augmented A-infinity algebra



The notion of augmented A 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).


Let RR be an E-∞ ring and AA an A-∞ algebra over RR.


An augmentation of AA is an RR-A-∞ algebra homomorphism

ϵ:AR. \epsilon \colon A \to R \,.

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.



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

ϵ:R𝕊 \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 Ring /𝕊E Ring nu E_\infty Ring_{/\mathbb{S}} \stackrel{\simeq}{\longrightarrow} E_\infty Ring^{nu}

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


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

See for instance (Arone-Lesh)


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

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

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 HH\mathbb{Z} appears in

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

See also


with comments on the relation to nonunital algebras.

Fully general discussion in higher algebra is in

Last revised on August 22, 2014 at 06:45:31. See the history of this page for a list of all contributions to it.