# nLab k-tuply monoidal n-groupoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

categorification

## Examples

#### Category theory

monoid theory in algebra:

# Contents

## Definition

### Homotopy-theoretic

Invoking the homotopy hypothesis, we define a $k$-tuply monoidal $n$-groupoid to be an $E_k$-$n$-truncated 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 periodic table

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

$k$↓\$n$→$-1$$0$$1$$2$...$\infty$
$0$trivialpointed setpointed groupoidpointed 2-groupoid ...pointed ∞-groupoid
$1$trivialmonoidmonoidal groupoidmonoidal 2-groupoid?...A-∞-space/E1-algebra
$2$\"commutative monoidbraided monoidal groupoidbraided monoidal 2-groupoid?...E2-algebra
$3$\"\"symmetric monoidal groupoidsylleptic monoidal 2-groupoid?...E3-algebra
$4$\"\"\"symmetric monoidal 2-groupoid?...E4-algebra
\"\"\"\"
$\infty$trivialcommutative monoidsymmetric monoidal groupoidsymmetric monoidal 2-groupoid? ...E-∞-algebra