A homotopy -type is a space where we consider its properties with regard to the fundamental groups of its components.
A continuous map is a homotopy -equivalence if it induces isomorphisms on and at each basepoint. Two spaces share the same homotopy -type if they are linked by a zig-zag chain of homotopy -equivalences.
For any space , you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space so that the inclusion of into is a homotopy -equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy -type. Accordingly, a homotopy -type may alternatively be defined as a space with trivial for , or as the unique (weak) homotopy type of such a space, or as its fundamental -groupoid (which will be a -groupoid).
See the general discussion in homotopy n-type.
A connected pointed homotopy 1-type is completely determined, up to (weak) homotopy equivalence, by the one group . A connected homotopy 1-type with is an Eilenberg-Mac Lane space and is often written . A general homotopy 1-type can then be written as a disjoint union of such s, and is completely determined by its fundamental groupoid.
In the other direction, for any (discrete) group one can construct its classifying space , which is a . In fact, many versions of this construction (such as the geometric realization of the simplicial nerve ; 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 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 -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 -types with , see the pages homotopy hypothesis, fundamental infinity-groupoid, cat-n-group, classifying space, crossed complex, and probably others.