nLab
homotopy 1-type

Idea

A homotopy $1$-type is a space where we consider its properties with regard to the fundamental groups ${\pi }_1$ of its components.

Definition

A continuous map $X\to Y$ is a homotopy $1$-equivalence if it induces isomorphisms on ${\pi }_0$ and ${\pi }_1$ at each basepoint. Two spaces share the same homotopy $1$-type if they are linked by a zig-zag chain of homotopy $1$-equivalences.

For any space $X$, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space $Y$ so that the inclusion of $X$ into $Y$ is a homotopy $1$-equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy $1$-type. Accordingly, a homotopy $1$-type may alternatively be defined as a space with trivial ${\pi }_i$ for $i>1$, or as the unique (weak) homotopy type of such a space, or as its fundamental $\mathrm{\infty }$-groupoid (which will be a $1$-groupoid).

See the general discussion in homotopy n-type.

Classification

A connected pointed homotopy 1-type is completely determined, up to (weak) homotopy equivalence, by the one group ${\pi }_1$. A connected homotopy 1-type with ${\pi }_1=G$ is an Eilenberg-Mac Lane space and is often written $K(G,1)$. A general homotopy 1-type can then be written as a disjoint union of such $K(G,1)$s, and is completely determined by its fundamental groupoid.

In the other direction, for any (discrete) group $G$ one can construct its classifying space $ℬG$, which is a $K(G,1)$. In fact, many versions of this construction (such as the geometric realization of the simplicial nerve $\mathrm{nerve}(G)$; see Dold-Kan correspondence) apply just as well to any groupoid. We can obtain any 1-type in this way, since a groupoid is up to homotopy type (of groupoids!) a disjoint union of groups. However this description is not natural in the category of groupoids, and is analogous to choosing a basis for a vector space.

Moreover, every continuous map between $K(G,1)$s is induced by a group homomorphism, every map between 1-types is induced by a functor between groupoids, and every homotopy is induced by a conjugation (aka a natural transformation between groupoids). In fact, one can show that the $(\mathrm{\infty },1)$-category of homotopy 1-types is equivalent to the 2-category Grpd of groupoids, via the above-described correspondence..

There are further aspects to this relationship; for instance, the van Kampen theorem for the fundamental groupoid shows how the algebra of groupoids models the gluing of spaces. The general result for non-connected spaces is possible because groupoids model homotopy 1-types, having structure in dimensions 0 and 1. For the search for algebraic structures that play an analogous role to groupoids for $n$-types with $n>1$, see the pages homotopy hypothesis, fundamental infinity-groupoid, cat-n-group, classifying space, crossed complex, and probably others.