prop - suspension of natural numbers monoid

(used in Spectral Superlines)

Prop The suspension of $\mathbb{N}\in CAlg(S)$ is the monoid $\mathbb{T}:=S^1$

Proof

By thm - Koszul Duality exchanges invertibility and categorification, we have a general formula for monoids

$\Sigma M\simeq (BM{)}^{ext}$ In particular $\Sigma \mathbb{N}\simeq (B\mathbb{N}{)}^{ext}\simeq B\mathbb{Z}\simeq \mathbb{T}$