homotopy 2-type


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

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Paths and cylinders

Homotopy groups

Basic facts




A homotopy 22-type is a view of a space where we consider its properties only up to the 22nd homotopy group π 2\pi_2. To make this precise, we look at maps that ‘see’ invariants in dimensions 0,1, and 2. These are the 2-equivalences:


A continuous map XYX \to Y is a homotopy 22-equivalence if it induces isomorphisms on π i\pi_i for i=0,1,2i = 0, 1, 2 at each basepoint. Two spaces share the same homotopy 22-type if they are linked by a zig-zag chain of homotopy 22-equivalences.

For any ‘nice’ 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 22-equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy 22-type. Accordingly, a homotopy 22-type may alternatively be defined as a space with trivial π i\pi_i for i>2i \gt 2, or as the unique (weak) homotopy type of such a space, or as its fundamental \infty-groupoid (which will be a 22-groupoid).

See the general discussion in homotopy n-type.


Homotopy 22-types can be classified by various different types of algebraic data.

Homotopy 22-types as crossed modules

Homotopy 22-types can be classified up to weak homotopy type by crossed modules of groupoids. These are the 22-truncated versions of crossed complexes. Such a CC consists of a morphism

δ:C 2C 1\delta: C_2 \to C_1

of groupoids with object set C 0C_0 such that C 2C_2 is totally disconnected, i.e. is a family of groups C 2(x),xC 0C_2(x), x \in C_0. Further the groupoid C 1C_1 operates on this family of groups so that (using right operations) if a:xya: x \to y in C 1C_1 and uC 2(x)u \in C_2(x) then u aC(y)u^a \in C(y); and the usual rules for an operation are satisfied, namely (uv) a=u av a(uv)^a=u^a v^a, u 1=uu^1=u, (u a) b=u ab(u^a)^b=u^{a b} when these are defined. Further the two basic crossed module rules hold:

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

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

for all aC 1,u,vC 2a \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 2Csk^2 C by adding trivial elements in dimensions higher than 2. Hence there is a simplicial nerve N ΔCN^\Delta C which in dimension nn is

Crs(Π(Δ * n),sk 2C). Crs(\Pi (\Delta^n_*), sk^2 C).

The geometric realisation of this is the classifying space BCBC. Its first and second homotopy groups at xC 0x \in C_0 are the cokernel and kernel of δ:C 2(x)C 1(x,x)\delta: C_2(x) \to C_1(x,x). Its components are those of the groupoid C 1C_1. All other homotopy groups are trivial.

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

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

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

Here the cotruncation cotr nDcotr^n D of a general crossed complex DD agree with DD up to dimension (n1)(n-1), is Cokδ n+1Cok \delta_{n+1} in dimension nn, and is trivial in higher dimensions.

It is in this sense that crossed modules of groupoids classify weak homotopy 22-types.

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

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

and with a unit interval object II so that (left) homotopies are determined as morphisms Crs 2(ID,E)Crs^2(I \otimes D,E) or as elements of CRS 2(D,E) 1CRS^2(D,E)_1.

Homotopy 22-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), GG, one can define a simplicial set called the classifying space W¯G\overline{W}G of the simplicial group, GG, for which construction see simplicial group. We thus can start with a crossed module CC form a simplicial group and then take W¯\overline{W} of that to get another model of C\mathcal{B}C.

Homotopy 22-types as 22-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


homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid

Revised on June 6, 2013 15:04:19 by Raeder? (