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\begin{document}

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\section*{infinity-group of units}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{unaugmented_definition}{Unaugmented definition}\dotfill \pageref*{unaugmented_definition} \linebreak
\noindent\hyperlink{AugmentedDefinition}{Augmented definition}\dotfill \pageref*{AugmentedDefinition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{AdjointnessToGroupRing}{Adjointness to $\infty$-group $\infty$-ring}\dotfill \pageref*{AdjointnessToGroupRing} \linebreak
\noindent\hyperlink{unaugmented_case}{Unaugmented case}\dotfill \pageref*{unaugmented_case} \linebreak
\noindent\hyperlink{augmented_case}{Augmented case}\dotfill \pageref*{augmented_case} \linebreak
\noindent\hyperlink{homotopy_groups}{Homotopy groups}\dotfill \pageref*{homotopy_groups} \linebreak
\noindent\hyperlink{CohomologyAndLogarithm}{Cohomology and logarithm}\dotfill \pageref*{CohomologyAndLogarithm} \linebreak
\noindent\hyperlink{RelationToPicardInfinityGroups}{Relation to Picard $\infty$-group and Brauer $\infty$-group}\dotfill \pageref*{RelationToPicardInfinityGroups} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SnatihTheorem}{Snaith's theorem and the units of K-theory and complex cobordism}\dotfill \pageref*{SnatihTheorem} \linebreak
\noindent\hyperlink{units_of_topological_modular_forms}{Units of topological modular forms}\dotfill \pageref*{units_of_topological_modular_forms} \linebreak
\noindent\hyperlink{inclusion_of_circle_bundles_into_higher_chromatic_cohomology}{Inclusion of circle $n$-bundles into higher chromatic cohomology}\dotfill \pageref*{inclusion_of_circle_bundles_into_higher_chromatic_cohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The generalization in [[(∞,1)-category theory]] of the notion of \emph{[[group of units]]} in ordinary category theory.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{unaugmented_definition}{}\subsubsection*{{Unaugmented definition}}\label{unaugmented_definition}

\begin{defn}
\label{PlainInfinityGroupOfUnits}\hypertarget{PlainInfinityGroupOfUnits}{}
Let $A$ be an [[A-∞ algebra|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]]

\begin{displaymath}
A^\times \coloneqq GL_1(A)
\end{displaymath}
of $A$ is the [[(∞,1)-pullback]] $GL_1(A)$ in

\begin{equation}
\itexarray{
    GL_1(A) &\to& \Omega^\infty A
    \\
    \downarrow &(pb)& \downarrow
    \\
    (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A
  }
  \,.
\label{GL1AsPullback}\end{equation}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In terms of [[derived algebraic geometry]] one has that

\begin{displaymath}
GL_1(A) \simeq \mathbb{G}_m(A) = Hom(Spec A, \mathbb{G}_m)
\end{displaymath}
is the [[mapping space]] from $Spec A$ into the [[multiplicative group]]. This point of view is adopted for instance in (\hyperlink{Lurie}{Lurie, p. 20}).

\end{remark}
\hypertarget{AugmentedDefinition}{}\subsubsection*{{Augmented definition}}\label{AugmentedDefinition}

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 ∞-group|augmented]] over the [[sphere spectrum]] (\hyperlink{Sagave11}{Sagave 11}).

\begin{defn}
\label{AugmentedGroupOfUnits}\hypertarget{AugmentedGroupOfUnits}{}
There is a functor

\begin{displaymath}
gl_1^J \colon CRing_\infty \to AbGrp_\infty/\mathbb{S}
  \,,
\end{displaymath}
given by \ldots{}

\end{defn}
This is (\hyperlink{Sagave11}{Sagave 11, def. 3.14 in view of example 3.8}, \hyperlink{SagaveSchlichtkrull11}{Sagave-Schlichtkrull 11, above theorem 1.8}). See also (\hyperlink{Sagave11}{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 (\hyperlink{SagaveSchlichtkrull11}{Sagave-Schlichtkrull 11, theorem 1.7- 1.8}).

\begin{prop}
\label{FiberSequenceForExtendedGl1}\hypertarget{FiberSequenceForExtendedGl1}{}
For $E$ an [[E-∞ ring]], there is a [[homotopy fiber sequence]] of [[abelian ∞-groups]]

\begin{displaymath}
gl_1(E) \to gl_1^J(E) \to \mathbb{S}
  \,,
\end{displaymath}
where on the left we have the ordinary $\infty$-group of units of def. \ref{PlainInfinityGroupOfUnits} and on the right we have the [[sphere spectrum]], regarded (being a [[connective spectrum]]) as an [[abelian ∞-group]].

