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The generalization in (∞,1)-category theory of the notion of group of units in ordinary category theory.
Let be an A-∞ ring spectrum.
For the underlying A-∞ space and the ordinary ring of connected components, write for its group of units.
Then the ∞-group of units
of is the (∞,1)-pullback in
There is slight refinement of the above definition, which essentially adds one 0-th “grading” homotopy group to and thereby makes the -group of units of E-∞ rings be canonically augmented over the sphere spectrum (Sagave 11).
There is a functor
(Sagave 11, def. 3.14)
Here the existence of the map is (Sagave 11, lemma 2.12 + lemma 3.16). The fact that the resulting sequence is a homotopy fiber sequence is (Sagave 11, prop. 4.1).
Using this, there is now a modified delooping of the ordinary -group of units:
Write for the homotopy cofiber of to yield
(Sagave 11, prop. 4.3)
Adjointness to -group -ring
This is (ABGHR 08, theorem 2.1/3.2, remark 3.4).
Also the augmented -group of units functor of def. 2 is a homotopy right adjoint. (Sagave 11, theorem 1.8).
Snaith's theorem asserts that
the K-theory spectrum for complex K-theory is the ∞-group ∞-ring of the circle 2-group localized away from the Bott element :
the periodic complex cobordism spectrum is the ∞-group ∞-ring of the classifying space for stable complex vector bundles (the classifying space for topological K-theory) localized away from the Bott element :
Inclusion of circle -bundles into higher chromatic cohomology
By Snaith’s theorem above there is a canonical map
that sends circle bundles to cocycles in topological K-theory.
At the next level there is a canonical map
that sends circle 2-bundles to tmf. See at tmf – Inclusion of circle 2-bundles.
Write for the ∞-group of units of the (a) Morava K-theory spectrum.
(Sati-Westerland 11, theorem 1)
A notion of spectrum of units of an -ring was originally described in
- Peter May, ring spaces and ring spectra Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave.
One explicit model was given in
- Christian Schlichtkrull, Units of ring spectra and their traces in algebraic K-theory, Geom. Topol. 8(2004) 645-673 (arXiv:math/0405079)
A general abstract discussion in stable (∞,1)-category theory is in
Remarks alluding to this are also on p. 20 of
Theorem 3.2 there is proven using classical results which are collected in
- Peter May, What precisely are -ring spaces and -ring spectra?, Geometry and Topology Monographs 16 (2009) 215–282 (pdf)
A survey of the situation in (∞,1)-category theory is also in section 3.1 of
A construction in terms of a model structure on spectra is in
- John Lind, Diagram spaces, diagram spectra, and spectra of units (arXiv:0908.1092)
A refinement of the construction of -groups of units to augmented ∞-groups over the sphere spectrum, such as to distinguish of a periodic E-∞ ring from its connective cover, is in
based on (Schlichtkrull 04).
The -group of units of Morava K-theory is discussed in