symmetric monoidal (∞,1)-category of spectra
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.
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
Write
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
This is (ABGHR 08, theorem 2.1/3.2, remark 3.4).
The left adjoint
is a higher analog of forming the group ring of an ordinary abelian group over the integers
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.
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)
There is a free-forgetful adjunction
The left adjoint functor $(-)_+$ adjoins basepoint. This is a strong monoidal functor (by this example) in that there is a natural isomorphism
Then there is the stabilization adjunction
(by this prop.)
Again the left adjoint is a strong monoidal functor in that there is a natural isomorphism
(by this prop)
Accordingly also the composite functor
given by
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
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
the H-group ring spectrum of $G$.
(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:
Since $\Sigma^\infty$ preserves wedge sum, this means that there is an isomorphism in $Ho(Spectra)$
(where the last isomorphism exhibits that wedge sum is the direct sum in the additive category $Ho(Spectra)$ (by this lemma)).
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.