# Download A Theory of Differentiation in Locally Convex Spaces / by S. Yamamuro PDF By S. Yamamuro

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Theorem. : C → CI op by ( M )i = M for all i and ( , lim) is an adjoint pair. ←− In particular, if C is abelian, then lim is left exact. 4. Let G : D → C be a functor and assume that inverse limits exist in D and C. For any preordered set I, the functor G induces a functor op op G : C I → DI with G(A)i = G(Ai ) and G(τ )i = G(τi ) for each inverse system A and morphism τ : A → B. We say that G preserves inverse op limits if for each preordered set I, lim G ∼ G lim as functors from DI to ←− = ←− C.

M such that → GB n−1 G ··· G ··· ← ). We write ∼ 1 ← 2 → ··· → m G B1 GC   GB 1 G0 1 GC G0 if there exist n-extensions ← . The relation ∼ is an equivalence relation on the class of n-extensions of C by A. We denote by [ ] the class of and by YextnA (C, A) the collection of all equivalence classes of n-extensions of C by A. We assume that YextnA (C, A) is a set (which is the case if A is a module category). If n ≥ 2, the Baer sum of [ ] and [ ] is the class of the n-extension + : ¯n → Bn−1 ⊕ B ¯ 0→A→B n−1 → · · · → B2 ⊕ B2 → B1 → C → 0, ¯n is the pushout of A → Bn , A → B and B ¯1 is the pullback of where B n B1 → C, B1 → C.

Let S be the pullback of g and g , S = {(b, b ) | g(b) = g (b )}, let D = {(f (a), −f (a)) | a ∈ A} ⊆ P, ¯ = S/D. and put B The Baer sum of [ ] and [ ] is the class of the extension + : f¯ g¯ ¯ → C → 0, 0→A→B where f¯(a) = (f (a), 0) and g¯((b, b )) = g (b ). An extension is split if it is equivalent to the extension ι π 0 → A → A ⊕ C → C → 0, where ι and π are the natural maps. Equivalently, the extension (as shown above) is split if there exists h : C → B such that gh = 1C or if there exists j : B → A such that jf = 1A .