Related papers: A Categorical Integration of Quantifiers:A Higher …
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
Contextuality is central to both the foundations of quantum theory and to the novel information processing tasks. Although it was recognized before Bell's nonlocality, despite some recent proposals, it still faces a fundamental problem: how…
We define generalized bialgebras and Hopf algebras and on this basis we introduce quantum categories and quantum groupoids. The quantization of the category of linear (super)spaces is constructed. We establish a criterion for the classical…
We develop a second-order extension of intuitionistic modal logic, allowing quantification over propositions, both syntactically and semantically. A key feature of second-order logic is its capacity to define positive connectives from the…
The bulk of this paper is devoted to the comparison of several models for the theory of (infinity,2)-categories: that is, higher categories in which all k-morphisms are invertible for k > 2 (the case of (infinity,n)-categories is also…
This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently…
We introduce a proof-theoretic approach to showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to prove that Taranovsky's "realizability disjunction" connective does not admit…
Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or…
We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical.…
We propose a type-theoretic framework for describing and proving properties of quantum computations, in particular those presented as quantum circuits. Our proposal is based on an observation that, in the polymorphic type system of Coq,…
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…
This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples…
Mackey functors provide the coefficient systems for equivariant cohomology theories. More generally, enriched presheaf categories provide a classification and organization for many stable model categories of interest. Changing enrichments…
Classical logic is embedded into constructive logic, through a definition of the classical connectives and quantifiers in terms of the constructive ones.
Various concerns suggest looking for internal co-categories in categories with strong logical structure. It turns out that in any coherent category, all co-categories are co-equivalence relations.
General coherence theorems are constructed that yield explicit presentations of categorical and algebraic objects. The categorical structures involved are finitary discrete Lawvere 2-theories, though they are approached within the language…
This paper is the first in a series of two papers, $\mathbf{Z}$-Categories I and $\mathbf{Z}$-Categories II, which develop the notion of $\mathbf{Z}$-category, the natural bi-infinite analog to strict $\omega$-categories, and show that the…
We propose an alternative framework for quantifying coherence. The framework is based on a natural property of coherence, the additivity of coherence for subspace-independent states, which is described by an operation-independent equality…
While behavioural equivalences among systems of the same type, such as Park/Milner bisimilarity of labelled transition systems, are an established notion, a systematic treatment of relationships between systems of different type is…
A careful reexamination of the quantization of systems with first- and second-class constraints from the point of view of coherent-state phase-space path integration reveals several significant distinctions from more conventional…