Contents

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 $(n+k)$-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.

## Definitions

### Category-theoretic

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.

### 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 $(n+1)$-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.

## 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-groupoid ...pointed ∞-groupoid
$1$trivialgroup2-group3-group...∞-group
$2$\"abelian groupbraided 2-groupbraided 3-group...braided ∞-group
$3$\"\"symmetric 2-groupsylleptic 3-group...groupal E3-algebra
$4$\"\"\"symmetric 3-group...groupal E4-algebra
\"\"\"\"

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