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Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver…
In this paper we introduce the notion of a categorical Mackey functor. This categorical notion allows us to obtain new Mackey functors by passing to Quillen's $K$-theory of the corresponding abelian categories. In the case of an action by…
We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by…
This paper determines the general formula for describing differentials of composite functions in terms of differentials of their factor functions. This generalises the formula commonly attributed to Faa di Bruno to functions in locally…
In this paper, we define the notion of crossed modules of groups with action and investigate related structures. Functions for computing of these structures have been written using the GAP computational discrete algebra programming…
Differential categories were introduced by Blute, Cockett, and Seely as categorical models of differential linear logic and have since lead to abstract formulations of many notions involving differentiation such as the directional…
In Part 1, we describe six projective-type model structures on the category of differential graded modules over a differential graded algebra A over a commutative ring R. When R is a field, the six collapse to three and are well-known, at…
A probabilistic model describes a system in its observational state. In many situations, however, we are interested in the system's response under interventions. The class of structural causal models provides a language that allows us to…
Generalized planning is concerned with the characterization and computation of plans that solve many instances at once. In the standard formulation, a generalized plan is a mapping from feature or observation histories into actions,…
Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalization of the theory of canonical extension to the setting of first order logic. We define a notion of…
Reverse differentiation is an essential operation for automatic differentiation. Cartesian reverse differential categories axiomatize reverse differentiation in a categorical framework, where one of the primary axioms is the reverse chain…
Category theory has been successfully applied in various domains of science, shedding light on universal principles unifying diverse phenomena and thereby enabling knowledge transfer between them. Applications to machine learning have been…
Generalized planning is concerned with the computation of plans that solve not one but multiple instances of a planning domain. Recently, it has been shown that generalized plans can be expressed as mappings of feature values into actions,…
The inferential model (IM) approach, like fiducial and its generalizations, depends on a representation of the data-generating process. Here, a particular variation on the IM construction is considered, one based on generalized…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We…
The cartesian structure possessed by relations, spans, profunctors, and other such morphisms is elegantly expressed by universal properties in double categories. Though cartesian double categories were inspired in part by the older program…
This paper is concerned with the foundations of the Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions by inductive data types. CAC generalizes inductive types equipped with higher-order primitive…
The cut and join operations play important roles in tensor models in general. We introduce a generalization of the cut operation associated with the higher order variations and demonstrate how they generate operators in the Aristotelian…
Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and…