\end{prop}
Here the existence of the map $gl_1(E) \to gl_1^J(E)$ is (\hyperlink{Sagave11}{Sagave 11, lemma 2.12 + lemma 3.16}). The fact that the resulting sequence is a homotopy fiber sequence is (\hyperlink{Sagave11}{Sagave 11, prop. 4.1}).

Using this, there is now a modified delooping of the ordinary $\infty$-group of units:

\begin{defn}
\label{}\hypertarget{}{}
Write $bgl_1^\ast(E)$ for the [[homotopy cofiber]] of $gl_1^J(E) \to \mathbb{S}$ to yield

\begin{displaymath}
gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E)
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Sagave11}{Sagave 11, prop. 4.3})

\begin{remark}
\label{}\hypertarget{}{}
It ought to be true that the non-connective delooping $bgl_1^\ast(E)$ sits inside the full [[Picard ∞-group]] of $E Mod$. (\hyperlink{Sagave11}{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 \emph{\href{twisted+cohomology#InfSections}{twisted cohomology -- by R-module bundles}}.

\end{remark}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{AdjointnessToGroupRing}{}\subsubsection*{{Adjointness to $\infty$-group $\infty$-ring}}\label{AdjointnessToGroupRing}

\hypertarget{unaugmented_case}{}\paragraph*{{Unaugmented case}}\label{unaugmented_case}

\begin{defn}
\label{GroupOfUnitsFunctor}\hypertarget{GroupOfUnitsFunctor}{}
Write

\begin{displaymath}
gl_1 \; \colon \;  CRing_\infty \to AbGrp_\infty
\end{displaymath}
for the [[(∞,1)-functor]] which sends a [[E-∞ ring|commutative ∞-ring]] to its [[∞-group of units]].

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
The [[∞-group of units]] [[(∞,1)-functor]] of def. \ref{GroupOfUnitsFunctor} is a right-[[adjoint (∞,1)-functor]]

\begin{displaymath}
CRing_\infty 
  \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}}
  AbGrp_\infty
  \,.
\end{displaymath}
\end{theorem}
This is (\hyperlink{ABGHR08}{ABGHR 08, theorem 2.1/3.2, remark 3.4}).

\begin{remark}
\label{}\hypertarget{}{}
The [[left adjoint]]

\begin{displaymath}
\mathbb{S}[-] \colon AbGrp_\infty \to CRing_\infty
\end{displaymath}
is a higher analog of forming the [[group ring]] of an ordinary [[abelian group]] over the [[integers]]

\begin{displaymath}
\mathbb{Z}[-] \colon Ab \to CRing
  \,,
\end{displaymath}
which is indeed [[left adjoint]] to forming the ordinary [[group of units]] of a ring.

We might call $\mathbb{S}[A]$ the \textbf{[[∞-group ∞-ring]]} of $A$ over the [[sphere spectrum]].

\end{remark}
\hypertarget{augmented_case}{}\paragraph*{{Augmented case}}\label{augmented_case}

Also the augmented $\infty$-group of units functor of def. \ref{AugmentedGroupOfUnits} is a homotopy right adjoint. (\hyperlink{Sagave11}{Sagave 11, theorem 1.7}).

\hypertarget{homotopy_groups}{}\subsubsection*{{Homotopy groups}}\label{homotopy_groups}

The [[homotopy groups]] of $GL_1(E)$ are

\begin{displaymath}
\pi_n(GL_1(E))
  = 
  \left\{
    \itexarray{
       \pi_0(E)^\times & |\, n = 0
       \\
       \pi_n(E) & | \, n \geq 1
    }
  \right.
\end{displaymath}
\hypertarget{CohomologyAndLogarithm}{}\subsubsection*{{Cohomology and logarithm}}\label{CohomologyAndLogarithm}

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]] [[Brown representability theorem|represented]] by $gl_1(E)$ in degree 0 is the ordinary [[group of units]] in the [[cohomology ring]] of $E$:

\begin{displaymath}
H^0(X, gl_1(E))  \simeq (E^0(X))^\times
  \,.
\end{displaymath}
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]]

\begin{displaymath}
\pi_{\bullet \geq 1} GL_1(E) \simeq \pi_{\bullet \geq 1} \Omega^\infty E
  \,.
\end{displaymath}
On cohomology elements this map

\begin{displaymath}
\pi_q(gl_1(E)) \simeq \tilde H^0(S^q, gl_1(E)) \simeq  (1+ \tilde R^0(S^q))^\times \subset (R^0(S^q))^\times
\end{displaymath}
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)-local stable homotopy theory|K(n)-localization]], where it is called then the [[logarithmic cohomology operation]], see there for more.

