- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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

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

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

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

Invoking the homotopy hypothesis, we define a **k-tuply monoidal n-groupoid** to be an *$E_k$-$n$-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 $k$-tuply groupal $n$-groupoids can then be identified with the grouplike? $E_k$-spaces?.

It is a classical theorem of homotopy theory that grouplike $E_k$-spaces are the same as $k$-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 $0$-tuply groupal $n$-groupoid is simply a pointed $n$-groupoid, that is an $n$-groupoid equipped with a chosen object (or a space equipped with a chosen basepoint). A $1$-tuply groupal $n$-groupoid may be called simply a **groupal $n$-groupoid**, or an **$(n+1)$-group**; topologically these can be identified with grouplike monoids that are $n$-types.

A **stably groupal $n$-groupoid**, or **symmetric groupal $n$-groupoid**, is an $(n+2)$-tuply groupal $n$-groupoid. This is also called a **symmetric $(n+1)$-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 $k$-tuply groupal for any value of $k$ in a consistent way. Topologically, these are called *grouplike $E_\infty$-spaces?* and can be identified with infinite loop spaces.

There is a periodic table of $k$-tuply groupal $n$-groupoids:

$k \,\backslash\, n$ | $-1$ | $0$ | $1$ | $2$ | $\cdots$ | $\infty$ |
---|---|---|---|---|---|---|

$0$ | trivial | pointed set | pointed groupoid | pointed 2-groupoid | $\cdots$ | pointed $\infty$-groupoid |

$1$ | trivial | group | 2-group | 3-group | $\cdots$ | $\infty$-group |

$2$ | “ | abelian group | braided 2-group | braided 3-group | $\cdots$ | braided $\infty$-group |

$3$ | “ | “ | symmetric 2-group | sylleptic 3-group | $\cdots$ | groupal $E_3$-space |

$4$ | “ | “ | “ | symmetric 3-group | $\cdots$ | groupal $E_4$-space |

$\vdots$ | “ | “ | “ | “ | $\ddots$ | $\vdots$ |

$\infty$ | “ | “ | “ | “ | $\cdots$ | groupal $E_\infty$-space / abelian $\infty$-group |

The homotopy theory of $k$-tuply groupal $n$-groupoids is discussed in

- A.R. Garzón, J.G. Miranda,
*Serre homotopy theory in subcategories of simplicial groups*, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

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