group theory

# Contents

## Idea

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 $\left(n+k\right)$-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 $\left(n,-1\right)$-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.

## Definitions

### Category-theoretic

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

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

### Homotopy-theoretic

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).

### Special cases

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 $\left(n+1\right)$-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 $\left(n+2\right)$-tuply groupal $n$-groupoid. This is also called a symmetric $\left(n+1\right)$-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.

## The periodic table

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

$k$↓\$n$$-1$$0$$1$$2$$\infty$
$0$trivialpointed setpointed groupoidpointed 2-groupoidpointed ∞-groupoid
$1$trivialgroup2-group3-group∞-group
$2$"abelian groupbraided 2-groupbraided 3-groupbraided ∞-group
$3$""symmetric 2-groupsylleptic 3-groupgroupal E3-algebra
$4$"""symmetric 3-groupgroupal E4-algebra
""""

## References

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

Revised on October 26, 2012 12:04:36 by Tim Porter (95.147.237.0)