# Categories for the Working Mathematician

Abstract: I. 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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### 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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### 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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