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Book

Categories for the Working Mathematician

01 Jan 1971-

AbstractI. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.

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Journal ArticleDOI
Abstract: In the first of two papers on MAGMA , a new system for computational algebra, we present the MAGMA language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets.

6,779 citations


Journal ArticleDOI
Abstract: A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 source gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

3,055 citations


Posted Content
TL;DR: The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance.
Abstract: UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology The result is a practical scalable algorithm that applies to real world data The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning

3,034 citations


Book
16 Mar 2009
Abstract: An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author's attempt to make it lovable. This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and ?; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences. This book will be of interest to practitioners in the field of pure and applied mathematics.

1,869 citations


Cites background from "Categories for the Working Mathemat..."

  • ...For a proof of this result, and its generalization to TorR n (A, B) for arbitrary rings R, see Mac Lane, Homology, pp. 150–159 and Mac Lane, “Slide and torsion products for modules,” Rendiconti del Sem. Mat. 15 (1955), 281–309....

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  • ...For a more complete discussion, see Mac Lane, Categories for the Working Mathematician, pp. 21–24, Douady– Douady, Algèbre et Théories Galoisiennes, pp. 24–25, and Herrlich–Strecker, Category Theory, Chapter II and the Appendix....

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  • ...123 Joseph J. Rotman Department of Mathematics University of Illinois at Urbana-Champaign Urbana IL 61801 USA rotman@math.uiuc.edu Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczynski, Case Western Reserve University ISBN: 978-0-387-24527-0 e-ISBN: 978-0-387-68324-9 DOI 10.1007/978-0-387-68324-9 Library of Congress Control Number: 2008936123 Mathematics Subject Classification (2000): 18-01 c© Springer Science+Business Media, LLC 2009 All rights reserved....

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  • ...The Adjoint Functor Theorem says that a functor G on an arbitrary category has a left adjoint [that is, there exists a functor F so that (F, G) is an adjoint pair] if and only if G preserves inverse limits and G satisfies a “solution set condition” [Mac Lane, Categories for the Working Mathematician, pp. 116–127 and 230]. One consequence is a proof of the existence of free objects when a forgetful functor has a left adjoint; see M. Barr, “The existence of free groups,” Amer. Math. Monthly, 79 (1972), 364–367....

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  • ...9The term functor was coined by the philosopher R. Carnap, and S. Mac Lane thought it was the appropriate term in this context....

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Journal ArticleDOI
TL;DR: This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources.
Abstract: An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology It is also clear that the nature of the data we are obtaining is significantly different For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it The data obtained is also often much noisier than in the past and has more missing information (missing data) This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data Geometry and topology are very natural tools to apply in this direction, since geometry can be regarded as the study of distance functions, and what one often works with are distance functions on large finite sets of data The mathematical formalism which has been developed for incorporating geometric and topological techniques deals with point clouds, ie finite sets of points equipped with a distance function It then adapts tools from the various branches of geometry to the study of point clouds The point clouds are intended to be thought of as finite samples taken from a geometric object, perhaps with noise Here are some of the key points which come up when applying these geometric methods to data analysis • Qualitative information is needed: One important goal of data analysis is to allow the user to obtain knowledge about the data, ie to understand how it is organized on a large scale For example, if we imagine that we are looking at a data set constructed somehow from diabetes patients, it would be important to develop the understanding that there are two types of the disease, namely the juvenile and adult onset forms Once that is established, one of course wants to develop quantitative methods for distinguishing them, but the first insight about the distinct forms of the disease is key

1,816 citations


Cites background or methods from "Categories for the Working Mathemat..."

  • ...We note that the correspondence X → π0(X) can actually be viewed as a functor (see [44]) from the category of topological spaces to the category of sets, in the sense that a continuous map f : X → Y induces a map of sets π0(f) : π0(X)→ π0(Y ), satisfying certain obvious conditions on composite maps and identity maps....

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  • ...We refer the reader to [44] for a treatment of categories and functors....

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  • ...Again, refer to [44] for material on categories, functors, and natural transformations....

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