symmetric monoidal (∞,1)-category of spectra
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).
Let $R$ be an E-∞ ring and $A$ an A-∞ algebra over $R$.
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. 5.2.3.14).
An augmentation of an E-∞ ring $R$, being an E-∞ algebra over the sphere spectrum $\mathbb{S}$, is a homomorphism
to the sphere spectrum, regarded as an E-∞ ring.
Forming augmentation ideals constitutes an equivalence of (∞,1)-categories
of $\mathbb{S}$-augmented $E_\infty$-rings and nonunital E-∞ rings (Lurie, prop. 5.2.3.15).
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)
For $A_\infty$-algebras in characteristic 0 (in chain complexes) augmentation appears for instance as def. 2.3.2.2 on p. 81 in
augmentation of $\mathbb{F}_p$ 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
Augmentation over the Eilenberg-MacLane spectrum $H\mathbb{Z}$ appears in
See also
and
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.