Related papers: Categorical Complexity
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
Intrinsic complexity of a relation on a given computable structure is captured by the notion of its degree spectrum - the set of Turing degrees of images of the relation in all computable isomorphic copies of that structure. We investigate…
In this paper we will prove that there exists a covariant functor from the category of schemes to the category of graphs. This functor provides a combination between algebraic varieties and combinatorial graphs so that the invariants…
We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent…
A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…
Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements &…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…
In this article, we show that the now classical protocol complex approach to distributed task solvability of Herlihy et al. can be understood in standard categorical terms. First, protocol complexes are functors, from chromatic (semi-)…
In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…
It is now very known how the subprojectivity of modules provides a fruitful new unified framework of the classical projectivity and flatness. In this paper, we extend this fact to the category of complexes by generalizing and unifying…
We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal…
It is well-known that biological phenomena are emergent. Emergent phenomena are quite interesting and amazing. However, they are difficult to be understood. Due to this difficulty, we propose a theory to describe emergence based on a…
In this dissertation we develop a new formal graphical framework for causal reasoning. Starting with a review of monoidal categories and their associated graphical languages, we then revisit probability theory from a categorical perspective…
We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems…
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes…
We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras,…
We develop a notion of computability and complexity of functions over the reals, which seems to be very natural when one tries to determine just how "difficult" a certain function is. This notion can be viewed as an extension of both BSS…
The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this…
We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform…
Modular categories are important algebraic structures in a variety of subjects in mathematics and physics. We provide an explicit, motivated and elementary definition of a modular category over a field of characteristic 0 as an equivalence…