Related papers: Sketches for arithmetic universes
A sketch is a category equipped with specified collections of cones and cocones. Its models are functors to the category of sets that send the distinguished cones and cocones to limit cones and colimit cocones, respectively. Sketches…
Traditional treatments of formal logic provide: 1. A syntax for formulas. 2. An inference relation between sets of formulas. 3. A rule for assigning meaning to formulas (semantics) that is sound with respect to the inference relation. First…
Reasoning in the 2-category Con of contexts, certain sketches for arithmetic universes (i.e. list arithmetic pretoposes; AUs), is shown to give rise to base-independent results of Grothendieck toposes, provided the base elementary topos has…
Generalizing different variants of "graph conditions and constraints" as well as "universal constraints" and "negative universal constraints" in the Diagram Predicate Framework (DPF), we introduce for arbitrary categories $\mathbf{Cxt}$ and…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
We present a sketch-based CAD modeling system, where users create objects incrementally by sketching the desired shape edits, which our system automatically translates to CAD operations. Our approach is motivated by the close similarities…
The integration of knowledge extracted from different models described by domain experts or from models generated by machine learning algorithms is strongly conditioned by the lack of an appropriated framework to specify and integrate…
Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal…
We introduce and study a purely syntactic notion of lax cones and $(\infty,\infty)$-limits on finite computads in \texttt{CaTT}, a type theory for $(\infty,\infty)$-categories due to Finster and Mimram. Conveniently, finite computads are…
We present the basic ideas of forms (a generalization of Ehresmann's sketches) and their theories and models, more explicitly than in previous expositions. Forms provide the ability to specify mathematical structures and data types in any…
An alternative foundation for 2-categories is explored by studying graph-theoretically a partial operation on 2-cells named juncture, which can replace vertical and horizontal composition. Juncture is a generalized vertical composition of…
We argue that locally Cartesian closed categories form a suitable doctrine for defining dependent type theories, including non-extensional ones. Using the theory of sketches, one may define syntactic categories for type theories in a style…
Data constraints are fundamental for practical data modelling, and a verifiable conformance of a data instance to a safety-critical constraint (satisfaction relation) is a corner-stone of safety assurance. Diagrammatic constraints are…
Recently, sketches have been introduced as a general language for representing the subgoal structure of instances drawn from the same domain. Sketches are collections of rules of the form C -> E over a given set of features where C…
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…
Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free…
We establish the equivalence between models of enhanced $2$-sketches and algebras over monads, including the (co)lax morphisms. More precisely, for any enhanced limit $2$-sketch $\mathbb{T}$ with tight cones, the enhanced $2$-category…
We construct the universal realized limit sketch associated to a given limit sketch. The construction uses factorization systems to organize the classical argument of [2], yielding a streamlined and conceptually unified formulation of the…
Parametric computer-aided design (CAD) is the dominant paradigm in mechanical engineering for physical design. Distinguished by relational geometry, parametric CAD models begin as two-dimensional sketches consisting of geometric primitives…
We describe a sketch interpretation system that detects and classifies clock numerals created by subjects taking the Clock Drawing Test, a clinical tool widely used to screen for cognitive impairments (e.g., dementia). We describe how it…