Related papers: When is a container a comonad?
In this work, we interpret the representations of multi-object scenes in vision encoders through the lens of structured representations. Structured representations allow modeling of individual objects distinctly and their flexible use based…
We introduce the notion of an ordered face structure. The ordered face structures to many-to-one computads are like positive face structures to positive-to-one computads. This allow us to give an explicit combinatorial description of…
We study monoidal profunctors as a tool to reason and structure pure functional programs both from a categorical perspective and as a Haskell implementation. From the categorical point of view we approach them as monoids in a certain…
In functional programming, datatypes a la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof…
We describe a new logical data model, called the concept-oriented model (COM). It uses mathematical functions as first-class constructs for data representation and data processing as opposed to using exclusively sets in conventional…
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…
Delta lenses are a kind of morphism between categories which are used to model bidirectional transformations between systems. Classical state-based lenses, also known as very well-behaved lenses, are both algebras for a monad and coalgebras…
In monadic programming, datatypes are presented as free algebras, generated by data values, and by the algebraic operations and equations capturing some computational effects. These algebras are free in the sense that they satisfy just the…
Many vision-related tasks benefit from reasoning over multiple modalities to leverage complementary views of data in an attempt to learn robust embedding spaces. Most deep learning-based methods rely on a late fusion technique whereby…
Design patterns are distilled from many real systems to catalog common programming practice. However, some object-oriented design patterns are distorted or overly complicated because of the lack of supporting programming language constructs…
We introduce dicodensity monads: a generalisation of pointwise codensity monads generated by functors to monads generated by mixed-variant bifunctors. Our construction is based on the notion of strong dinaturality (also known as Barr…
Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces…
Function is defined as the ensemble of tasks that enable the product to complete the designed purpose. Functional tools, such as functional modeling, offer decision guidance in the early phase of product design, where explicit design…
Various 3D semantic attributes such as segmentation masks, geometric features, keypoints, and materials can be encoded as per-point probe functions on 3D geometries. Given a collection of related 3D shapes, we consider how to jointly…
Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication…
Deep data types are those that are constructed from other data types, including, possibly, themselves. In this case, they are said to be truly nested. Deep induction is an extension of structural induction that traverses all of the…
Databases have been studied category-theoretically for decades. The database schema -- whose purpose is to arrange high-level conceptual entities -- is generally modeled as a category or sketch. The data itself, often called an instance, is…
Monads govern computational side-effects in programming semantics. They can be combined in a ''bottom-up'' way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we equip…
In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$…
Standard methods of using categorical variables as predictors either endow them with an ordinal structure or assume they have no structure at all. However, categorical variables often possess structure that is more complicated than a linear…