Related papers: Cartesian Differential Storage Categories
Dictionary learning methods can be split into: i) class specific dictionary learning ii) class shared dictionary learning. The difference between the two categories is how to use discriminative information. With the first category, samples…
The Causal Set approach to quantum gravity asserts that spacetime, at its smallest length scale, has a discrete structure. This discrete structure takes the form of a locally finite order relation, where the order, corresponding with the…
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…
We introduce a version of skein categories of surfaces which depends on a tensor ideal in a linear ribbon category, thereby extending the existing theory to the setting of non-semisimple TQFTs. We obtain modified notions of skein algebras…
It is proved that equalities between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equality in the language of free cartesian categories collapses a cartesian category into a preorder. An…
For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for…
We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding…
Connectionist approaches to machine learning, \emph{i.e.} neural networks, are enjoying a considerable vogue right now. However, these methods require large volumes of data and produce models that are uninterpretable to humans. An…
We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use…
The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems…
Optics and lenses are abstract categorical gadgets that model systems with bidirectional data flow. In this paper we observe that the denotational definition of optics - identifying two optics as equivalent by observing their behaviour from…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…
Tannaka duality and its extensions by Lurie, Sch\"appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete…
Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major…
Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found,…
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…
This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every…
The linear classifier is widely used in various image classification tasks. It works by optimizing the distance between a sample and its corresponding class center. However, in real-world data, one class can contain several local clusters,…
This paper seeks to apply categorical logic to the design of artificial intelligent agents that reason symbolically about objects more richly structured than sets. Using Johnstone's sequent calculus of terms- and formulae-in-context, we…