(\hyperlink{Rezk06}{Rezk 06})

\hypertarget{RelationToPicardInfinityGroups}{}\subsubsection*{{Relation to Picard $\infty$-group and Brauer $\infty$-group}}\label{RelationToPicardInfinityGroups}

Given an [[E-∞ ring]] $E$, the [[looping]] of the Brauer $\infty$-group is the [[Picard ∞-group]] (\hyperlink{Szymik11}{Szymik 11, theorem 5.7}).

\begin{displaymath}
\Omega Br(E) \simeq Pic(E).
\end{displaymath}
The [[looping]] of that is the ∞-group of units (\hyperlink{Sagave11}{Sagave 11, theorem 1.2}).

\begin{displaymath}
\Omega^2 Br(E) \simeq \Omega Pic(E) \simeq GL_1(E)
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{SnatihTheorem}{}\subsubsection*{{Snaith's theorem and the units of K-theory and complex cobordism}}\label{SnatihTheorem}

[[Snaith's theorem]] asserts that

\begin{enumerate}%
\item the [[K-theory spectrum]] for [[complex K-theory]] is the [[∞-group ∞-ring]] of the [[circle 2-group]] localized away from the [[Bott element]] $\beta$:

\begin{displaymath}
KU \simeq (\mathbb{S}[B U(1)])[\beta^{-1}]
  \,;
\end{displaymath}

\item 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]] $\beta$:

\begin{displaymath}
MU \simeq (\mathbb{S}[B U])[\beta^{-1}]
  \,.
\end{displaymath}


\end{enumerate}
\hypertarget{units_of_topological_modular_forms}{}\subsubsection*{{Units of topological modular forms}}\label{units_of_topological_modular_forms}

Analysis of the $\infty$-group of units of [[tmf]] is in (\hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, from section 12 on}).

\hypertarget{inclusion_of_circle_bundles_into_higher_chromatic_cohomology}{}\subsubsection*{{Inclusion of circle $n$-bundles into higher chromatic cohomology}}\label{inclusion_of_circle_bundles_into_higher_chromatic_cohomology}

By \hyperlink{SnatihTheorem}{Snaith's theorem} above there is a canonical map

\begin{displaymath}
B U(1) \to \mathbb{S}[B U(1)] \to  KU
\end{displaymath}
that sends [[circle bundles]] to cocycles in [[topological K-theory]].

At the next level there is a canonical map

\begin{displaymath}
B^2 U(1) \to \mathbb{S}[B^2 U(1)] \to tmf
\end{displaymath}
that sends [[circle 2-bundles]] to [[tmf]]. See at \emph{\href{tmf#InclusionOfCircle2Bundles}{tmf -- Inclusion of circle 2-bundles}}.

Write $gl_1(K(n))$ for the [[∞-group of units]] of the (a) [[Morava K-theory]] spectrum.

\begin{prop}
\label{}\hypertarget{}{}
For $p = 2$ and all $n \in \mathbb{N}$, there is an [[equivalence in an (infinity,1)-category|equivalence]]

\begin{displaymath}
Maps(B^{n+1}U(1), B gl_1(K(n)))
  \simeq
  \mathbb{Z}/(2)
\end{displaymath}
between the [[mapping space]] from the [[classifying space]] for [[circle n-bundle|circle (n+1)-bundles]] to the [[delooping]] of the [[∞-group of units]] of $K(n)$.

\end{prop}
(\hyperlink{SatiWesterland11}{Sati-Westerland 11, theorem 1})

\begin{remark}
\label{}\hypertarget{}{}
By the discussion at [[(∞,1)-vector bundle]] this means that for each such map there is a type of [[twisted cohomology|twist]] of Morava K-theory (at $p = 2$).

