nLab ∞-group ∞-ring

Idea

Given a group object in the context of homotopy theory then the free construction of a ring object from it in stable homotopy category is the corresponding “group ring” construction generalized to homotopy theory.

Definition

One may consider the construction at various levels of algebraic structure:

• for ∞-groups and E-∞ rings: ∞-Group E-∞ rings

• for H-groups and ring spectra: H-group ring spectra

$\infty$-Group $\infty$-rings

Definition

Write

$$gl_1 \; \colon \; CRing_\infty \to AbGrp_\infty$$

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

Theorem

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

$$CRing_\infty \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}} AbGrp_\infty \,.$$

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

Remark

The left adjoint

$$\mathbb{S}[-] \colon AbGrp_\infty \to CRing_\infty$$

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

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

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.

H-Group ring spectra

We consider here the simpler concept after passage to equivalence classes.

Recall

the classical homotopy category $(Ho(Top), \times, \ast)$ which is a symmetric monoidal category with respect to forming Cartesian product spaces (tensor unit is the point space)

its pointed objects version $(Ho(Top^{\ast/}), \wedge, S^0)$, which is a symmetric monoidal category with respect to smash product of pointed topological spaces (tensor unit is the 0-sphere)

the stable homotopy category $(Ho(Spectra), \wedge, \mathbb{S})$ which is a symmetric monoidal category with respect to the smash product of spectra (tensor unit is the sphere spectrum)

There is a free-forgetful adjunction

$$Ho(Top^{\ast/}) \underoverset {\underset{U}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} Ho(Top)$$

The left adjoint functor $(-)_+$ adjoins basepoint. This is a strong monoidal functor (by this example) in that there is a natural isomorphism

$$(X \times Y)_+ \simeq X_+ \wedge Y_+ \,.$$

Then there is the stabilization adjunction

$$Ho(Spectra) \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty}{\longleftarrow}} {} Ho(Top^{\ast/})$$

(by this prop.)

Again the left adjoint is a strong monoidal functor in that there is a natural isomorphism

$$\Sigma^\infty( (X,x) \wedge (Y,y) ) \simeq \left(\Sigma^\infty (X,x)\right) \wedge \left( \Sigma^\infty (Y,y) \right)$$

(by this prop)

Accordingly also the composite functor

$$\mathbb{S}[-] \coloneqq \Sigma^\infty(-)_+$$

given by

$$Ho(Spectra) \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty}{\longleftarrow}} {} Ho(Top^{\ast/}) \underoverset {\underset{U}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} Ho(Top)$$

are strong monoidal.

That $\mathbb{S}[-] = \Sigma^\infty((-)_+)$ is strong monoidal functor for a monoid $(A,\mu)$ in $(Ho(Top), \times , \ast)$ (an H-space), then also

$$\Sigma^\infty (A_+) \;\in\; Ho(Spectra)$$

is a monoid. This is the “monoid ring spectrum” of $A$.

If $A = G$ is in fact a group object in $(Ho(Top),\times, \ast)$ (hence an H-group), then we may call the ring spectrum

$$\Sigma^\infty(G_+) \;\in \; Ho(Spectra)$$

the H-group ring spectrum of $G$.

Remark

(H-group ring spectrum is a direct sum with the sphere spectrum)

Notice that an H-group $G$ already is canonically a pointed object itself, pointed by its neutral element $e \colon \ast \to G$. Regarded as an object $(G,e) \in Ho(Top^{\ast/})$ this way then the pointed object $G_+$ above is equivalently the wedge sum of $G$ with the 0-sphere:

$$G_+ \simeq (G,e) \vee S^0 \,.$$

Since $\Sigma^\infty$ preserves wedge sum, this means that there is an isomorphism in $Ho(Spectra)$

$$\begin{aligned} \Sigma^\infty(G_+) & \simeq \Sigma^\infty \left( (G,e) \vee S^0 \right) \\ & \simeq \left( \Sigma^\infty(G,e) \right) \vee \mathbb{S} \\ & \simeq \left( \Sigma^\infty(G,e) \right) \oplus \mathbb{S} \end{aligned}$$

(where the last isomorphism exhibits that wedge sum is the direct sum in the additive category $Ho(Spectra)$ (by this lemma)).

Examples

• The localization of the $E_\infty$-group ring $\Sigma^\infty(BU(1)_+)$ of the circle 2-group at the Bott element $\beta$ is equivalently the representing spectrum KU of complex topological K-theory:

  $\mathbb{S}[B U(1)][\beta^{-1}] \simeq KU$

  This is Snaith's theorem.

References

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)