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analagously, finitary enriched monads on…
A new generalisation of the notion of space, called "vectoid", is suggested in this work. Basic definitions, examples and properties are presented, as well as a construction of direct product of vectoids. Proofs of more complicated…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative…
In earlier work, Batanin has shown that an important class of definitions of higher categories could be apprehended together simply as monads over globular sets. This allowed him to generalize the notion of polygraph, initially introduced…
We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n+k generators and k relations and has an n-element system of generators, then this algebra is a free…