nLab k-tuply associative n-groupoid

Idea

A version of the k-tuply monoidal n-groupoid but for $A_n$ spaces.

Definition

Homotopy-theoretic

Invoking the homotopy hypothesis, we define a $k$-tuply associative $n$-groupoid to be an $A_k$-$n$-truncated type: a topological space which is a homotopy n-type with a multiplication that is associative up to higher homotopies involving up to nn variables.

Periodic table

There is a periodic table of $k$-tuply monoidal $n$-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	pointed set	pointed groupoid	pointed 2-groupoid	$\cdots$	pointed $\infty$-groupoid $A_1$-space
$2$	trivial	unital magma	magmoidal groupoid with unit	magmoidal 2-groupoid with unit?	$\cdots$	H-space/A2-space
$3$	“	monoid	H-monoidal groupoid	H-monoidal 2-groupoid?	$\cdots$	H-monoid?/A3-space
$4$	“	“	monoidal groupoid	A4 2-groupoid?	$\cdots$	A4-space
$5$	“	“	“	monoidal 2-groupoid?	$\cdots$	A5-space
$\vdots$	“	“	“	“	$\ddots$	$\vdots$
$\infty$	trivial	monoid	monoidal groupoid	monoidal 2-groupoid?	$\cdots$	$A_\infty$-space

See also

• k-tuply monoidal n-groupoid