|
Abstract: It is well known that the classifying space BG of a group G classifies principal homogeneous G-spaces, that is, spaces that are "locally like G." This fact goes hand in hand with BG being a connected space whose only non trivial homotopy group is its fundamental group, which is equal to G itself. A space with this property is called a (connected) homotopy 1-type. G itself should then be considered as a 0-type.
One can ask whether similar ideas apply to homotopy n-types, that is, spaces whose non-trivial homotopy groups occur in the interval 1,...,n. It turns out one has to work with things that are more general than groups. They are non-connected homotopy (n-1)-types which carry a (weak) group law. To describe them concretely, one must resort to extensions of the concept of space. There are various alternatives, all of them leading to equivalent theories. Typically, using stacks is the one that has the most geometric flavor. I will give an introduction to this state of affairs for n=2,3. In particular I will try to describe group laws on categories (and stacks) and describe their morphisms. |