Author: Eiko

Tags: higher topos theory

Time: 2024-12-02 17:32:28 - 2024-12-02 17:32:36 (UTC)

Motivation

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 $K(G,n)$

$${\pi }_i(K(G,n))=\{\begin{array}{ll}G& i=n\\ 0& i\ne n\end{array}$$

$n=1$ the Classifying Space

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 $\bar{X}=EG{\times }_{BG}X$ is a $G$-torsor over $X$, it is a space with free $G$ action and $\bar{X}/G\cong X$.

Adventure Map

• Chapter 1 presents the basics of $\mathrm{\infty }$-categories. This chapter serves as user’s guide, filling definitions and explanations that classical category theory extends to $\mathrm{\infty }$-categories. Some proofs are delayed to later chapters.

• Chapter 2 studies families of $\mathrm{\infty }$-categories parametrized by a $\mathrm{\infty }$-category, what condition is needed to make the fibers $_{D} of $F:\mathcal{C}\to \mathcal{D}$ well-behaved.