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Elements of ?-Category Theory
This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.
Elements of ∞-Category Theory
The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.
Simplicial Methods for Higher Categories
This monograph presents a new model of mathematical structures called weak n-categories. These structures find their motivation in a wide range of fields, from algebraic topology to mathematical physics, algebraic geometry and mathematical logic. While strict n-categories are easily defined in terms associative and unital composition operations they are of limited use in applications, which often call for weakened variants of these laws. The author proposes a new approach to this weakening, whose generality arises not from a weakening of such laws but from the very geometric structure of its cells; a geometry dubbed weak globularity. The new model, called weakly globular n-fold categories, i...
Diagrammatic Morphisms and Applications
The technique of diagrammatic morphisms is an important ingredient in comprehending and visualizing certain types of categories with structure. It was widely used in this capacity in many areas of algebra, low-dimensional topology and physics. It was also applied to problems in classical and quantum information processing and logic. This volume contains articles based on talks at the Special Session, ``Diagrammatic Morphisms in Algebra, Category Theory, and Topology'', at the AMS Sectional Meeting in San Francisco. The articles describe recent achievements in several aspects of diagrammatic morphisms and their applications. Some of them contain detailed expositions on various diagrammatic techniques. The introductory article by D. Yetter is a thorough account of the subject in a historical perspective.
Towards Higher Categories
This IMA Volume in Mathematics and its Applications TOWARDS HIGHER CATEGORIES contains expository and research papers based on a highly successful IMA Summer Program on n-Categories: Foundations and Applications. We are grateful to all the participants for making this occasion a very productive and stimulating one. We would like to thank John C. Baez (Department of Mathematics, University of California Riverside) and J. Peter May (Department of Ma- ematics, University of Chicago) for their superb role as summer program organizers and editors of this volume. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Fadil Santosa, Director of ...
Moderate Deviations for the Range of Planar Random Walks
Given a symmetric random walk in ${\mathbb Z}^2$ with finite second moments, let $R_n$ be the range of the random walk up to time $n$. The authors study moderate deviations for $R_n -{\mathbb E}R_n$ and ${\mathbb E}R_n -R_n$. They also derive the corresponding laws of the iterated logarithm.
Category Theory in Context
Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.
Uniqueness and Stability in Determining a Rigid Inclusion in an Elastic Body
The authors consider the inverse problem of determining a rigid inclusion inside an isotropic elastic body $\Omega$, from a single measurement of traction and displacement taken on the boundary of $\Omega$. For this severely ill-posed problem they prove uniqueness and a conditional stability estimate of log-log type.
Philosophy of Information
Information is a recognized fundamental notion across the sciences and humanities, which is crucial to understanding physical computation, communication, and human cognition. The Philosophy of Information brings together the most important perspectives on information. It includes major technical approaches, while also setting out the historical backgrounds of information as well as its contemporary role in many academic fields. Also, special unifying topics are high-lighted that play across many fields, while we also aim at identifying relevant themes for philosophical reflection. There is no established area yet of Philosophy of Information, and this Handbook can help shape one, making sure...
The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions
The author defines the correlation of holes on the triangular lattice under periodic boundary conditions and studies its asymptotics as the distances between the holes grow to infinity. He proves that the joint correlation of an arbitrary collection of triangular holes of even side-lengths (in lattice spacing units) satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of right-pointing and left-pointing unit triangles in each hole. The author details this parallel by indicating tha...
