Related papers: Functor is to Lens as Applicative is to Biplate: I…
We present an expository overview of the monoidal structures in the category of linearly compact vector spaces. Bimonoids in this category are the natural duals of infinite-dimensional bialgebras. We classify the relations on words whose…
We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of ``entwined comodules'' and ``entwined contramodules'' over a triple $(\mathscr C,A,\psi)$, where…
In the context of abstract coinduction in complete lattices, the notion of compatible function makes it possible to introduce enhancements of the coinduction proof principle. The largest compatible function, called the companion, subsumes…
To any affine scheme with a $\mathbb{G}_m$-action, we provide a Bousfield colocalization on the equivariant derived category of modules by constructing, via homotopical methods, an idempotent integral kernel. This endows the equivariant…
A shelf is a set with a binary operation~$\op$ satisfying $a \op (b \op c) = (a \op b) \op (a \op c)$. Racks are shelves with invertible translations $b \mapsto a \op b$; many of their aspects, including cohomological, are better understood…
Cofunctors are a kind of map between categories which lift morphisms along an object assignment. In this paper, we introduce cofunctors between categories enriched in a distributive monoidal category. We define a double category of enriched…
It is shown that the one-lens system in para-axial optics can serve as an optical computer for contraction of Wigner's little groups and an analogue computer which transforms analytically computations on a spherical surface to those on a…
This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving…
We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of `regular fibrations' is constructed, as well as a 2-category of `regular proarrow equipments', and it is shown that the two…
Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to…
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…
The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There…
We introduce a category-theoreticabstraction of a syntax with auxiliary functions, called an admissiblemonad morphism. Relying on an abstract form of structural recursion,we then design generic tools to construct admissible monad…
We show that the double category $\mathbb{C}\mathbf{at}^\#$ of comonoids in the category of polynomial functors (previously shown by Ahman-Uustalu and Garner to be equivalent to the double category of categories, cofunctors, and…
We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an abelian monoidal category. Under some further conditions we show that such a cotensor coalgebra exists and satisfies a meaningful universal…
In previous work, we introduced the notion of functional strategies: first-class generic functions that can traverse terms of any type while mixing uniform and type-specific behaviour. Functional strategies transpose the notion of term…
Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E…
Categories can be identified -- up to isomorphism -- with polynomial comonads on Set. The left Kan extension of a functor along itself is always a comonad -- called the density comonad -- so it defines a category when its carrier is…
Bimorphic lenses are a simplification of polymorphic lenses that (like polymorphic lenses) have a type defined by 4 parameters, but which are defined in a monomorphic type system (i.e. an ordinary category with finite products). We show…
Given a 2-category $\twocat{K}$ admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category $\twocat{L}$ with a 2-monad S on it such that: (1)S has the adjoint-pseudo-algebra property.…