symmetric monoidal (∞,1)-category of spectra
The generalization in (∞,1)-category theory of the notion of group of units in ordinary category theory.
Let $A$ be an A-∞ ring spectrum.
For $\Omega^\infty A$ the underlying A-∞ space and $\pi_0 \Omega^\infty A$ the ordinary ring of connected components, write $(\pi_0 \Omega^\infty A)^\times$ for its group of units.
Then the ∞-group of units
of $A$ is the (∞,1)-pullback $GL_1(A)$ in
In terms of derived algebraic geometry one has that
is the mapping space from $Spec A$ into the multiplicative group. This point of view is adopted for instance in (Lurie, p. 20).
There is slight refinement of the above definition, which essentially adds one 0-th “grading” homotopy group to $B gl_1(E)$ and thereby makes the $\infty$-group of units of E-∞ rings be canonically augmented over the sphere spectrum (Sagave 11).
There is a functor
given by …
This is (Sagave 11, def. 3.14 in view of example 3.8, Sagave-Schlichtkrull 11, above theorem 1.8). See also (Sagave 11, section 1.4) for comments on how this yields an $\infty$-version of $\mathbb{Z}$-grading on an abelian group.
In fact this grading extends form the group of units to the full $\infty$-ring (Sagave-Schlichtkrull 11, theorem 1.7- 1.8).
For $E$ an E-∞ ring, there is a homotopy fiber sequence of abelian ∞-groups
where on the left we have the ordinary $\infty$-group of units of def. 1 and on the right we have the sphere spectrum, regarded (being a connective spectrum) as an abelian ∞-group.
Here the existence of the map $gl_1(E) \to gl_1^J(E)$ 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 $\infty$-group of units:
Write $bgl_1^\ast(E)$ for the homotopy cofiber of $gl_1^J(E) \to \mathbb{S}$ to yield
It ought to be true that the non-connective delooping $bgl_1^\ast(E)$ sits inside the full Picard ∞-group of $E Mod$. (Sagave 11, remark 4.11). (Apparently it’s the full inclusion on those degree-0 twists which are grading twists, i.e. on the elements $(-)\wedge\Sigma^n E$.)
See also at twisted cohomology – by R-module bundles.
Write
for the (∞,1)-functor which sends a commutative ∞-ring to its ∞-group of units.
The ∞-group of units (∞,1)-functor of def. 4 is a right-adjoint (∞,1)-functor
This is (ABGHR 08, theorem 2.1/3.2, remark 3.4).
The left adjoint
is a higher analog of forming the group ring of an ordinary abelian group over the integers
which is indeed left adjoint to forming the ordinary group of units of a ring.
We might call $\mathbb{S}[A]$ the ∞-group ∞-ring of $A$ over the sphere spectrum.
Also the augmented $\infty$-group of units functor of def. 2 is a homotopy right adjoint. (Sagave 11, theorem 1.7).
The homotopy groups of $GL_1(E)$ are
Given $E$ an E-∞ ring, then write $gl_1(E)$ for its $\infty$-group of units regarded as a connective spectrum. For $X$ the homotopy type of a topological space, then the cohomology represented by $gl_1(E)$ in degree 0 is the ordinary group of units in the cohomology ring of $E$:
In positive degree the canonical map of pointed homotopy types $GL_1(E) = \Omega^\infty gl_1(E) \to \Omega^\infty E$ is in fact an isomorphism on all homotopy groups
On cohomology elements this map
is logarithm-like, in that it sends $1 + x \mapsto x$.
But there is not a homomorphism of spectra of this form. This only exists after K(n)-localization, where it is called then the logarithmic cohomology operation, see there for more.
(Rezk 06)
Given an E-∞ ring $E$, the looping of the Brauer $\infty$-group is the Picard ∞-group (Szymik 11, theorem 5.7).
The looping of that is the ∞-group of units (Sagave 11, theorem 1.2).
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 $\beta$:
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 $\beta$:
Analysis of the $\infty$-group of units of tmf is in (Ando-Hopkins-Rezk 10, from section 12 on).
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 $gl_1(K(n))$ for the ∞-group of units of the (a) Morava K-theory spectrum.
For $p = 2$ and all $n \in \mathbb{N}$, there is an equivalence
between the mapping space from the classifying space for circle (n+1)-bundles to the delooping of the ∞-group of units of $K(n)$.
(Sati-Westerland 11, theorem 1)
By the discussion at (∞,1)-vector bundle this means that for each such map there is a type of twist of Morava K-theory (at $p = 2$).
A notion of spectrum of units of an $E_\infty$-ring was originally described in
One explicit model was given in
A general abstract discussion in stable (∞,1)-category theory is in
Charles Rezk, section II of The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)
Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)
Remarks alluding to this are also on p. 20 of
Theorem 3.2 there is proven using classical results which are collected in
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
A refinement of the construction of $\infty$-groups of units to augmented ∞-groups over the sphere spectrum, such as to distinguish $gl_1$ of a periodic E-∞ ring from its connective cover, is in
based on (Schlichtkrull 04). See also
Steffen Sagave, Christian Schlichtkrull, Diagram spaces and symmetric spectra, Advances in Mathematics, Volume 231, Issues 3–4, October–November 2012, Pages 2116–2193 (arXiv:1103.2764)
Markus Szymik, Brauer spaces for commutative rings and structured ring spectra (arXiv:1110.2956)
Andrew Baker, Birgit Richter, Markus Szymik, Brauer groups for commutative $\mathbb{S}$-algebras, J. Pure Appl. Algebra 216 (2012) 2361–2376 (arXiv:1005.5370)
The $\infty$-group of units of Morava K-theory is discussed in
The cohomology with coefficients in $gl_1(E)$ and the corresponding logarithmic cohomology operations are discussed in
The group of units of tmf is analyzed from section 12 on in