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B-mod category equivalence theorem (Theorem)
Theorem 0.1   B-mod category equivalence theorem.

Let $\mathcal{A}$ be an Abelian category with arbitrary direct sums (or coproducts). Also, let $P$ in $\mathcal{A}$ be a compact projective generator and set $B = (End_{\mathcal{A}} P)^{op}$. The functor $hom_\mathcal{A}(P,--)$ yields an equivalence of categories between $\mathcal{A}$ and the category $B-mod$.

Proof. The proof proceeds in two steps. At the first step one shows that the functor

$\displaystyle F(X) = hom_{\mathcal{A}}(P,X)$
is fully faithful, and therefore, at the second step one can apply the Abelian category equivalence lemma to yield the sought for equivalence of categories.



"B-mod category equivalence theorem" is owned by bci1.

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See Also: Yoneda lemma, Morita equivalence

Also defines:  coproduct, compact projective generator
Keywords:  B-mod category equivalence, compact projective generator, abelian category with arbitrary direct sums, coproducts

Cross-references: Abelian category equivalence lemma, categories, functor, Abelian category
There is 1 reference to this object.

This is version 10 of B-mod category equivalence theorem, born on 2009-06-15, modified 2009-06-15.
Object id is 804, canonical name is BModCategoryEquivalenceTheorem.
Accessed 801 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
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