infinity-group of units




Group Theory



The generalization in (∞,1)-category theory of the notion of group of units in ordinary category theory.


Unaugmented definition


Let AA be an A-∞ ring spectrum.

For Ω A\Omega^\infty A the underlying A-∞ space and π 0Ω A\pi_0 \Omega^\infty A the ordinary ring of connected components, write (π 0Ω A) ×(\pi_0 \Omega^\infty A)^\times for its group of units.

Then the ∞-group of units

A ×GL 1(A) A^\times \coloneqq GL_1(A)

of AA is the (∞,1)-pullback GL 1(A)GL_1(A) in

(1)GL 1(A) Ω A (pb) (π 0Ω A) × π 0Ω A. \array{ GL_1(A) &\to& \Omega^\infty A \\ \downarrow &(pb)& \downarrow \\ (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A } \,.

In terms of derived algebraic geometry one has that

GL 1(A)𝔾 m(A)=Hom(SpecA,𝔾 m) GL_1(A) \simeq \mathbb{G}_m(A) = Hom(Spec A, \mathbb{G}_m)

is the mapping space from SpecASpec A into the multiplicative group. This point of view is adopted for instance in (Lurie, p. 20).

Augmented definition

There is slight refinement of the above definition, which essentially adds one 0-th “grading” homotopy group to Bgl 1(E)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

gl 1 J:CRing AbGrp /𝕊, gl_1^J \colon CRing_\infty \to AbGrp_\infty/\mathbb{S} \,,

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 EE an E-∞ ring, there is a homotopy fiber sequence of abelian ∞-groups

gl 1(E)gl 1 J(E)𝕊, gl_1(E) \to gl_1^J(E) \to \mathbb{S} \,,

where on the left we have the ordinary \infty-group of units of def. 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)gl 1 J(E)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 *(E)bgl_1^\ast(E) for the homotopy cofiber of gl 1 J(E)𝕊gl_1^J(E) \to \mathbb{S} to yield

gl 1(E)gl 1 J(E)𝕊bgl 1 *(E). gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E) \,.

(Sagave 11, prop. 4.3)


It ought to be true that the non-connective delooping bgl 1 *(E)bgl_1^\ast(E) sits inside the full Picard ∞-group of EModE 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 ()Σ nE(-)\wedge\Sigma^n E.)

See also at twisted cohomology – by R-module bundles.


Adjointness to \infty-group \infty-ring

Unaugmented case



gl 1:CRing AbGrp gl_1 \; \colon \; CRing_\infty \to AbGrp_\infty

for the (∞,1)-functor which sends a commutative ∞-ring to its ∞-group of units.


The ∞-group of units (∞,1)-functor of def. is a right-adjoint (∞,1)-functor

CRing gl 1𝕊[]AbGrp . CRing_\infty \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}} AbGrp_\infty \,.

This is (ABGHR 08, theorem 2.1/3.2, remark 3.4).


The left adjoint

𝕊[]:AbGrp CRing \mathbb{S}[-] \colon AbGrp_\infty \to CRing_\infty

is a higher analog of forming the group ring of an ordinary abelian group over the integers

[]:AbCRing, \mathbb{Z}[-] \colon Ab \to CRing \,,

which is indeed left adjoint to forming the ordinary group of units of a ring.

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

Augmented case

Also the augmented \infty-group of units functor of def. is a homotopy right adjoint. (Sagave 11, theorem 1.7).

Homotopy groups

The homotopy groups of GL 1(E)GL_1(E) are

π n(GL 1(E))={π 0(E) × |n=0 π n(E) |n1 \pi_n(GL_1(E)) = \left\{ \array{ \pi_0(E)^\times & |\, n = 0 \\ \pi_n(E) & | \, n \geq 1 } \right.

