recollection - mode theory

Definition (CSY) A Mode is an idempotent presentably symmetric monoidal $\infty$-category. That is, the subcategory $CAl{g}^{Idem}(P{r}^L)\subset CAlg(P{r}^L)$ spanned by idempotent $E_{\infty }$ algebras.

The notion of an idempotent algebra makes sense in any ambient symmetric monoidal $\infty$-category $C^{\otimes }$. The fundamental feature of this notion is that when $A\in CAlg$ is idempotent, the functor $U:Mo{d}_A(C)\to C$ is fully faithful.

is an equivalence.

We can detect idempotence of $R\in CAlg(V)$ by the following simple condition: $Spec(R)$ is $(-1)$-truncated.

This also means, mapping spaces in subcategory of idempotent algebras are $(-1)$-truncated: i.e. this subcategory is a poset.