nLab augmented A-infinity algebra

Idea

The notion of augmented $A_\infty$-algebra is the analogue in higher algebra of the notion of augmented algebra in ordinary algebra: an A-∞ algebra euipped with a homomorphism to the base E-∞ ring (which might be a plain commutative ring).

Definition

Let $R$ be an E-∞ ring and $A$ an A-∞ algebra over $R$.

Definition

An augmentation of $A$ is an $R$-A-∞ algebra homomorphism

$$\epsilon \colon A \to R \,.$$

Remark

In as far as one considers A-∞ algebras are presented by simplicial objects or similar, there might also be a (less intrinsic) notion of augmentation as in augmented simplicial sets. This is not what the above defines.

Fully generally, a definition of augmentation of ∞-algebras over an (∞,1)-operad is in (Lurie, def. 5.2.3.14).

Examples

Example

An augmentation of an E-∞ ring $R$, being an E-∞ algebra over the sphere spectrum $\mathbb{S}$, is a homomorphism

$$\epsilon \colon R \to \mathbb{S}$$

to the sphere spectrum, regarded as an E-∞ ring.

Forming augmentation ideals constitutes an equivalence of (∞,1)-categories

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

of $\mathbb{S}$-augmented $E_\infty$-rings and nonunital E-∞ rings (Lurie, prop. 5.2.3.15).

Example

A bipermutative category $\mathcal{C}$ induces (as discussed there) an E-∞ ring $\vert \mathcal{C}\vert$. If $\mathcal{C}$ is equipped with a bi-monoidal functor $\mathcal{C} \to \mathcal{Z}$ then this induces an augmentation of $\vert \mathcal{C}\vert$ over $H \mathbb{Z}$, the Eilenberg-MacLane spectrum of the integers.

See for instance (Arone-Lesh)

Related concepts

• augmented algebra, augmentation ideal

• augmentation, augmented simplicial set

• augmented ∞-group

References

For $A_\infty$-algebras in characteristic 0 (in chain complexes) augmentation appears for instance as def. 2.3.2.2 on p. 81 in

• Kenji Lefèvre-Hasegawa, Sur les A-infini catégories (arXiv:math/0310337)

augmentation of $\mathbb{F}_p$ E-∞ algebras is considered in definition 7.1 of

• Michael Mandell, $E_\infty$-Algebras and $p$-adic homotopy theory (pdf)

The following articles discuss (just) augmented ∞-groups.

Augmentation (of ∞-groups of units of E-∞ rings) over the sphere spectrum appears in

• Steffen Sagave, Spectra of units for periodic ring spectra (arXiv:1111.6731)

Augmentation over the Eilenberg-MacLane spectrum $H\mathbb{Z}$ appears in

• Gregory Arone, Kathryn Lesh, Augmented $\Gamma$-spaces, the stable rank filtration, and a $b u$-analogue of the Whitehead conjecture (pdf)

See also

• Benoit Fresse, The Bar Complex of an E-infinity Algebra, Adv. Math. 223 (2010), pages 2049-2096 (arXiv:math/0601085)

and

• Stefan Schwede, section 7.8 of Stable Homotopy of Algebraic Theories, 2001 (pdf)

with comments on the relation to nonunital algebras.

Fully general discussion in higher algebra is in

• Jacob Lurie, section 5.2.3 in Higher Algebra