1,078 research outputs found
Filtered colimit elimination from Birkhoff's variety theorem
Birkhoff's variety theorem, a fundamental theorem of universal algebra,
asserts that a subclass of a given algebra is definable by equations if and
only if it satisfies specific closure properties. In a generalized version of
this theorem, closure under filtered colimits is required. However, in some
special cases, such as finite-sorted equational theories and ordered algebraic
theories, the theorem holds without assuming closure under filtered colimits.
We call this phenomenon "filtered colimit elimination," and study a sufficient
condition for it. We show that if a locally finitely presentable category
satisfies a noetherian-like condition, then filtered colimit
elimination holds in the generalized Birkhoff's theorem for algebras relative
to .Comment: 22 page
C*-Algebras over Topological Spaces: Filtrated K-Theory
We define the filtrated K-theory of a C*-algebra over a finite topological
space X and explain how to construct a spectral sequence that computes the
bivariant Kasparov theory over X in terms of filtrated K-theory. For finite
spaces with totally ordered lattice of open subsets, this spectral sequence
becomes an exact sequence as in the Universal Coefficient Theorem, with the
same consequences for classification. We also exhibit an example where
filtrated K-theory is not yet a complete invariant. We describe a space with
four points and two C*-algebras over this space in the bootstrap class that
have isomorphic filtrated K-theory but are not KK(X)-equivalent. For this
particular space, we enrich filtrated K-theory by another K-theory functor, so
that there is again a Universal Coefficient Theorem. Thus the enriched
filtrated K-theory is a complete invariant for purely infinite, stable
C*-algebras with this particular spectrum and belonging to the appropriate
bootstrap class.Comment: Changes to theorem and equation numbering
- …