nLab homotopy 2-type

Idea

A homotopy $2$-type is a view of a space where we consider its properties only up to the $2$nd homotopy group $\pi_2$. To make this precise, we look at maps that ‘see’ invariants in dimensions 0,1, and 2. These are the 2-equivalences:

Definition

A continuous map $X \to Y$ is a homotopy $2$-equivalence if it induces isomorphisms on $\pi_i$ for $i = 0, 1, 2$ at each basepoint. Two spaces share the same homotopy $2$-type if they are linked by a zig-zag chain of homotopy $2$-equivalences.

For any ‘nice’ 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 $2$-equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy $2$-type. Accordingly, a homotopy $2$-type may alternatively be defined as a space with trivial $\pi_i$ for $i \gt 2$, or as the unique (weak) homotopy type of such a space, or as its fundamental $\infty$-groupoid (which will be a $2$-groupoid).

See the general discussion in homotopy n-type.

Classification

Homotopy $2$-types can be classified by various different types of algebraic data.

Homotopy $2$-types as crossed modules

Homotopy $2$-types can be classified up to weak homotopy type by crossed modules of groupoids. These are the $2$-truncated versions of crossed complexes. Such a $C$ consists of a morphism

$$\delta: C_2 \to C_1$$

of groupoids with object set $C_0$ such that $C_2$ is totally disconnected, i.e. is a family of groups $C_2(x), x \in C_0$. Further the groupoid $C_1$ operates on this family of groups so that (using right operations) if $a: x \to y$ in $C_1$ and $u \in C_2(x)$ then $u^a \in C(y)$; and the usual rules for an operation are satisfied, namely $(uv)^a=u^a v^a$, $u^1=u$, $(u^a)^b=u^{a b}$ when these are defined. Further the two basic crossed module rules hold:

CM1) $\delta(u^a)= a^{-1} (\delta u) a$;

CM2) $v^{-1} u v = u ^{\delta v}$;

for all $a \in C_1, u,v \in C_2$ when the rules make sense.

Such a crossed module may be extended to a crossed complex $sk^2 C$ by adding trivial elements in dimensions higher than 2. Hence there is a simplicial nerve $N^\Delta C$ which in dimension $n$ is

$$Crs(\Pi (\Delta^n_*), sk^2 C).$$

The geometric realisation of this is the classifying space $BC$. Its first and second homotopy groups at $x \in C_0$ are the cokernel and kernel of $\delta: C_2(x) \to C_1(x,x)$. Its components are those of the groupoid $C_1$. All other homotopy groups are trivial.

If $X$ is a CW-complex then there is a bijection of homotopy classes

$$[X,BC] \cong [\Pi X_*, sk^2 C],$$

and hence there is a map $X \to B(cotr^2 \Pi X_*)$ inducing isomorphisms of homotopy groups in dimensions 1 and 2.

Here the cotruncation $cotr^n D$ of a general crossed complex $D$ agree with $D$ up to dimension $(n-1)$, is $Cok \delta_{n+1}$ in dimension $n$, and is trivial in higher dimensions.

It is in this sense that crossed modules of groupoids classify weak homotopy $2$-types.

The category $Crs^2$ of such crossed modules of groupoids is equivalent to that of strict 2-groupoids. Further, $Crs^2$ is monoidal closed:

$$Crs^2(C \otimes D, E) \cong Crs^2(C, CRS^2(D,E))$$

and with a unit interval object $I$ so that (left) homotopies are determined as morphisms $Crs^2(I \otimes D,E)$ or as elements of $CRS^2(D,E)_1$.

Homotopy $2$-types as simplicial group(oid)s

As a crossed module give rise to an internal groupoid in the category of groups (or groupoids), we can take the nerve of that structure and get a simplicial group (or simplicially enriched groupoid). From a simplicial group(oid), $G$, one can define a simplicial set called the classifying space $\overline{W}G$ of the simplicial group, $G$, for which construction see simplicial group. We thus can start with a crossed module $C$ form a simplicial group and then take $\overline{W}$ of that to get another model of $\mathcal{B}C$.

Homotopy $2$-types as $2$-groupoids

With respect to the standard homotopy theory-structure on 2-groupoids (2-truncated infinity-groupoids) these are equivalent to homotopy 2-types. See at homotopy hypothesis for more on this.

Homotopy 2-types as double groupoids

see

• Antonio Martínez Cegarra; Benjamın A. Heredia ; Josué Remedios,

  Double groupoids and homotopy 2-types Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also arXiv:1003.3820.

Related concepts

• fundamental 2-groupoid

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