In the previous lecture, we outlined some approaches to describing the cohomology of the classifying space of G-bundles M on a Riemann surface X. For example, we asserted that the cochain complex C ∗ (M; Q) is quasi-isomorphic to a continuous tensor product ⊗x∈XC ∗ (BGx; Q). Here it is vital that we work at the level of cochains, rather than cohomology: there is no corresponding procedure to recover the cohomology ring H ∗ (M; Q) from the graded rings H ∗ (BGx; Q). Consequently, even if our ultimate interest is in understanding the cohomology ring H ∗ (M; Q), it will be helpful to have a good way of thinking about chain-level constructions in homological algebra. Throughout this lecture, let Chain denote the abelian category whose objects are chain complexe
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