symmetric monoidal (∞,1)-category of spectra
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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
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 the ∞-group ∞-ring of over the sphere spectrum.
We consider here the simpler concept after passage to equivalence classes.
Recall
the classical homotopy category which is a symmetric monoidal category with respect to forming Cartesian product spaces (tensor unit is the point space)
its pointed objects version , 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 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 is strong monoidal functor for a monoid in (an H-space), then also
is a monoid. This is the “monoid ring spectrum” of .
If is in fact a group object in (hence an H-group), then we may call the ring spectrum
the H-group ring spectrum of .
(H-group ring spectrum is a direct sum with the sphere spectrum)
Notice that an H-group already is canonically a pointed object itself, pointed by its neutral element . Regarded as an object this way then the pointed object above is equivalently the wedge sum of with the 0-sphere:
Since preserves wedge sum, this means that there is an isomorphism in
(where the last isomorphism exhibits that wedge sum is the direct sum in the additive category (by this lemma)).
The localization of the -group ring of the circle 2-group at the Bott element is equivalently the representing spectrum KU of complex topological K-theory:
This is Snaith's theorem.
Last revised on March 20, 2023 at 19:49:54. See the history of this page for a list of all contributions to it.