1,505 research outputs found
Ind- and Pro- definable sets
We describe the ind- and pro- categories of the category of definable sets,
in some first order theory, in terms of points in a sufficiently saturated
model.Comment: 8 pages; Part of author's phd thesi
Mac Lane method in the investigation of magnetic translation groups
Central extensions of the three-dimensional translation group T=Z^3 by the
unitary group U(1) (a group of factors) are considered within the frame of the
Mac~Lane method. All nonzero vectors t in T are considered to be generators of
T. This choice leads to very illustrative relations between the Mac~Lane method
and Zak's approach to magnetic translation groups. It is shown that factor
systems introduced by Zak and Brown can be realized only for the unitary group
U(1) and for some of its finite subgroups.Comment: 8 pages, 1 fig. in text, romp_sty.tex attached at the beginning
Presented at 28 Symp. on Math. Phys., Torun 2-6 Dec 199
Finite Products are Biproducts in a Compact Closed Category
If a compact closed category has finite products or finite coproducts then it
in fact has finite biproducts, and so is semi-additive.Comment: 9 pages. Introduction further expanded, minor errors correcte
Database queries and constraints via lifting problems
Previous work has demonstrated that categories are useful and expressive
models for databases. In the present paper we build on that model, showing that
certain queries and constraints correspond to lifting problems, as found in
modern approaches to algebraic topology. In our formulation, each so-called
SPARQL graph pattern query corresponds to a category-theoretic lifting problem,
whereby the set of solutions to the query is precisely the set of lifts. We
interpret constraints within the same formalism and then investigate some basic
properties of queries and constraints. In particular, to any database we
can associate a certain derived database \Qry(\pi) of queries on . As an
application, we explain how giving users access to certain parts of
\Qry(\pi), rather than direct access to , improves ones ability to
manage the impact of schema evolution
Finite domination and Novikov rings. Iterative approach
Suppose C is a bounded chain complex of finitely generated free modules over
the Laurent polynomial ring L = R[x,1/x]. Then C is R-finitely dominated, ie,
homotopy equivalent over R to a bounded chain complex of finitely generated
projective R-modules, if and only if the two chain complexes C((x)) and
C((1/x)) are acyclic, as has been proved by Ranicki. Here C((x)) is the tensor
product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as
the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor
product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x].
In this paper, we prove a generalisation of this criterion which allows us to
detect finite domination of bounded below chain complexes of projective modules
over Laurent rings in several indeterminates.Comment: 15 pages; diagrams typeset with Paul Taylor's "diagrams" macro
package. Version 2: clarified proof of main theorem, fixed minor typos;
Version 3: expanded introduction, now 16 pages; Version 4: corrected mistake
on functoriality of mapping tor
Spherical Categories
This paper is a study of monoidal categories with duals where the tensor
product need not be commutative. The motivating examples are categories of
representations of Hopf algebras and the motivating application is the
definition of 6j-symbols as used in topological field theories.
We introduce the new notion of a spherical category. In the first section we
prove a coherence theorem for a monoidal category with duals following MacLane
(1963). In the second section we give the definition of a spherical category,
and construct a natural quotient which is also spherical.
In the third section we define spherical Hopf algebras so that the category
of representations is spherical. Examples of spherical Hopf algebras are
involutory Hopf algebras and ribbon Hopf algebras. Finally we study the natural
quotient in these cases and show it is semisimple.Comment: 16 pages. Minor correction
Explicit tensor network representation for the ground states of string-net models
The structure of string-net lattice models, relevant as examples of
topological phases, leads to a remarkably simple way of expressing their ground
states as a tensor network constructed from the basic data of the underlying
tensor categories. The construction highlights the importance of the fat
lattice to understand these models.Comment: 5 pages, pdf figure
Magnetic translation groups as group extension
Extensions of a direct product T of two cyclic groups Z_n1 and Z_n2 by an
Abelian (gauge) group G with the trivial action of T on G are considered. All
possible (nonequivalent) factor systems are determined using the Mac Lane
method. Some of resulting groups describe magnetic translation groups. As
examples extensions with G=U(1) and G=Z_n are considered and discussed.Comment: 10 page
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