# nLab ∞-group ∞-ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

group theory

# Contents

## 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:

### $\infty$-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

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

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 $(Ho(Top^{\ast/}), \wedge, S^1)$, 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)

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

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.