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 $i \gt 1$, or as the unique (weak) homotopy type of such a space, or as its fundamental $\infty$-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 $\mathcal{B}G$, which is a $K(G,1)$. In fact, many versions of this construction (such as the geometric realization of the simplicial nerve $nerve(G)$; 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 $(\infty,1)$-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 $n\gt 1$, see the pages homotopy hypothesis, fundamental infinity-groupoid, cat-n-group, classifying space, crossed complex, and probably others.

Related concepts

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid