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Tag Archives: coproducts
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 30
Tensor Product of AAlgebras Proposition 1. Let B, C be Aalgebras. Their tensor product has a natural structure of an Aalgebra which satisfies . Proof Fix . The map is Abilinear so it induces an Alinear map Now varying (b, c) gives … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, algebras, coproducts, fibres, tensor product, varieties
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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