nLab k-tuply groupal n-groupoid




A kk-tuply groupal nn-groupoid is an nn-groupoid in which objects can be multiplied in kk invertible ways, all of which interchange with each other up to isomorphism. By the Eckmann-Hilton argument, this implies that these kk ways all end up being equivalent, but that the single resulting operation is more and more commutative as kk increases. The stabilization hypothesis states that by the time we reach k=n+2k = n + 2, the multiplication has become “maximally commutative.”

There is as yet no general category-theoretic definition of kk-tuply groupal nn-groupoid, but the delooping hypothesis says that a kk-tuply groupal nn-groupoid can be interpreted as a special kind of (n+k)(n+k)-groupoid, and if we wish we can take this hypothesis as a definition. A kk-tuply groupal nn-groupoid can also be thought of as a kk-tuply monoidal (n,1)(n,-1)-category. An alternate approach is to invoke instead the homotopy hypothesis to identify nn-groupoids with homotopy n-types, in which case we can apply classical homotopy theory.



Invoking the delooping hypothesis, we define a kk-tuply groupal nn-groupoid to be a pointed (n+k)(n+k)-groupoid such that any two parallel jj-morphisms are equivalent, for j<kj \lt k. One usually relabels the jj-morphisms as (jk)(j-k)-morphisms. You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of (n+k)(n+k)-groupoid.

The given point serves as an equivalence between (1)(-1)-morphisms (for now, see (n,r)(n,r)-category for these), so there is nothing to say if k0k \leq 0 except that the category is pointed. Thus we may as well assume that k0k \geq 0. Also, according to the stabilisation hypothesis, every kk-tuply groupal nn-groupoid for k>n+2k \gt n + 2 may be reinterpreted as an (n+2)(n+2)-tuply groupal nn-groupoid. Unlike the restriction k0k \geq 0, this one is not trivial.


Invoking the homotopy hypothesis, we define a k-tuply monoidal n-groupoid to be an E kE_k-nn-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 kk-tuply groupal nn-groupoids can then be identified with the grouplike? E kE_k-spaces?.

It is a classical theorem of homotopy theory that grouplike E kE_k-spaces are the same as kk-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).

Special cases

A 00-tuply groupal nn-groupoid is simply a pointed nn-groupoid, that is an nn-groupoid equipped with a chosen object (or a space equipped with a chosen basepoint). A 11-tuply groupal nn-groupoid may be called simply a groupal nn-groupoid, or an (n+1)(n+1)-group; topologically these can be identified with grouplike monoids that are nn-types.

A stably groupal nn-groupoid, or symmetric groupal nn-groupoid, is an (n+2)(n+2)-tuply groupal nn-groupoid. This is also called a symmetric (n+1)(n+1)-group, with the numbering off as before. Although the general definition above won't give it, there is a notion of stably groupal \infty-groupoid, basically an \infty-groupoid that can be made kk-tuply groupal for any value of kk in a consistent way. Topologically, these are called grouplike E E_\infty-spaces? and can be identified with infinite loop spaces.

The periodic table

There is a periodic table of kk-tuply groupal nn-groupoids:

k\nk \,\backslash\, n1-1001122\cdots\infty
00trivialpointed setpointed groupoidpointed 2-groupoid\cdotspointed \infty -groupoid
11trivialgroup2-group3-group\cdots \infty -group
22abelian groupbraided 2-groupbraided 3-group\cdotsbraided \infty -group
33symmetric 2-groupsylleptic 3-group\cdotsgroupal E 3 E_3 -space
44symmetric 3-group\cdotsgroupal E 4 E_4 -space
\infty\cdotsgroupal E E_\infty -space
/ abelian \infty -group

See also


The homotopy theory of kk-tuply groupal nn-groupoids is discussed in

Last revised on June 25, 2022 at 19:07:11. See the history of this page for a list of all contributions to it.