Some classical theory
which reduces the homotopy theory of 2-types to a "computable" algebraic theory. Furthermore, a simplicial group with Moore complex of length 1 can be represented by a group H with two endomorphisms s:H-->H and t:H-->H satisfying the axioms
The homotopy groups of a cat1-group H are defined as
number of standard group-theoretic constructions can be viewed
naturally as a cat1-group.
For instance, the following commands begin by constructing the cat1-group of the last example for the group G=SmallGroup(64,134). They then construct the fundamental group of H and then the second homotopy group of H as a pi_1-module. These have orders 8 and 2 respectively.
cohomology H3(pi_1,pi_2) classifies those cat1-groups
with fundamental group equal to pi_1 and second homotopy module equal
to pi_2. The classification is up to a Yoneda equivalence.
The following additional commands show that there are 1024 Yoneda equivalence classes of cat1-groups with homotopy group pi_1 and homotopy module pi_2 equal to that in our example.
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
|A 3-cocycle f : pi_1 × pi_1 × pi_1 -----> pi_2 corresponding to the second cohomology class in H3(pi_1,pi_2) can be produced using the following command,|