Tag Archives: category theory

Commutative Algebra 50

Posted on May 8, 2020 by limsup

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 →

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Commutative Algebra 49

Posted on May 7, 2020 by limsup

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 →

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Commutative Algebra 48

Posted on May 6, 2020 by limsup

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 →

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Commutative Algebra 20

Posted on April 7, 2020 by limsup

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 right-composing with f. Proof Let be a morphism in . We need … Continue reading →

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Commutative Algebra 19

Posted on April 6, 2020 by limsup

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 →

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Commutative Algebra 18

Posted on April 5, 2020 by limsup

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 →

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