Related papers: What is an equivalence in a higher category?
For a differential graded k-quiver Q we define the free A-infinity-category FQ generated by Q. The main result is that for an arbitrary A-infinity-category A the restriction A-infinity-functor A_\infty(FQ,A) -> A_1(Q,A) is an equivalence,…
We give a model-independent definition of limits for diagrams valued in an $(\infty,n)$-category. We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of…
Sentence similarity is considered the basis of many natural language tasks such as information retrieval, question answering and text summarization. The semantic meaning between compared text fragments is based on the words semantic…
We investigate the concept of strong equivalence within the extended framework of Answer Set Programming with constraints. Two groups of rules are considered strongly equivalent if, informally speaking, they have the same meaning in any…
We devise a condition strictly between the existence of an $n$-ary and an $n{+}1$-ary near-unanimity term. We evaluate exactly the distributivity and modularity levels implied by such a condition.
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…
Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a…
The problem of identifying to which of a given set of classes objects belong is ubiquitous, occurring in many research domains and application areas, including medical diagnosis, financial decision making, online commerce, and national…
Knowing when a classifier's prediction can be trusted is useful in many applications and critical for safely using AI. While the bulk of the effort in machine learning research has been towards improving classifier performance,…
Machine learning systems have been shown to propagate the societal errors of the past. In light of this, a wealth of research focuses on designing solutions that are "fair." Even with this abundance of work, there is no singular definition…
We generalize the notion of stable equivalence of Morita type and define what is called "singular equivalence of Morita type with level". Such an equivalence of induces an equivalence between singular categories. We will also prove that a…
Internal categories feature notions of limit and completeness, as originally proposed in the context of the effective topos. This paper sets out the theory of internal completeness in a general context, spelling out the details of the…
Argumentation is a process of evaluating and comparing a set of arguments. A way to compare them consists in using a ranking-based semantics which rank-order arguments from the most to the least acceptable ones. Recently, a number of such…
The main purpose of the present article is to give some new Hilbert's sum type inequalities, which in special cases yield the classical Hilbert's inequalities. Our results provide some new estimates to these types of inequalities.
The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed…
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories,…
We prove existence of equalizers in certain categories of cocomplete cocategories. This allows us to complete the proof of the fact that A-infinity functor categories arise as internal Hom-objects in the category of differential graded…
The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically…
We define a generalized likelihood function based on uncertainty measures and show that maximizing such a likelihood function for different measures induces different types of classifiers. In the probabilistic framework, we obtain…