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[(∞,1)-vector bundle]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A notion of spectrum of units of an $E_\infty$-ring was originally described in

\begin{itemize}%
\item [[Peter May]], \emph{$E_\infty$ ring spaces and $E_\infty$ ring spectra} Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. With contributions by [[Frank Quinn]], Nigel Ray, and J\o{}rgen Tornehave.

\end{itemize}
One explicit model was given in

\begin{itemize}%
\item [[Christian Schlichtkrull]], \emph{Units of ring spectra and their traces in algebraic K-theory}, Geom. Topol. 8(2004) 645-673 (\href{http://arxiv.org/abs/math/0405079}{arXiv:math/0405079})

\end{itemize}
A general abstract discussion in [[stable (∞,1)-category]] theory is in

\begin{itemize}%
\item [[Charles Rezk]], section II of \emph{The units of a ring spectrum and a logarithmic cohomology operation}, J. Amer. Math. Soc. 19 (2006), 969-1014 (\href{http://arxiv.org/abs/math/0407022}{arXiv:math/0407022})


\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], \emph{Units of ring spectra and Thom spectra} (\href{http://arxiv.org/abs/0810.4535}{arXiv:0810.4535})


\item [[Jacob Lurie]], construction 3.9.4 of \emph{Elliptic Cohomology I: Spectral Abelian Varieties} (\href{http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf}{pdf})



\end{itemize}
Remarks alluding to this are also on p. 20 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology]]}

\end{itemize}
Theorem 3.2 there is proven using classical results which are collected in

\begin{itemize}%
\item [[Peter May]], \emph{What precisely are $E_\infty$-ring spaces and $E_\infty$-ring spectra?}, Geometry and Topology Monographs 16 (2009) 215--282 (\href{http://www.math.uchicago.edu/~may/PAPERS/Final1.pdf}{pdf})

\end{itemize}
A survey of the situation in [[(∞,1)-category theory]] is also in section 3.1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology]]}

\end{itemize}
A construction in terms of a [[model structure on spectra]] is in

\begin{itemize}%
\item John Lind, \emph{Diagram spaces, diagram spectra, and spectra of units} (\href{http://arxiv.org/abs/0908.1092}{arXiv:0908.1092})

\end{itemize}
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

\begin{itemize}%
\item [[Steffen Sagave]], \emph{Spectra of units for periodic ring spectra}, Algebr. Geom. Topol. 16 (2016) 1203-1251 (\href{http://arxiv.org/abs/1111.6731}{arXiv:1111.6731})

\end{itemize}
based on (\hyperlink{Schlichtkrull04}{Schlichtkrull 04}). See also

\begin{itemize}%
\item [[Steffen Sagave]], [[Christian Schlichtkrull]], \emph{Diagram spaces and symmetric spectra}, Advances in Mathematics, Volume 231, Issues 3--4, October--November 2012, Pages 2116--2193 (\href{https://arxiv.org/abs/1103.2764}{arXiv:1103.2764})


\item [[Markus Szymik]], \emph{Brauer spaces for commutative rings and structured ring spectra} (\href{http://arxiv.org/abs/1110.2956}{arXiv:1110.2956})


\item [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], \emph{Brauer groups for commutative $\mathbb{S}$-algebras}, J. Pure Appl. Algebra 216 (2012) 2361--2376 (\href{http://arxiv.org/abs/1005.5370}{arXiv:1005.5370})



\end{itemize}
The $\infty$-group of units of [[Morava K-theory]] is discussed in

\begin{itemize}%
\item [[Hisham Sati]], [[Craig Westerland]], \emph{Twisted Morava K-theory and E-theory} (\href{http://arxiv.org/abs/1109.3867}{arXiv:1109.3867})

\end{itemize}
The [[cohomology]] with coefficients in $gl_1(E)$ and the corresponding [[logarithmic cohomology operations]] are discussed in

\begin{itemize}%
\item [[Charles Rezk]], \emph{The units of a ring spectrum and a logarithmic cohomology operation}, J. Amer. Math. Soc. 19 (2006), 969-1014 (\href{http://arxiv.org/abs/math/0407022}{arXiv:math/0407022})

\end{itemize}
The group of units of [[tmf]] is analyzed from section 12 on in

\begin{itemize}%
\item [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf})

\end{itemize}
[[!redirects ∞-group of units]]

[[!redirects ∞-groups of units]] [[!redirects infinity-groups of units]]



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