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Morita Theorems for Functor Categories

D. C. Newell

1972
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Transactions of the American Mathematical Society
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We generalize the Morita theorems to certain functor categories using properties of adjoint functors. Let sé and 38 be categories. Recall that sé and 38 are said to be equivalent if there are functors u: sé -> {% and v:3S-^sé and natural isomorphisms of functors •n: I => vu and e: uv => /', where /and /' denote the identity functors on sé and 38 respectively. When jé = JtR and 38 = Jis, where R and S are unitary rings and JtR and Jis denote the categories of right /î-modules and right S-modules
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... respectively, the Morita theorems state that the pairs of functors (v, u),u: sé^ 38 and v : 3S -+sé, which define an equivalence of categories between sé and 38 are in one-to-one correspondence with bimodules RES which are R-S progenerators, the correspondence being given by isomorphisms u^ B E and P£ Homs (E, ) give a one-to-one correspondence between bimodules RES and adjoint pairs (v, u) from ^#s to JiR i.e. functors v: Jfs -> JiR having « as a left adjoint. Since the class of pairs (v, u) which give an equivalence of categories is a subclass of the class of adjoint pairs, one may view the Morita theorems as characterizing the corresponding subclass of bimodules. In [4], we have shown that the class of adjoint pairs between certain functor categories is in one-to-one correspondence with bifunctors in a manner generalizing the above correspondence on module categories. In this paper, we generalize the Morita theorems to characterize those bifunctors which give an equivalence of functor categories. Our exposition and results parallel [2], but our proofs use properties of adjoint functors and results of [4] . §1 summarizes these properties and results and introduces the machinery for the rest of the paper, §2 generalizes the notion of "projectives" and "generators" to bifunctors, §3 generalizes the notion of "Morita context" to bifunctors, while §4 is devoted to our theorem. Our results will be stated in the following setting: Ab will denote the category of abelian groups, all of our categories will be preadditive, and all of our functors will be additive. However, it is clear that our results will hold for ^"-categories, where y is a closed category satisfying certain "completion" conditions.

doi:10.2307/1996184
fatcat:suyrzx2s4zeqbncchunfqpzblq