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homotopy 2-type

Idea

A homotopy $2$-type is a space where we consider its properties only up to the $2$nd homotopy group ${\pi }_2$.

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 $i>2$, or as the unique (weak) homotopy type of such a space, or as its fundamental $\mathrm{\infty }$-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. So such a $C$ consists of a morphism

$$\delta :C_2\to C_1$$\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 $(\mathrm{uv}{)}^a=u^{\mathrm{av}}{}^a$, $u^1=u$, $(u^a{)}^b=u^{\mathrm{ab}}$ when these are defined. Further the two basic crossed module rules hold:

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

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

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 ${\mathrm{sk}}^{2}C$ by adding trivial elements in dimensions higher than 2. Hence there is a simplicial nerve $N^{\Delta }C$ which in dimension $n$ is

$$\mathrm{Crs}(\Pi ({\Delta }_{*}^n),{\mathrm{sk}}^{2}C).$$Crs(\Pi (\Delta^n_*), sk^2 C).

The geometric realisation of this is the classifying space $\mathrm{BC}$. Its first and second homotopy groups at $x\in C_0$ are the cokernel and kernel of $\delta :C_2(x)\to C_1(x,x)$. It 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

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

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

Here the cotruncation ${\mathrm{cotr}}^{n}D$ of a general crossed complex $D$ agree with $D$ up to dimension $(n-1)$, is $\mathrm{Cok}{\delta }_{n+1}$ 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 ${\mathrm{Crs}}^2$ of such crossed modules of groupoids is equivalent to that of strict 2-groupoids. Further, ${\mathrm{Crs}}^2$ is monoidal closed:

$${\mathrm{Crs}}^2(C\otimes D,E)\cong {\mathrm{Crs}}^2(C,{\mathrm{CRS}}^2(D,E))$$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 ${\mathrm{Crs}}^2(I\otimes D,E)$ or as elements of ${\mathrm{CRS}}^2(D,E{)}_1$.

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 $ℬC$.

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

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