A -tuply groupal -groupoid is an -groupoid in which objects can be multiplied in invertible ways, all of which interchange with each other up to isomorphism. By the Eckmann-Hilton argument, this implies that these ways all end up being equivalent, but that the single resulting operation is more and more commutative as increases. The stabilization hypothesis states that by the time we reach , the multiplication has become “maximally commutative.”
There is as yet no general category-theoretic definition of -tuply groupal -groupoid, but the delooping hypothesis says that a -tuply groupal -groupoid can be interpreted as a special kind of -groupoid, and if we wish we can take this hypothesis as a definition. A -tuply groupal -groupoid can also be thought of as a -tuply monoidal -category. An alternate approach is to invoke instead the homotopy hypothesis to identify -groupoids with homotopy n-types, in which case we can apply classical homotopy theory.
Invoking the delooping hypothesis, we define a -tuply groupal -groupoid to be a pointed -groupoid such that any two parallel -morphisms are equivalent, for . One usually relabels the -morphisms as -morphisms. You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of -groupoid.
The given point serves as an equivalence between -morphisms (for now, see -category for these), so there is nothing to say if except that the category is pointed. Thus we may as well assume that . Also, according to the stabilisation hypothesis, every -tuply groupal -groupoid for may be reinterpreted as an -tuply groupal -groupoid. Unlike the restriction , this one is not trivial.
Invoking the homotopy hypothesis, we define a -tuply monoidal -groupoid to be an --type: a topological space which is a homotopy n-type and which is equipped with an action by the little k-cubes operad (or some operad equivalent to it). The -tuply groupal -groupoids can then be identified with the grouplike? -spaces?.
It is a classical theorem of homotopy theory that grouplike -spaces are the same as -fold loop spaces (see J.P. May, The Geometry of Iterated Loop Spaces). This is the topological version of the delooping hypothesis (from which, of course, it takes its name).
A -tuply groupal -groupoid is simply a pointed -groupoid, that is an -groupoid equipped with a chosen object (or a space equipped with a chosen basepoint). A -tuply groupal -groupoid may be called simply a groupal -groupoid, or an -group; topologically these can be identified with grouplike monoids that are -types.
A stably groupal -groupoid, or symmetric groupal -groupoid, is an -tuply groupal -groupoid. This is also called a symmetric -group, with the numbering off as before. Although the general definition above won't give it, there is a notion of stably groupal -groupoid, basically an -groupoid that can be made -tuply groupal for any value of in a consistent way. Topologically, these are called grouplike -spaces? and can be identified with infinite loop spaces.
There is a periodic table of -tuply groupal -groupoids:
|trivial||pointed set||pointed groupoid||pointed 2-groupoid||…||pointed ∞-groupoid|
|"||abelian group||braided 2-group||braided 3-group||…||braided ∞-group|
|"||"||symmetric 2-group||sylleptic 3-group||…||groupal E3-algebra|
|"||"||"||symmetric 3-group||…||groupal E4-algebra|
The homotopy theory of -tuply groupal -groupoids is discussed in