Related papers: Internalizing decorated bicategories: The globular…
Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of…
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can…
This is the author's Ph.D. Thesis. It contains results from four years of research into realizability and categorical logic. The main subjects are the axiomatisation of realizable propositions, and a characterization of realizability…
This article surveys recent advances and future challenges in the $2$-representation theory of finitary $2$-categories with a particular emphasis on problems related to classification of various classes of $2$-representations.
Providing a human-understandable explanation of classifiers' decisions has become imperative to generate trust in their use for day-to-day tasks. Although many works have addressed this problem by generating visual explanation maps, they…
We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving…
The familiar construction of categories of fractions, due to Gabriel and Zisman, allows one to invert a class W of arrows in a category in a universal way. Similarly, bicategories of fractions allow one to invert a collection of arrows in a…
We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise…
This paper introduces the concept of gluing in a general category, enabling us to define categories that admit glued-up objects. To achieve this, we introduce the notion of a gluing index category. Subsequently, we provide an entirely…
Graph data structures are fundamental for studying connected entities. With an increase in the number of applications where data is represented as graphs, the problem of graph generation has recently become a hot topic. However, despite its…
Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories. In a poly-bicategory, the existence of enough 2-cells satisfying certain universal properties (representability) induces…
We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent…
Definable subcategories may be extended along a ring homomorphism directly, by using their defining conditions in the new module category, or by tensoring up with the new ring. We investigate what is preserved and reflected by these…
Whilst cooking is a very important human activity, there has been little consideration given to how we can formalize recipes for use in a reasoning framework. We address this need by proposing a graphical formalization that captures the…
A convenient bicategory of topological stacks is constructed which is both complete and Cartesian closed. This bicategory, called the bicategory of compactly generated stacks, is the analogue of classical topological stacks, but for a…
We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are…
A fruitful idea, when providing subdifferential formulae and dual representations for convex risk measures, is to make use of the conjugate duality theory in convex optimization. In this paper we underline the outstanding role played by the…
In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for…
The Bipartite Polarization Problem is an optimization problem where the goal is to find the highest polarized bipartition on a weighted and labelled graph that represents a debate developed through some social network, where nodes represent…