Cohomology and logarithm

Given EE an E-∞ ring, then write gl 1(E)gl_1(E) for its \infty-group of units regarded as a connective spectrum. For XX the homotopy type of a topological space, then the cohomology represented by gl 1(E)gl_1(E) in degree 0 is the ordinary group of units in the cohomology ring of EE:

H 0(X,gl 1(E))(E 0(X)) ×. H^0(X, gl_1(E)) \simeq (E^0(X))^\times \,.

In positive degree the canonical map of pointed homotopy types GL 1(E)=Ω gl 1(E)Ω EGL_1(E) = \Omega^\infty gl_1(E) \to \Omega^\infty E is in fact an isomorphism on all homotopy groups

π 1GL 1(E)π 1Ω E. \pi_{\bullet \geq 1} GL_1(E) \simeq \pi_{\bullet \geq 1} \Omega^\infty E \,.

On cohomology elements this map

π q(gl 1(E))H˜ 0(S q,gl 1(E))(1+R˜ 0(S q)) ×(R 0(S q)) × \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

is logarithm-like, in that it sends 1+xx1 + 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)

Relation to Picard \infty-group and Brauer \infty-group

Given an E-∞ ring EE, the looping of the Brauer \infty-group is the Picard ∞-group (Szymik 11, theorem 5.7).

ΩBr(E)Pic(E). \Omega Br(E) \simeq Pic(E).

The looping of that is the ∞-group of units (Sagave 11, theorem 1.2).

Ω 2Br(E)ΩPic(E)GL 1(E). \Omega^2 Br(E) \simeq \Omega Pic(E) \simeq GL_1(E) \,.


Snaith’s theorem and the units of K-theory and complex cobordism

Snaith's theorem asserts that

  1. the K-theory spectrum for complex K-theory is the ∞-group ∞-ring of the circle 2-group localized away from the Bott element β\beta:

    KU(𝕊[BU(1)])[β 1]; KU \simeq (\mathbb{S}[B U(1)])[\beta^{-1}] \,;
  2. 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 β\beta:

    MU(𝕊[BU])[β 1]. MU \simeq (\mathbb{S}[B U])[\beta^{-1}] \,.

Units of topological modular forms

Analysis of the \infty-group of units of tmf is in (Ando-Hopkins-Rezk 10, from section 12 on).

Inclusion of circle nn-bundles into higher chromatic cohomology

By Snaith’s theorem above there is a canonical map

BU(1)𝕊[BU(1)]KU B U(1) \to \mathbb{S}[B U(1)] \to KU

that sends circle bundles to cocycles in topological K-theory.

At the next level there is a canonical map

B 2U(1)𝕊[B 2U(1)]tmf B^2 U(1) \to \mathbb{S}[B^2 U(1)] \to tmf

that sends circle 2-bundles to tmf. See at tmf – Inclusion of circle 2-bundles.

Write gl 1(K(n))gl_1(K(n)) for the ∞-group of units of the (a) Morava K-theory spectrum.


For p=2p = 2 and all nn \in \mathbb{N}, there is an equivalence

Maps(B n+1U(1),Bgl 1(K(n)))/(2) Maps(B^{n+1}U(1), B gl_1(K(n))) \simeq \mathbb{Z}/(2)

between the mapping space from the classifying space for circle (n+1)-bundles to the delooping of the ∞-group of units of K(n)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=2p = 2).


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

  • Peter May, p. 54 of: E E_\infty ring spaces and E E_\infty ring spectra, Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave (pdf)

One explicit model was given in

Further discussion in

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 E E_\infty-ring spaces and E E_\infty-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 \infty-groups of units to augmented ∞-groups over the sphere spectrum, such as to distinguish gl 1gl_1 of a periodic E-∞ ring from its connective cover, is in

based on (Schlichtkrull 04). See also

The \infty-group of units of Morava K-theory is discussed in

The cohomology with coefficients in gl 1(E)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

Last revised on December 14, 2021 at 15:52:19. See the history of this page for a list of all contributions to it.