Cohomology groups \(H^\bullet(X;G)\) of singular cochains in coefficients \(G\) have been extremely useful, because they
Have good formal properties
Are computable
But the usual definition by singular \(G\)-valued cochains is not satisfactory. Can we understand the cohomology groups in a more intrinsic way?
It turnsout that \(H^\bullet(X;G)\) is a representable functor, there exists Eilenberg-MacLane spaces \(K(G,n)\) and a universal class \(H^n(K(G,n);G)\) such that pullback of \(\eta\) determines a bijection
\[ [X,K(G,n)] \to H^n(X;G) \]
and the space \(K(G,n)\) is characterized by that property. It can also be characterized by the homotopy groups of \(K(G,n)\)
\[ \pi_i(K(G,n)) = \begin{cases} G & i = n \\ 0 & i \neq n \end{cases} \]
When \(n=1\) we use \(BG=K(G,1)\) the classifying space of \(G\), the universal cover \(EG\) of \(BG\) is the classifying bundle of \(G\) which is contractible and has free \(G\) action by deck transformations.
\[\pi: EG \to BG\]
For every \(f\in \mathrm{Hom}(X,BG)\), the product \(\overline{X} = EG\times_{BG} X\) is a \(G\)-torsor over \(X\), it is a space with free \(G\) action and \(\overline{X}/G\cong X\).
Chapter 1 presents the basics of \(\infty\)-categories. This chapter serves as user’s guide, filling definitions and explanations that classical category theory extends to \(\infty\)-categories. Some proofs are delayed to later chapters.
Chapter 2 studies families of \(\infty\)-categories parametrized by a \(\infty\)-category, what condition is needed to make the fibers $_{D} of \(F:\mathcal{C}\to \mathcal{D}\) well-behaved.