Related papers: Directed Containers as Categories
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…
Containers are used to carve out a class of strictly positive data types in terms of shapes and positions. They can be interpreted via a fully-faithful functor into endofunctors on Set. Monadic containers are those containers whose…
Containers represent a wide class of type constructions relevant for functional programming and (co)inductive reasoning. Indexed containers generalize this notion to better fit the scope of dependently typed programming. When interpreting…
The paper establishes an equivalence between directed homotopy categories of (diagrams of) cubical sets and (diagrams of) directed topological spaces. This equivalence both lifts and extends an equivalence between classical homotopy…
A double category is constructed from a `fattened' version of a given category, motivated in part by a context of parallel transport. We also study monoidal structures on the underlying category and on the fattened category.
In this short note, we argue that directed homotopy can be given the structure of generalized modules, over particular monoids. This is part of a general attempt for refoundation of directed topology.
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…
Containers conveniently represent a wide class of inductive data types. Their derivatives compute representations of types of one-hole contexts, useful for implementing tree-traversal algorithms. In the category of containers and cartesian…
Given a category, one may construct slices of it. That is, one builds a new category whose objects are the morphisms from the category with a fixed codomain and morphisms certain commutative triangles. If the category is a groupoid, so that…
We generalize the constructions and results of Chapter 10 in Baldwin's "Categoricity" to coherent accessible categories with concrete directed colimits and concrete monomorphisms. In particular, we prove that if any category of this form is…
We develop and investigate a general theory of representations of second-order functionals, based on a notion of a right comodule for a monad on the category of containers. We show how the notion of comodule representability naturally…
We observe that the process of associating an action to any Schreier extension of monoids with commutative and cancellative kernel is functorial. We show that this functor is a generalisation of the direction functor, used to give a…
This paper provides a mathematical framework for client-server communication that results in a modular and type-safe architecture. It is informed and motivated by the software engineering practice of developing server backends with a…
One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a…
A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to…
This paper introduces an inherently strict presentation of categories with products, coproducts, or symmetric monoidal products that is inspired by file systems and directories. Rather than using nested binary tuples to combine objects or…
Fong developed `decorated cospans' to model various kinds of open systems: that is, systems with inputs and outputs. In this framework, open systems are seen as the morphisms of a category and can be composed as such, allowing larger open…
We introduce semidirect products of skew monoidal categories as a categorification of semidirect products of monoids (or, perhaps more familiarly, of groups). We also discuss how this construction interacts with monoidal, autonomous and…
Coherence in a monoidal category asserts that all morphisms built from structural isomorphisms with a fixed source and target coincide. These structural isomorphisms include, in particular, the associators. Linearly distributive categories…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…