We establish a canonical and unique tensor product for commutative monoids and groups in
an
–category
which
generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that
–(semi)ring objects
in
give rise to
–ring spectrum
objects in
. In
the case that
is
the
–category of
spaces this produces a multiplicative infinite loop space machine which can be applied to the
algebraic
–theory
of rings and ring spectra.
The main tool we use to establish these results is the theory of smashing localizations of presentable
–categories.
In particular, we identify preadditive and additive
–categories
as the local objects for certain smashing localizations. A central theme is the
stability of algebraic structures under basechange; for example, we show
. Lastly, we
also consider these algebraic structures from the perspective of Lawvere algebraic theories in
–categories.
Keywords
infinite loop space machines, structured ring spectra,
K-theory