Related papers: A Yoneda-Style Embedding for Virtual Equipments

The tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces…

This work presents an exposition of both the internal structure of derived category of an abelian category D*(A) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented…

What is a good vector representation of an object? We believe that it should be generative in 3D, in the sense that it can produce new 3D objects; as well as be predictable from 2D, in the sense that it can be perceived from 2D images. We…

In this work, we introduce GELATO (Geometry-preserving Embeddings via Locked Aligned TOwers), a novel approach to multimodal embedding models. We build on the VLM-style architecture, in which non-text encoders are adapted to produce input…

In this paper, we first introduce a technique that we call "Yoneda representation of flat functors", based on ideas from indexed category theory; then we provide applications of this technique to the theory of classifying toposes.…

Oftentimes, patterns can be represented through different modalities. For example, leaf data can be in the form of images or contours. Handwritten characters can also be either online or offline. To exploit this fact, we propose the use of…

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell embedding theorem states the existence of a ring $R$ and an exact full embedding $\mathcal{A} \rightarrow R$-Mod. This theorem is useful as it allows one to prove general…

Embeddings are ubiquitous in machine learning, appearing in recommender systems, NLP, and many other applications. Researchers and developers often need to explore the properties of a specific embedding, and one way to analyze embeddings is…

In this paper we explore a family of type isomorphisms in System F whose validity corresponds, semantically, to some form of the Yoneda isomorphism from category theory. These isomorphisms hold under theories of equivalence stronger than…

As in many tasks combining vision and language, both modalities play a crucial role in Visual Question Answering (VQA). To properly solve the task, a given model should both understand the content of the proposed image and the nature of the…

A 2-categorical generalisation of elementary topos is provided and some of the properties of the yoneda structure it generates are explored. Examples relevant to the globular approach to higher category theory are discussed. This paper also…

Many object pose estimation algorithms rely on the analysis-by-synthesis framework which requires explicit representations of individual object instances. In this paper we combine a gradient-based fitting procedure with a parametric neural…

In this work, we propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional computational grids with curvilinear edge elements. The approximation by means of rectilinear edges of a domain with curvilinear geometrical…

We design the conforming virtual element method for the numerical approximation of the two dimensional elastodynamics problem. We prove stability and convergence of the semi-discrete approximation and derive optimal error estimates under…

The rapid evolution of technology has transformed business operations and customer interactions worldwide, with personalization emerging as a key opportunity for e-commerce companies to engage customers more effectively. The application of…

In this paper, we analyze the embedding cell method, an algorithm which has been developed for the numerical homogenization of metal-ceramic composite materials. We show the convergence of the iteration scheme of this algorithm and the…

We make use of a higher version of the Yoneda embedding to construct, from a given quasicategory, a tribe, as a subcategory of a well-behaved simplicial model category, that presents the same $(\infty,1)$-category as the former…

Transformer architectures rely on explicit position encodings in order to preserve a notion of word order. In this paper, we argue that existing work does not fully utilize position information. For example, the initial proposal of a…

This work concerns adaptive refinement procedures for meshes of polygonal virtual elements. Specifically, refinement procedures previously proposed by the authors for structured meshes are generalized for the challenging case of arbitrary…

There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and…