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Tag Archives: category theory
Commutative Algebra 50
Adjoint Functors Adjoint functors are a general construct often used for describing universal properties (among other things). Take two categories and . Definition. Covariant functors and are said to be adjoint if we have isomorphisms which are natural in A and … Continue reading
Posted in Advanced Algebra
Tagged adjoint functors, category theory, colimits, hom functor, leftexact, limits, rightexact, tensor products, universal properties
2 Comments
Commutative Algebra 49
Morphism of Diagrams Throughout this article denotes a category and J is an index category. Definition Given diagrams , a morphism is a natural transformation . Thus we have the category of all diagrams in of type J, which we … Continue reading
Posted in Advanced Algebra
Tagged category theory, colimits, coproducts, limits, products, pullbacks, pushouts, universal properties
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Commutative Algebra 48
Introduction For the next few articles we are back to discussing category theory to develop even more concepts. First we will look at limits and colimits, which greatly generalize the concept of products and coproducts and cover loads of interesting … Continue reading
Posted in Advanced Algebra
Tagged category theory, colimits, coproducts, fibres, functors, pullbacks, pushouts
4 Comments
Commutative Algebra 20
Yoneda Lemma For an object , define the covariant functor Proposition 1. Any morphism in gives us a natural transformation In summary, the natural transformation is obtained by rightcomposing with f. Proof Let be a morphism in . We need … Continue reading
Posted in Advanced Algebra
Tagged category theory, coproducts, functors, hom functor, natural transformations, products, yoneda lemma
4 Comments
Commutative Algebra 19
Natural Transformations “I didn’t invent categories to study functors; I invented them to study natural transformations.” – Saunders Mac Lane, one of the founders of category theory A natural transformation is, loosely speaking, a homomorphism between functors. Its definition may … Continue reading
Commutative Algebra 18
Basics of Category Theory As we proceed, we should cover some rudimentary category theory or many of the subsequent constructions would seem unmotivated. The essence of category is in studying algebraic objects and the homomorphisms between them. By now we have … Continue reading
Posted in Advanced Algebra
Tagged algebra, category theory, contravariant, coslice categories, covariant, functors, morphisms, rings
2 Comments