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de Rham's theorem states the integral of forms on chains induce an isomorphism, while the exact sequence proof not give the isomorphism explicitly. I have see some tedious explanation the two canonical isomorphisms coincide. Are there some general principle guarantee the coincidence, similar with universal $\delta$-functor or acyclic model in topology?

MiGang
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  • See Theorem 5.36 in "Foundations of Differentiable Manifolds and Lie Groups" by Frank Warner (or Chapter 5 of that book more generally; it uses soft sheaves and the crutch of paracompact Hausdorff spaces to set up the abelian sheaf theory without reference to injective resolutions, but by acyclicity considerations from the latter point of view one can see that the abstract "$\delta$-functorial" constructions made therein coincide with the ones which would be made via injective resolutions). – nfdc23 Dec 29 '16 at 19:28
  • @nfdc23 Thanks for your tips. I know Warner's book. In some sense I need a simple general principle to avoid the bothering checking. – MiGang Jan 06 '17 at 15:11
  • The general principle you want is Grothendieck's theorem on the universality of erasable (sometimes called "effaceable") $\delta$-functors. This ensures that one only has to check in degree 0 to prove that two isomorphisms between suitable $\delta$-functors coincide (this is discussed in Lang's Algebra, for example). Warner does prove the integration isomorphism is an instance of a general $\delta$-functor isomorphism that he builds, so by the erasability the comparison you seek can be checked in degree 0. – nfdc23 Jan 06 '17 at 18:09
  • @nfdc23 The problem is integration only acting on constant sheaf (\bb R), while the universal (\delta)-functor principle sitting in the Abelian categories. – MiGang Jan 19 '17 at 04:59
  • Warner's book gives a direct proof that the integration map with that constant sheaf coincides with the general isomorphism for acyclic resolutions. That is exactly what is shown in the result 5.36 of his book that I mentioned in my first comment. The "general principle" I refer to ensures that the abstract cohomological formalism set up using soft resolutions in Warner's book (on nice enough topological spaces) coincides with the one based on injective resolutions as for abelian categories quite generally. So my two comments really do address your question. – nfdc23 Jan 19 '17 at 22:20
  • @nfdc23 Thank you for your patiently tips. Now I got it. – MiGang Feb 16 '17 at 06:25

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