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 ground fields: Chains on loop spaces

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

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

Chains on loop spaces

Every ∞-group is equivalently (the homotopy type of) based loop space $\Omega X$ (see there). Under the stable Dold-Kan correspondence the singular homology chain complex $C_\bullet(\Omega X)$ of loop space models the spectrum $H k(\Omega X)$, where $H k$ is the Eilenberg-MacLane spectrum for the ground field.

This $C_\bullet(\Omega X)$ generallu inherits A-infinity algebra structure (Stasheff 1963 Thm. 2.3).

If the loop space is equivalently modeled by Moore loops and as such becomes a strict topological monoid, then this rigidifies to a dg-algebra structure on $C_\bullet(\Omega_{Moore} X)$ (cf. Rivera 2022).

$\infty$-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

Example

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

The $A_\infty$-algebra structure on the singular homology chain complex of an $A_\infty$-space is due to:

• Jim Stasheff, Thm. 2.3 of: Homotopy associativity of H-spaces II 108 2 (1963) 293-312 [doi:10.2307/1993609, doi:10.1090/S0002-9947-1963-0158400-5]

If the loop infinity-group is rigidified to a topological monoid by using Moore loops then the $A_\infty$-algebra structure on its chains rigidified to a dg-algebra stucture. As such this is discussed in relation to the cobar construction:

• John Frank Adams: On the cobar construction, Proc Natl Acad Sci USA 42 7 (1956) 409–412 [doi:10.1073/pnas.42.7.409]

• Camilo Arias Abad, Florian Schätz: Flat $\mathbb{Z}$-graded connections and loop spaces, International Mathematics Research Notices 2018 4 (2018) 961–1008 [doi:10.1093/imrn/rnw258, arXiv:1510.04975]

• Manuel Rivera, Mahmoud Zeinalian: Cubical rigidification, the cobar construction, and the based loop space, Algebr. Geom. Topol. 18 (2018) 3789-3820 [arXiv:1612.04801, doi:10.2140/agt.2018.18.3789]

• Manuel Rivera: Adams’ cobar construction revisited, Documenta Math. 27 (2022) 1213-1223 [arxiv:1910.08455, pdf]

The above abstract stable homotopy theory of commutative group $\infty$-rings is due to:

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