group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
of the triangulated category $KK$ of KK-theory generated by the tensor unit is called the bootstrap category (Rosenberg-Schochet 87), because on this subcategory KK-theory has a good axiomatic description (see Blackadar 22.3).
The canonical functor from KK to (the homotopy category of) KU-module spectra restricts to a full and faithful functor on the bootstrap category (prop. below), and so the KK-bootstrap category may also be thought of as a full subcategory of $KU Mod$
Many C*-algebras that “appear in practice” are in the bootstrap category, notably the groupoid convolution algebras of amenable Lie groupoids (prop. below). Generally, a C*-algebra is in the bootstrap category precisely if it is KK-equivalent to a commutative C*-algebra.
See (Blackadar, def. 22.3.4).
The groupoid convolution algebra of an amenable topological groupoid is in the bootstrap category.
This is due to (Tu 99, prop. 10.7), recalled for instance as (Uuye 11, example 3.6).
In the bootstrap category a Künneth theorem for operator K-theory is true. (Rosenberg-Schochet 87).
On the bootstrap category the operator K-theory spectrum functor $KK \to Ho(KU Mod)$ is
The first statement is (DEKM 11, section 3), the second is (DEKM 11, prop. 4.2).
See at KK-theory the section Triangulated and spectrum-enriched structure for more details.
The notion originates in
A texbook account is around def. 22.3.4 of
The corresponding localizing subcategories are discussed in
The full embedding into the homotopy category of KU-modules is discussed in section 3 of
The inclusion of groupoid convolution algebras of amenable topological groupoids into the bootstrap category is discussed in
Discussion of the bootstrap category in equivariant K-theory is in
This show that there are algebras in the plain bootstrap category with circle group-equivariant structure which are not $U(1)$-equivariant KK-equivalent to a commutative $U(1)$-$C^\ast$-alegbra.
Last revised on August 15, 2013 at 06:17:17. See the history of this page for a list of all contributions to it.