Algebraic
-theory,
introduced by Cortiñas and Thom (2007), is a bivariant
-theory defined
on the category
of algebras over a commutative unital ring
. It consists of a triangulated
category
endowed
with a functor from
to
that is the universal excisive, homotopy invariant and matrix-stable
homology theory. Moreover, one can recover Weibel’s homotopy
-theory
from
since we
have
for any
algebra
. We
prove that
with the split surjections as fibrations and the
-equivalences
as weak equivalences is a stable category of fibrant objects, whose homotopy category
is
.
As a consequence of this, we prove that the Dwyer–Kan localization
of the
-category of algebras at
the set of
-equivalences
is a stable infinity category whose homotopy category is
.
Keywords
bivariant algebraic $K$-theory, $\infty$-categories,
categories of fibrant objects
