Related papers: Paranatural Category Theory
We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed…
Deep learning, despite its remarkable achievements, is still a young field. Like the early stages of many scientific disciplines, it is marked by the discovery of new phenomena, ad-hoc design decisions, and the lack of a uniform and…
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs,…
This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax…
Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a…
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external…
Given a pair of pseudo double categories $\mathbb A$ and $\mathbb B$, the lax functors from $\mathbb A$ to $\mathbb B$, along with their transformations, modules, and multimodulations, assemble into a virtual double category…
We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured…
We state a Yoneda-type lemma which leads to various functor categories being compact closed.
Dinatural transformations, which generalise the ubiquitous natural transformations to the case where the domain and codomain functors are of mixed variance, fail to compose in general; this has been known since they were discovered by Dubuc…
Based on the monoid classifier, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and…
Based on Gandy's principles for models of computation we give category-theoretic axioms describing locally deterministic updates to finite objects. Rather than fixing a particular category of states, we describe what properties such a…
Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…
In this paper we introduce the notion of (pointed) prenormal category, modelled after regular categories, but with the key notions of coequaliser and kernel pair replaced by those of cokernel and kernel. This framework provides a natural…
We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo…
We identify morphisms of strong profunctors as a categorification of quantum supermaps. These black-box generalisations of diagrams-with-holes are hence placed within the broader field of profunctor optics, as morphisms in the category of…
A paraconsistent type theory (an extension of a fragment of intuitionistic type theory by adding opposite types) is here extended by adding co-function types. It is shown that, in the extended paraconsistent type system, the opposite type…
A dagger category is a category equipped with a functorial way of reversing morphisms, i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories with additional structure have been studied under different names e.g.…
Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to…
We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…