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Two two categories of algebras and corresponding categories of coalgebras and their relationship with measuring coalgebras and comodules.

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The universal measuring coalgebras has been used for over thirty years to provide a context for linear maps between algebras which are not algebra homomorphisms but still in some sense preserve the bilinear structure of the algebra, such as derivations. The two categories F˙ and F.of algebras introduced here offer an alternative context for such structure preserving maps. The universal measuring coalgebra is recovered as a limit of the category of morphisms F.(A,B). Furthermore, the cofinite dual of the the corresponding limit for the category F˙(A,B) is the universal measuring coalgebra P(A,B). This facilitates calculation of P(A,B). I will describe applications to number fields and Dirac operators.

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