nLab
∞-group ∞-ring

Contents

Context

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:CRing AbGrp 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 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).

Remark

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.

H-Group ring spectra

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

Recall

the classical homotopy category (Ho(Top),×,*)(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 */),,S 1)(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),,𝕊)(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 */)U() +Ho(Top) 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×Y) +X +Y +. (X \times Y)_+ \simeq X_+ \wedge Y_+ \,.

Then there is the stabilization adjunction

Ho(Spectra)Ω Σ Ho(Top */) 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

Σ ((X,x)(Y,y))(Σ (X,x))(Σ (Y,y)) \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

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

given by

Ho(Spectra)Ω Σ Ho(Top */)U() +Ho(Top) 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 𝕊[]=Σ (() +)\mathbb{S}[-] = \Sigma^\infty((-)_+) is strong monoidal functor for a monoid (A,μ)(A,\mu) in (Ho(Top),×,*)(Ho(Top), \times , \ast) (an H-space), then also

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

is a monoid. This is the “monoid ring spectrum” of AA.

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

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

the H-group ring spectrum of GG.

Remark

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

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

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

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

Σ (G +) Σ ((G,e)S 0) (Σ (G,e))𝕊 (Σ (G,e))𝕊 \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)Ho(Spectra) (by this lemma)).

Examples

References

Last revised on June 2, 2017 at 08:25:07. See the history of this page for a list of all contributions to it.