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). See also (Sagave 11, section 1.4) for comments on how this yields an $\infty$-version of $\mathbb{Z}$-grading on an abelian group.
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 spring ∞-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
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