103,530 research outputs found
On properties of (weakly) small groups
A group is small if it has countably many complete -types over the empty
set for each natural number n. More generally, a group is weakly small if
it has countably many complete 1-types over every finite subset of G. We show
here that in a weakly small group, subgroups which are definable with
parameters lying in a finitely generated algebraic closure satisfy the
descending chain conditions for their traces in any finitely generated
algebraic closure. An infinite weakly small group has an infinite abelian
subgroup, which may not be definable. A small nilpotent group is the central
product of a definable divisible group with a definable one of bounded
exponent. In a group with simple theory, any set of pairwise commuting elements
is contained in a definable finite-by-abelian subgroup. First corollary : a
weakly small group with simple theory has an infinite definable
finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal
solvable group A of derived length n is contained in an A-definable almost
solvable group of class n
Pauli topological subsystem codes from Abelian anyon theories
We construct Pauli topological subsystem codes characterized by arbitrary
two-dimensional Abelian anyon theories--this includes anyon theories with
degenerate braiding relations and those without a gapped boundary to the
vacuum. Our work both extends the classification of two-dimensional Pauli
topological subsystem codes to systems of composite-dimensional qudits and
establishes that the classification is at least as rich as that of Abelian
anyon theories. We exemplify the construction with topological subsystem codes
defined on four-dimensional qudits based on the anyon
theory with degenerate braiding relations and the chiral semion theory--both of
which cannot be captured by topological stabilizer codes. The construction
proceeds by "gauging out" certain anyon types of a topological stabilizer code.
This amounts to defining a gauge group generated by the stabilizer group of the
topological stabilizer code and a set of anyonic string operators for the anyon
types that are gauged out. The resulting topological subsystem code is
characterized by an anyon theory containing a proper subset of the anyons of
the topological stabilizer code. We thereby show that every Abelian anyon
theory is a subtheory of a stack of toric codes and a certain family of twisted
quantum doubles that generalize the double semion anyon theory. We further
prove a number of general statements about the logical operators of translation
invariant topological subsystem codes and define their associated anyon
theories in terms of higher-form symmetries.Comment: 67 + 35 pages, single column forma
Morita homotopy theory of C*-categories
In this article we establish the foundations of the Morita homotopy theory of
C*-categories. Concretely, we construct a cofibrantly generated simplicial
symmetric monoidal Quillen model structure M_Mor on the category C*cat1 of
small unital C*-categories. The weak equivalences are the Morita equivalences
and the cofibrations are the *-functors which are injective on objects. As an
application, we obtain an elegant description of the Brown-Green-Rieffel Picard
group in the associated Morita homotopy category Ho(M_Mor). We then prove that
the Morita homotopy category is semi-additive. By group completing the induced
abelian monoid structure at each Hom-set we obtain an additive category
Ho(M_Mor)^{-1} and a canonical functor C*cat1 {\to} Ho(M_Mor)^{-1} which is
characterized by two simple properties: inversion of Morita equivalences and
preservation of all finite products. Finally, we prove that the classical
Grothendieck group functor becomes co-represented in Ho(M_Mor)^{-1} by the
tensor unit object.Comment: 35 page
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