nLab
homotopy 1-type

Idea

A homotopy 11-type is a space where we consider its properties with regard to the fundamental groups π 1\pi_1 of its components.

Definition

A continuous map XYX \to Y is a homotopy 11-equivalence if it induces isomorphisms on π 0\pi_0 and π 1\pi_1 at each basepoint. Two spaces share the same homotopy 11-type if they are linked by a zig-zag chain of homotopy 11-equivalences.

For any space XX, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space YY so that the inclusion of XX into YY is a homotopy 11-equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy 11-type. Accordingly, a homotopy 11-type may alternatively be defined as a space with trivial π i\pi_i for i>1i \gt 1, or as the unique (weak) homotopy type of such a space, or as its fundamental \infty-groupoid (which will be a 11-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 π 1\pi_1. A connected homotopy 1-type with π 1=G\pi_1 = G is an Eilenberg-Mac Lane space and is often written K(G,1)K(G,1). A general homotopy 1-type can then be written as a disjoint union of such K(G,1)K(G,1)s, and is completely determined by its fundamental groupoid.

In the other direction, for any (discrete) group GG one can construct its classifying space G\mathcal{B}G, which is a K(G,1)K(G,1). In fact, many versions of this construction (such as the geometric realization of the simplicial nerve nerve(G)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)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 (,1)(\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 nn-types with n>1n\gt 1, see the pages homotopy hypothesis, fundamental infinity-groupoid, cat-n-group, classifying space, crossed complex, and probably others.

Revised on November 4, 2009 18:05:09 by Mike Shulman (128.135.239.87)