symmetric monoidal (∞,1)-category of spectra
The notion of augmented -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 be an E-∞ ring and an A-∞ algebra over .
An augmentation of is an -A-∞ algebra homomorphism
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 , being an E-∞ algebra over the sphere spectrum , is a homomorphism
to the sphere spectrum, regarded as an E-∞ ring.
Forming augmentation ideals constitutes an equivalence of (∞,1)-categories
of -augmented -rings and nonunital E-∞ rings (Lurie, prop. 5.2.3.15).
A bipermutative category induces (as discussed there) an E-∞ ring . If is equipped with a bi-monoidal functor then this induces an augmentation of over , the Eilenberg-MacLane spectrum of the integers.
See for instance (Arone-Lesh)
For -algebras in characteristic 0 (in chain complexes) augmentation appears for instance as def. 2.3.2.2 on p. 81 in
augmentation of 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 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.