Related papers: Cartesian double theories: A double-categorical fr…
The main result of this paper may be stated as a construction of "almost representations" for the canonical presheaves of object extensions of length n on the C-systems defined by locally cartesian closed universe categories with binary…
Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets…
In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have…
We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category…
Using duality in optimization theory we formulate a dual approach to the S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix. We…
A concept of "evolving categories" is suggested to build a simple, scalable, mathematically consistent framework for representing in uniform way both data and algorithms. A state machine for executing algorithms becomes clear, rich and…
We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of promonoidal category. Since the theory of promonoidal categories has not been developed…
In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the classical notions of Kan extension, Yoneda embedding $\text y_A\colon A \to \hat A$, exact square, total category and…
Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the…
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…
In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category endowed with two reciprocally stable factorization systems such that X \to 1 is in…
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this…
Arbitrarily many pairwise inequivalent modular categories can share the same modular data. We exhibit a family of examples that are module categories over twisted Drinfeld doubles of finite groups, and thus in particular integral modular…
Quantum computation can be formulated through various models, each highlighting distinct structural and resource-theoretic aspects of quantum computational power. This paper develops a unified categorical framework that encompasses these…
We develop the Morita theory of fusion 2-categories. In order to do so, we begin by proving that the relative tensor product of modules over a separable algebra in a fusion 2-category exists. We use this result to construct the Morita…
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 show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories.…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems…
We give an introduction to constructive category theory by answering two guiding computational questions. The first question is: how do we compute the set of all natural transformations between two finitely presented functors like…