576 research outputs found
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
Introduction to Categories and Categorical Logic
The aim of these notes is to provide a succinct, accessible introduction to
some of the basic ideas of category theory and categorical logic. The notes are
based on a lecture course given at Oxford over the past few years. They contain
numerous exercises, and hopefully will prove useful for self-study by those
seeking a first introduction to the subject, with fairly minimal prerequisites.
The coverage is by no means comprehensive, but should provide a good basis for
further study; a guide to further reading is included. The main prerequisite is
a basic familiarity with the elements of discrete mathematics: sets, relations
and functions. An Appendix contains a summary of what we will need, and it may
be useful to review this first. In addition, some prior exposure to abstract
algebra - vector spaces and linear maps, or groups and group homomorphisms -
would be helpful.Comment: 96 page
Endomorphisms and automorphisms of locally covariant quantum field theories
In the framework of locally covariant quantum field theory, a theory is
described as a functor from a category of spacetimes to a category of
*-algebras. It is proposed that the global gauge group of such a theory can be
identified as the group of automorphisms of the defining functor. Consequently,
multiplets of fields may be identified at the functorial level. It is shown
that locally covariant theories that obey standard assumptions in Minkowski
space, including energy compactness, have no proper endomorphisms (i.e., all
endomorphisms are automorphisms) and have a compact automorphism group.
Further, it is shown how the endomorphisms and automorphisms of a locally
covariant theory may, in principle, be classified in any single spacetime. As
an example, the endomorphisms and automorphisms of a system of finitely many
free scalar fields are completely classified.Comment: v2 45pp, expanded to include additional results; presentation
improved and an error corrected. To appear in Rev Math Phy
Causal sites as quantum geometry
We propose a structure called a causal site to use as a setting for quantum
geometry, replacing the underlying point set. The structure has an interesting
categorical form, and a natural "tangent 2-bundle," analogous to the tangent
bundle of a smooth manifold. Examples with reasonable finiteness conditions
have an intrinsic geometry, which can approximate classical solutions to
general relativity. We propose an approach to quantization of causal sites as
well.Comment: 21 pages, 3 eps figures; v2: added references; to appear in JM
Higher-dimensional Algebra and Topological Quantum Field Theory
The study of topological quantum field theories increasingly relies upon
concepts from higher-dimensional algebra such as n-categories and n-vector
spaces. We review progress towards a definition of n-category suited for this
purpose, and outline a program in which n-dimensional TQFTs are to be described
as n-category representations. First we describe a "suspension" operation on
n-categories, and hypothesize that the k-fold suspension of a weak n-category
stabilizes for k >= n+2. We give evidence for this hypothesis and describe its
relation to stable homotopy theory. We then propose a description of
n-dimensional unitary extended TQFTs as weak n-functors from the "free stable
weak n-category with duals on one object" to the n-category of "n-Hilbert
spaces". We conclude by describing n-categorical generalizations of deformation
quantization and the quantum double construction.Comment: 36 pages, LaTeX; this version includes all 36 figure
A categorical framework for the quantum harmonic oscillator
This paper describes how the structure of the state space of the quantum
harmonic oscillator can be described by an adjunction of categories, that
encodes the raising and lowering operators into a commutative comonoid. The
formulation is an entirely general one in which Hilbert spaces play no special
role. Generalised coherent states arise through the hom-set isomorphisms
defining the adjunction, and we prove that they are eigenstates of the lowering
operators. Surprisingly, generalised exponentials also emerge naturally in this
setting, and we demonstrate that coherent states are produced by the
exponential of a raising morphism acting on the zero-particle state. Finally,
we examine all of these constructions in a suitable category of Hilbert spaces,
and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure
Abstract Tensor Systems as Monoidal Categories
The primary contribution of this paper is to give a formal, categorical
treatment to Penrose's abstract tensor notation, in the context of traced
symmetric monoidal categories. To do so, we introduce a typed, sum-free version
of an abstract tensor system and demonstrate the construction of its associated
category. We then show that the associated category of the free abstract tensor
system is in fact the free traced symmetric monoidal category on a monoidal
signature. A notable consequence of this result is a simple proof for the
soundness and completeness of the diagrammatic language for traced symmetric
monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda
Algebras for parameterised monads
Parameterised monads have the same relationship to adjunctions with parameters as monads do to adjunctions. In this paper, we investigate algebras for parameterised monads. We identify the Eilenberg-Moore category of algebras for parameterised monads and prove a generalisation of Beck’s theorem characterising this category. We demonstrate an application of this theory to the semantics of type and effect systems
Monoids, Embedding Functors and Quantum Groups
We show that the left regular representation \pi_l of a discrete quantum
group (A,\Delta) has the absorbing property and forms a monoid
(\pi_l,\tilde{m},\tilde{\eta}) in the representation category Rep(A,\Delta).
Next we show that an absorbing monoid in an abstract tensor *-category C gives
rise to an embedding functor E:C->Vect_C, and we identify conditions on the
monoid, satisfied by (\pi_l,\tilde{m},\tilde{\eta}), implying that E is
*-preserving. As is well-known, from an embedding functor E: C->\mathrm{Hilb}
the generalized Tannaka theorem produces a discrete quantum group (A,\Delta)
such that C is equivalent to Rep_f(A,\Delta). Thus, for a C^*-tensor category C
with conjugates and irreducible unit the following are equivalent: (1) C is
equivalent to the representation category of a discrete quantum group
(A,\Delta), (2) C admits an absorbing monoid, (3) there exists a *-preserving
embedding functor E: C->\mathrm{Hilb}.Comment: Final version, to appear in Int. Journ. Math. (Added some references
and Subsection 1.2.) Latex2e, 21 page
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