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Categories and Modules
by Takahiro Kato - viXra.org , 2015
Modules and morphisms among them subsume categories and functors and provide more general framework to explore the theory of structures. In this book we generalize the basic notions and results of category theory using this framework of modules.
Combinatorics and Algebra of Tensor Calculus
by Sen Hu, Xuexing Lu, Yu Ye - arXiv , 2015
In this paper, we reveal the combinatorial nature of tensor calculus for strict tensor categories and show that there exists a monad which is described by the coarse-graining of graphs and characterizes the algebraic nature of tensor calculus.
Category Theory for Scientists
by David I. Spivak - arXiv , 2013
We attempt to show that category theory can be applied throughout the sciences as a framework for modeling phenomena and communicating results. In order to target the scientific audience, this book is example-based rather than proof-based.
Categories and Homological Algebra
by Pierre Schapira - UPMC , 2011
These notes introduce the language of categories and present the basic notions of homological algebra, first from an elementary point of view, next with a more sophisticated approach, with the introduction of triangulated and derived categories.
Category Theory for Computing Science
by Michael Barr, Charles Wells - Prentice Hall , 1998
This book is a textbook in basic category theory, written specifically to be read by researchers and students in computing science. We expound the constructions basic to category theory in the context of applications to computing science.
Banach Modules and Functors on Categories of Banach Spaces
by J. Cigler, V. Losert, P.W. Michor - Marcel Dekker Inc , 1979
This book is the final outgrowth of a sequence of seminars about functors on categories of Banach spaces (held 1971 - 1975) and several doctoral dissertations. It has been written for readers with a general background in functional analysis.
Functors and Categories of Banach Spaces
by Peter W. Michor - Springer , 1978
The aim of this book is to develop the theory of Banach operator ideals and metric tensor products along categorical lines: these two classes of mathematical objects are endofunctors on the category Ban of all Banach spaces in a natural way.
Category Theory and Functional Programming
by Mikael Vejdemo-Johansson - University of St. Andrews , 2012
An introduction to category theory that ties into Haskell and functional programming as a source of applications. Topics: definition of categories, special objects and morphisms, functors, natural transformation, (co-)limits and special cases, etc.
Higher Topos Theory
by Jacob Lurie - Princeton University Press , 2009
Jacob Lurie presents the foundations of higher category theory, using the language of weak Kan complexes, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language.
by Jacob Lurie - Harvard University , 2011
Contents: Stable infinite-Categories; infinite-Operads; Algebras and Modules over infinte-Operads; Associative Algebras and Their Modules; Little Cubes and Factorizable Sheaves; Algebraic Structures on infinite-Categories; and more.
Introduction to Categories and Categorical Logic
by Samson Abramsky, Nikos Tzevelekos - arXiv , 2011
These notes provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions.
Category Theory Lecture Notes
by Daniele Turi - University of Edinburgh , 2001
These notes were written for a course in category theory. The course was designed to be self-contained, drawing most of the examples from category theory itself. It was intended for post-graduate students in theoretical computer science.
An Introduction to Category Theory in Four Easy Movements
by A. Schalk, H. Simmons - Manchester University , 2005
Notes for a course offered as part of the MSc. in Mathematical Logic. From the table of contents: Development and exercises; Functors and natural transformations; Limits and colimits, a universal solution; Cartesian closed categories.
Category Theory Lecture Notes
by Michael Barr, Charles Wells , 1999
Categories originally arose in mathematics out of the need of a formalism to describe the passage from one type of mathematical structure to another. These notes form a short summary of some major topics in category theory.
- Wikibooks , 2010
This book is an introduction to category theory, written for those who have some understanding of one or more branches of abstract mathematics, such as group theory, analysis or topology. It contains examples drawn from various branches of math.
Basic Category Theory
by Jaap van Oosten - University of Utrecht , 2007
Contents: Categories and Functors; Natural transformations; (Co)cones and (Co)limits; A little piece of categorical logic; Adjunctions; Monads and Algebras; Cartesian closed categories and the lambda-calculus; Recursive Domain Equations.
Abelian Categories: an Introduction to the Theory of Functors
by Peter Freyd - Harper and Row , 1964
From the table of contents: Fundamentals (Contravariant functors and dual categories); Fundamentals of Abelian categories; Special functors and subcategories; Metatheorems; Functor categories; Injective envelopes; Embedding theorems.
Model Categories and Simplicial Methods
by Paul Goerss, Kristen Schemmerhorn - Northwestern University , 2004
There are many ways to present model categories, each with a different point of view. Here we would like to treat model categories as a way to build and control resolutions. We are going to emphasize the analog of projective resolutions.
Notes on Categories and Groupoids
by P. J. Higgins - Van Nostrand Reinhold , 1971
A self-contained account of the elementary theory of groupoids and some of its uses in group theory and topology. Category theory appears as a secondary topic whenever it is relevant to the main issue, and its treatment is by no means systematic.
Seminar on Triples and Categorical Homology Theory
by B. Eckmann - Springer , 1969
This volume concentrates a) on the concept of 'triple' or standard construction with special reference to the associated 'algebras', and b) on homology theories in general categories, based upon triples and simplicial methods.
Higher Operads, Higher Categories
by Tom Leinster - arXiv , 2003
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from topology, quantum algebra, mathematical physics, logic, and computer science.
Higher-Dimensional Categories: an illustrated guide book
by Eugenia Cheng, Aaron Lauda - University of Sheffield , 2004
This work gives an explanatory introduction to various definitions of higher-dimensional category. The emphasis is on ideas rather than formalities; the aim is to shed light on the formalities by emphasizing the intuitions that lead there.
by Marc Levine - American Mathematical Society , 1998
This book combines foundational constructions in the theory of motives and results relating motivic cohomology to more explicit constructions. Prerequisite for understanding the work is a basic background in algebraic geometry.
A Gentle Introduction to Category Theory: the calculational approach
by Maarten M. Fokkinga , 1994
These notes present the important notions from category theory. The intention is to provide a fairly good skill in manipulating with those concepts formally. This text introduces category theory in the calculational style of the proofs.
Computational Category Theory
by D.E. Rydeheard, R.M. Burstall , 2001
The book is a bridge-building exercise between computer programming and category theory. Basic constructions of category theory are expressed as computer programs. It is a first attempt at connecting the abstract mathematics with concrete programs.
Categories, Types, and Structures
by Andrea Asperti, Giuseppe Longo - MIT Press , 1991
Here is an introduction to category theory for the working computer scientist. It is a self-contained introduction to general category theory and the mathematical structures that constitute the theoretical background.
Abstract and Concrete Categories: The Joy of Cats
by Jiri Adamek, Horst Herrlich, George Strecker - John Wiley & Sons , 1990
A modern introduction to the theory of structures via the language of category theory, the emphasis is on concrete categories. The first five chapters present the basic theory, while the last two contain more recent research results.
Basic Concepts of Enriched Category Theory
by Max Kelly - Cambridge University Press , 2005
The book presents a selfcontained account of basic category theory, assuming as prior knowledge only the most elementary categorical concepts. It is designed to supply a connected account of the theory, or at least of a substantial part of it.
Toposes, Triples and Theories
by Michael Barr, Charles Wells - Springer-Verlag , 2005
Introduction to toposes, triples and theories and the connections between them. The book starts with an introduction to category theory, then introduces each of the three topics of the title. Exercises provide examples or develop the theory further.