237 research outputs found
Note on star-autonomous comonads
We develop an alternative approach to star-autonomous comonads via linearly
distributive categories. It is shown that in the autonomous case the notions of
star-autonomous comonad and Hopf comonad coincide.Comment: 9 page
When is a container a comonad?
Abbott, Altenkirch, Ghani and others have taught us that many parameterized
datatypes (set functors) can be usefully analyzed via container representations
in terms of a set of shapes and a set of positions in each shape. This paper
builds on the observation that datatypes often carry additional structure that
containers alone do not account for. We introduce directed containers to
capture the common situation where every position in a data-structure
determines another data-structure, informally, the sub-data-structure rooted by
that position. Some natural examples are non-empty lists and node-labelled
trees, and data-structures with a designated position (zippers). While
containers denote set functors via a fully-faithful functor, directed
containers interpret fully-faithfully into comonads. But more is true: every
comonad whose underlying functor is a container is represented by a directed
container. In fact, directed containers are the same as containers that are
comonads. We also describe some constructions of directed containers. We have
formalized our development in the dependently typed programming language Agda
A category of quantum categories
Quantum categories were introduced in [4] as generalizations of both
bi(co)algebroids and small categories. We clarify details of that work. In
particular, we show explicitly how the monadic definition of a quantum category
unpacks to a set of axioms close to the definitions of a bialgebroid in the
Hopf algebraic literature. We define notions of functor and natural
transformation for quantum categories.Comment: Revised and expanded. A lot of diagrams. 40 page
What is a categorical model of arrows?
We investigate what the correct categorical formulation of Hughes’ Arrows should be. It has long been folklore that Arrows, a functional programming construct, and Freyd categories, a categorical notion due to Power, Robinson and Thielecke, are somehow equivalent. In this paper, we show that the situation is more subtle. By considering Arrows wholly within the base category we derive two alternative formulations of Freyd category that are equivalent to Arrows—enriched Freyd categories and indexed Freyd categories. By imposing a further condition, we characterise those indexed Freyd categories that are isomorphic to Freyd categories. The key differentiating point is the number of inputs available to a computation and the structure available on them, where structured input is modelled using comonads
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