# dlicata335/hott-agda

stuff

 @@ -42,17 +42,18 @@ \section{Introduction} One basic question in algebraic topology is calculating the \emph{homotopy groups} of a space. Given a space $X$ with a -distinguished point $x_0$, the \emph{fundamental group of $X$ at the point $x_0$} (denoted -$\pi_1(X,x_0)$) is the group of loops at $x_0$ up to homotopy, with composition as the group operation. -This fundamental group is the first in a sequence of \emph{homotopy - groups}, which provide higher-dimensional information about a space: -the homotopy groups $\pi_n(X,x_0)$ count'' the $n$-dimensional loops in -that space up to homotopy. $\pi_2(X,x_0)$ is the group of homotopies between +distinguished point $x_0$, the \emph{fundamental group of $X$ at the + point $x_0$} (denoted $\pi_1(X,x_0)$) is the group of loops at $x_0$ +up to homotopy, with composition as the group operation. This +fundamental group is the first in a sequence of \emph{homotopy groups}, +which provide higher-dimensional information about a space: the homotopy +groups $\pi_n(X,x_0)$ count'' the $n$-dimensional loops in that space +up to homotopy. $\pi_2(X,x_0)$ is the group of homotopies between $\dsd{id}_{{x_0}}$ and itself, $\pi_3(X,x_0)$ is the group of homotopies between $\dsd{id}_{\dsd{id}_{{x_0}}}$ and itself, and so on. -\emph{Calculating a homotopy group $\pi_n(X,x_0)$} is to construct a group isomorphism -between $\pi_n(X,x_0)$ and some explicit description of a group, -such as \Z\/ or $\Z_k$ (\Z\/ mod $k$). +\emph{Calculating a homotopy group $\pi_n(X,x_0)$} is to construct a +group isomorphism between $\pi_n(X,x_0)$ and some explicit description +of a group, such as \Z\/ or $\Z_k$ (\Z\/ mod $k$). The homotopy groups of a space can be difficult to calculate. This is true even for spaces as simple as the $n$-dimensional spheres (the