Higher Category Theory (Lecture 5)

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∈X C * (BG x ; 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 * (BG x ; 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 complexes of abelian groups. We will always employ homological conventions when discussing chain complexes (so that differential on a chain complex lowers degree). If V * is a chain complex, then its homology H * (V *) is given by H n (V *) = {x ∈ V n : dx = 0}/{x ∈ V n : (∃y ∈ V n−1)[x = dy]}. Any map of chain complexes α : V * → W * induces a map H * (V *) → H * (W *). We say that α is a quasi-isomorphism if it induces an isomorphism on homology. For many purposes, it is convenient to treat quasi-isomorphisms as if they are isomorphisms (emphasizing the idea that a chain complex is just a vessel for carrying information about its homology). One can make this idea explicit using Verdier's theory of derived categories. The derived category of abelian groups can be described as the category D obtained from Chain by formally inverting all quasi-isomorphisms. The theory of derived categories is a very useful tool in homological algebra, but has a number of limitations. Many of these stem from the fact that D is not very well-behaved from a categorical point of view. The category D does not generally have limits or colimits, even of very simple types. For example, a morphism f : X → Y in D generally does not have a cokernel in D. However, there is a substitute: every morphism f in D fits into a " distinguished triangle " X f → Y → Cn(f) → X[1]. Here we …