Book•
Categories for the Working Mathematician
01 Jan 1971-
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
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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01 Jul 1997TL;DR: In this paper, the authors introduce the category of Structured ring and module spectra under $S$ and show how to construct a monadic bar construction on a function spectra.
Abstract: Introduction Prologue: the category of ${\mathbb L}$-spectra Structured ring and module spectra The homotopy theory of $R$-modules The algebraic theory of $R$-modules $R$-ring spectra and the specialization to $MU$ Algebraic $K$-theory of $S$-algebras $R$-algebras and topological model categories Bousfield localizations of $R$-modules and algebras Topological Hochschild homology and cohomology Some basic constructions on spectra Spaces of linear isometries and technical theorems The monadic bar construction Epilogue: The category of ${\mathbb L}$-spectra under $S$ Appendix A. Twisted half-smash products and function spectra Bibliography Index.
817 citations
Cites background from "Categories for the Working Mathemat..."
...We first recall from [42] the definition of a coend, or tensor product of functors....
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TL;DR: In this article, the correctness of appropriate string diagrams for various kinds of monoidal categories with duals has been proved for various classes of classes of subject classes, including algebra, geometry, physics, and astronomy.
778 citations
Book•
01 Jan 1990
TL;DR: This chapter discusses Categories and Functors, Topological Categories, Partial Morphisms, Quasitopoi, and Topological Universes, as well as partial Morphisms in Abstract Categories and Cartesian Closed Categories.
Abstract: Motivation. Foundations. CATEGORIES, FUNCTORS, AND NATURAL TRANSFORMATIONS. Categories and Functors. Subcategories. Concrete Categories and Concrete Functors. Natural Transformations. OBJECTS AND MORPHISMS. Objects and Morphisms in Abstract Categories. Objects and Morphisms in Concrete Categories. Injective Objects and Essential Embeddings. SOURCES AND SINKS. Sources and Sinks. Limits and Colimits. Completeness and Cocompleteness. Functors and Limits. FACTORIZATION STRUCTURES. Factorization Structures for Morphisms. Factorization Structures for Sources. E-Reflective Subcategories. Factorization Structures for Functors. ADJOINTS AND MONADS. Adjoint Functors. Adjoint Situations. Monads. TOPOLOGICAL AND ALGEBRAIC CATEGORIES. Topological Categories. Topological Structure Theorems. Algebraic Categories. Algebraic Structure Theorems. Topologically Algebraic Categories. Topologically Algebraic Structure Theorems. CARTESIAN CLOSEDNESS AND PARTIAL MORPHISMS. Cartesian Closed Categories. Partial Morphisms, Quasitopoi, and Topological Universes. Bibliography. Tables. Table of Categories. Table of Symbols. Index.
765 citations
01 Feb 1992
TL;DR: This paper explores the use monads to structure functional programs and describes the relation between monads and the continuation-passing style in a compiler for Haskell that is written in Haskell.
Abstract: This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and the continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.
738 citations
TL;DR: In this article, a reference guide to various notions of monoidal categories and their associated string diagrams is presented, which is useful not only to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning.
Abstract: This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study Where possible, we provide pointers to more rigorous treatments in the literature Where we include results that have only been proved in special cases, we indicate this in the form of caveats
732 citations
Cites background from "Categories for the Working Mathemat..."
...ssed in Section 5. 8 3.1 (Planar) monoidal categories A monoidal category (also sometimes called tensor category) is a category with an associative unital tensor product. More specifically: Definition ([29, 23]). A monoidal category is a category with the following additional structure: • a new operation A ⊗B on objects and a new object constant I; • a new operation on morphisms: if f : A → C and g : B → D,...
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...rlier draft. 4 2 Categories We only give the most basic definitions of categories, functo rs, and natural transformations. For a gentler introduction, with more details and examples, see e.g. Mac Lane [29]. Definition. A category Cconsists of: • a class |C| of objects, denoted A, B, C, ...; • for each pair of objects A,B, a set homC(A,B) of morphisms, which are denoted f : A → B; • identity morphisms id...
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