Related papers: A type-theoretic definition of lax $(\infty,\infty…
We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal's…
Given a 2-category $\mathcal{A}$, a $2$-functor $\mathcal{A} \overset {F} {\longrightarrow} \mathcal{C}at$ and a distinguished 1-subcategory $\Sigma \subset \mathcal{A}$ containing all the objects, a $\sigma$-cone for $F$ (with respect to…
We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces…
Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development of homotopy…
Abstract clones serve as an algebraic presentation of the syntax of a simple type theory. From the perspective of universal algebra, they define algebraic theories like those of groups, monoids and rings. This link allows one to study the…
For a diagram of simplicial combinatorial model categories, we show that the associated lax limit, endowed with the projective model structure, is a presentation of the lax limit of the underlying $\infty$-categories. Our approach can also…
Language models are typically applied at the sentence level, without access to the broader document context. We present a neural language model that incorporates document context in the form of a topic model-like architecture, thus…
Contextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to…
This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…
Language models (LMs) have been used in cognitive modeling as well as engineering studies -- they compute information-theoretic complexity metrics that simulate humans' cognitive load during reading. This study highlights a limitation of…
Techniques in which words are represented as vectors have proved useful in many applications in computational linguistics, however there is currently no general semantic formalism for representing meaning in terms of vectors. We present a…
Type theories with higher-order subtyping or singleton types are examples of systems where computation rules for variables are affected by type information in the context. A complication for these systems is that bounds declared in the…
This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…
We first study Clarke's tangent cones at infinity to unbounded subsets of $\mathbb{R}^n.$ We prove that these cones are closed convex and show a characterization of their interiors. We then study subgradients at infinity for extended real…
We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to…
We propose semantic anchoring, a unified account of how large language models turn pretrained capacity into goal-directed behavior: external structure (in-context examples, retrieval, or light tuning) binds the model's latent patterns to…
Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computer-scientific information. That is, one wants to…
We propose a theory for modeling concepts that uses the state-context-property theory (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. This theory enables us to incorporate context…
We introduce a model of simple type theory with potential infinite carrier sets. The functions in this model are automatically continuous, as defined in this paper. This notion of continuity does not rely on topological concepts, including…
We describe several new ways of specifying the behaviour of Lemaitre-Tolman (LT) models, in each case presenting the method for obtaining the LT arbitrary functions from the given data, and the conditions for existence of such solutions. In…