Related papers: RCF1: Theories of PR Maps and Partial PR Maps
We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…
The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…
Consider a rational map from a projective space to a product of projective spaces, induced by a collection of linear projections. Motivated by the the theory of limit linear series and Abel-Jacobi maps, we study the basic properties of the…
In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.
We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the…
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand "theories of indiscernibles" to be special…
Morphisms between schemes arising from multigraded rings are essential for understanding geometric relationships in algebraic geometry, yet a systematic theory for such maps has been lacking. In this paper, we develop a comprehensive…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…
Specifying a Bayesian prior is notoriously difficult for complex models such as neural networks. Reasoning about parameters is made challenging by the high-dimensionality and over-parameterization of the space. Priors that seem benign and…
We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an…
A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of…
Even if path planning can be solved using standard techniques from dynamic programming and control, the problem can also be approached using probabilistic inference. The algorithms that emerge using the latter framework bear some appealing…
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain…
We complete the foundational architecture of Algebraic Phase Theory by developing a categorical and $2$-categorical framework for algebraic phases. Building on the structural notions introduced in Papers~I-III, we define phase morphisms,…
We define pullback and separated presentations of modules over pullback rings, and, if the ring is a pullback of epimorphisms over a semisimple ring, an algorithm reducing such a presentation of a module to an $R$-diagram. The latter is the…
Interventional causal models describe several joint distributions over some variables used to describe a system, one for each intervention setting. They provide a formal recipe for how to move between the different joint distributions and…
We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…
In this article we introduce generalized projective spaces (Definitions $[2.1, 2.5]$) and prove three main theorems in two different contexts. In the first context we prove, in main Theorem $A$, the surjectivity of the Chinese remainder…
We describe a notion of categorical model for unitless fragments of (multiplicative) linear logic. The basic definition uses promonoidal categories, and we also give an equivalent elementary axiomatisation.