Defn - omega-monoids

Construction

Definition

We define an $\infty$-category, an $\omega$-monoid is an object in the $\infty$-category

$$\omega Monoid\simeq lim(CAlg(0Cat)\stackrel{{\omega }_0}{\leftarrow }CAlg(1Cat)\stackrel{{\omega }_1}{\leftarrow }CAlg(2Cat)\stackrel{{\omega }_0}{\leftarrow }\cdots )$$

where the notation is explained in setting - presentable infinity categories.

Repeating this in the linear/stable setting, we’ll call the resulting objects $\omega$-algebras: $\omega Alg\simeq lim(CAlg(0St)\stackrel{{\omega }_0}{\leftarrow }CAlg(1St)\stackrel{{\omega }_1}{\leftarrow }CAlg(2St)\stackrel{{\omega }_0}{\leftarrow }\cdots )$

These are symmetric monoidal, and coproducts coincide with pushouts.

Remark These are “symmetric monoidal categorification towers”.

Remark An object $\mathcal{E}:\ast \to \omega Alg$ is data

• ${\mathcal{E}}_n\in CAlg(nCat)$ for each each $n\in \mathbb{N}\cup \{0\}$
• equivalences ${\mathcal{E}}_n\stackrel{\sim }{\to }{\omega }_n({\mathcal{E}}_{n+1})$

Remark this is an instance of what we’ll call a “soul spectrum” (or,… “spectre & spectres?“) - a spectrum object in $(\infty ,\infty )$-categories, at a certain $\infty$-connective limit

Brauer groups for omega-monoids constr - module omega-monoids

Example ()

Actions:

• Consider traces in this context
• What’s the relevant GOODWILLIE CALCULUS in this setting? (e.g., 1-excision should be relaxed to a lax-version). What thing “lax-stabilizes” to $\omega Alg$? What “lax-semiaddivity” property is enjoyed by $\omega Alg$?
  • The fact that coproducts coincide tensor product in commutative algebras is an instance of this stability.
• Remark: this might be a good axiomatic characterization for that limit!