Open AccessBook
Categories for the Working Mathematician
TLDR
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.read more
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Journal ArticleDOI
Representation and character theory in 2-categories
Nora Ganter,Mikhail Kapranov +1 more
TL;DR: Hopkins, Kuhn and Ravenel as mentioned in this paper developed a (2-)categorical generalization of the theory of group representations and characters, and defined an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors.
Proceedings ArticleDOI
Applying quantitative semantics to higher-order quantum computing
TL;DR: In this paper, a denotational semantics for a quantum lambda calculus with recursion and an infinite data type is proposed, using constructions from the quantitative semantics of linear logic.
Book ChapterDOI
Configuration spaces with summable labels
TL;DR: An n-monoid is the appropriate extension of an A ∞-space for the theory of n-fold loop spaces as mentioned in this paper, and it is defined spaces of configurations on n-manifolds with summable labels in partial N-monoids, which cover symmetric products, spaces of rational curves and spaces of labelled subsets.
Book ChapterDOI
The java memory model: operationally, denotationally, axiomatically
TL;DR: A semantics to a small fragment of Java capturing the new memory model (JMM) described in the Language Specification is given by combining operational, denotational and axiomatic techniques in a novel semantic framework.
Journal ArticleDOI
Operated semigroups, Motzkin paths and rooted trees
TL;DR: In this article, the concept of operated semigroups with intuitive and convenient combinatorial descriptions is introduced, and a construction of free Rota-Baxter algebras in terms of Motzkin paths and rooted trees is given.
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