By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's answer organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward outcome, and additionally, a similarity category [AJ within the Brauer workforce Br(k) of Azumaya algebras over a commutative ring okay contains all algebras B such that the corresponding different types mod-A and mod-B including k-linear morphisms are identical by way of a k-linear functor. (For fields, Br(k) contains similarity sessions of straightforward primary algebras, and for arbitrary commutative ok, this can be subsumed below the Azumaya [51]1 and Auslander-Goldman [60J Brauer workforce. ) quite a few different situations of a marriage of ring conception and type (albeit a shot­ gun wedding!) are inside the textual content. moreover, in. my try to additional simplify proofs, significantly to cast off the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside of ring idea. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre­ spondence theorem for projective modules (Theorem four. 7) recommended by way of the Morita context. As a derivative, this offers starting place for a slightly whole conception of straightforward Noetherian rings-but extra approximately this within the advent.

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One therefore comes with a general definition of “interpretation” in a category C of a logical system, which generalizes Tarski’s truth definition: this will then be the special case of classical logic and the category Set. In this chapter I treat, for reasons of space, only a fragment of first order logic: regular logic. On this fragment the valid statements of classical and intuitionistic logic coincide. For an interpretation of a term calculus like the λ-calculus, which is of paramount importance in theoretical computer science, the reader is referred to chapter 7.

61 7 Cartesian closed categories and the λ-calculus Many set-theoretical constructions are completely determined (up to isomorphism, as always) by their categorical properties in Set. We are therefore tempted to generalize them to arbitrary categories, by taking the characteristic categorical property as a definition. Of course, this procedure is not really well-defined and it requires sometimes a real insight to pick the ‘right’ categorical generalization. For example, the category of sets has very special properties: • f : X → Y is mono if and only if f g = f h implies g = h for any two maps g, h : 1 → X, where 1 is a terminal object (we say 1 is a generator); • objects X and Y are isomorphic if there exist monos f : X → Y and g : Y → X (the Cantor-Bernstein theorem); f • every mono X → Y is part of a coproduct diagram Xd dd f dd dd Z g /Y And if you believe the axiom of choice, there is its categorical version: • Every epi is split None of these properties is generally valid, and categorical generalizations based on them are usually of limited value.

We have maps r : P → Q and i : Q → P such that ri is the identity on Q and ir = T : P → P . For x ∈ P , y ∈ Q we have x ≤ i(y) ⇔ r(x) ≤ y (check); so r i and the operation T arises from this adjunction. 1 Algebras for a monad Given a monad (T, η, µ) on a category C, we define the category T -Alg of T algebras as follows: • Objects are pairs (X, h) where X is an object of C and h : T (X) → X is an arrow in C such that T 2 (X) T (h) / T (X) µX  T (X) h  /X h ηX / T (X) Xi ii ii and ii ii h idX "  X commute; f • Morphisms: (X, h) → (Y, k) are morphisms X → Y in C for which T (X) T (f ) / T (Y ) h k  X f  /Y commutes.