Related papers: Categorical Comprehensions and Recursion
The aim of these notes is to introduce the intuition motivating the notion of a "complicial set", a simplicial set with certain marked "thin" simplices that witness a composition relation between the simplices on their boundary. By varying…
This paper proposes a formal cognitive framework for problem solving based on category theory. We introduce cognitive categories, which are categories with exactly one morphism between any two objects. Objects in these categories are…
We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…
This dissertation builds a compositional cyber-physical systems theory to develop concrete semantics relating the above diverse views necessary for safety and security assurance. In this sense, composition can take two forms. The first is…
Simple optics are defined using actions of monoidal categories. Compound optics arise, for instance, as natural transformations between polynomial functors. Since a monoidal category is a special case of a bicategory, we formulate complex…
Constructive type theory combines logic and programming in one language. This is useful both for reasoning about programs written in type theory, as well as for reasoning about other programming languages inside type theory. It is…
In a coherent category, the posets of subobjects have very strong properties. We emphasize the validity of these properties, in general categories, for well-behaved classes of subobjects. As an example of application, we investigate the…
Cognition does not only depend on bottom-up sensor feature abstraction, but also relies on contextual information being passed top-down. Context is higher level information that helps to predict belief states at lower levels. The main…
For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional…
Abstracting from a low level to a more explanatory high level of description, and ideally while preserving causal structure, is fundamental to scientific practice, to causal inference problems, and to robust, efficient and interpretable AI.…
Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its…
Many classification problems require decisions among a large number of competing classes. These tasks, however, are not handled well by general purpose learning methods and are usually addressed in an ad-hoc fashion. We suggest a general…
Categories can be identified -- up to isomorphism -- with polynomial comonads on Set. The left Kan extension of a functor along itself is always a comonad -- called the density comonad -- so it defines a category when its carrier is…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
Systematic compositionality is the ability to recombine meaningful units with regular and predictable outcomes, and it's seen as key to humans' capacity for generalization in language. Recent work has studied systematic compositionality in…
As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is…
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a…
We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.
We consider a reinforcement learning framework where agents have to navigate from start states to goal states. We prove convergence of a cycle-detection learning algorithm on a class of tasks that we call reducible. Reducible tasks have an…