nLab nonunital Ek-algebra

Idea

The nonunital variant of Ek-algebra, hence the generalization of the concept of nonunital ring from algebra to higher algebra.

Properties

Relation to augmented $\mathbb{E}_k$-algebra

Nonunital $\mathbb{E}_k$-algebras are equivalent to unital $\mathbb{E}_k$-algebras sliced over the free $\mathbb{E}_k$-algebra on a single generator.

In particular nonunital E-∞ algebras are equivalent to unital $E_\infty$-algebras sliced over the sphere spectrum:

$$E_\infty Ring^{nu} \simeq E_\infty Ring_{/\mathbb{S}} \,,$$

where the map from right to left is given by $\mathbb{E}_k$-unitalization and the converse by forming augmentation ideals.

(Lurie, prop. 5.4.4.10)

References

• Jacob Lurie, section 5.4.4 of Higher Algebra