Tag Archives: yoneda lemma

Commutative Algebra 29

Posted on April 16, 2020 by limsup

Distributivity Finally, tensor product is distributive over arbitrary direct sums. Proposition 1. Given any family of modules , we have: Proof Take the map which takes . Note that this is well-defined: since only finitely many are non-zero, only finitely … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , | 2 Comments

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 →

Posted in Advanced Algebra | Tagged , , , , , , | 4 Comments

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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