1,523 research outputs found
Open Graphs and Monoidal Theories
String diagrams are a powerful tool for reasoning about physical processes,
logic circuits, tensor networks, and many other compositional structures. The
distinguishing feature of these diagrams is that edges need not be connected to
vertices at both ends, and these unconnected ends can be interpreted as the
inputs and outputs of a diagram. In this paper, we give a concrete construction
for string diagrams using a special kind of typed graph called an open-graph.
While the category of open-graphs is not itself adhesive, we introduce the
notion of a selective adhesive functor, and show that such a functor embeds the
category of open-graphs into the ambient adhesive category of typed graphs.
Using this functor, the category of open-graphs inherits "enough adhesivity"
from the category of typed graphs to perform double-pushout (DPO) graph
rewriting. A salient feature of our theory is that it ensures rewrite systems
are "type-safe" in the sense that rewriting respects the inputs and outputs.
This formalism lets us safely encode the interesting structure of a
computational model, such as evaluation dynamics, with succinct, explicit
rewrite rules, while the graphical representation absorbs many of the tedious
details. Although topological formalisms exist for string diagrams, our
construction is discreet, finitary, and enjoys decidable algorithms for
composition and rewriting. We also show how open-graphs can be parametrised by
graphical signatures, similar to the monoidal signatures of Joyal and Street,
which define types for vertices in the diagrammatic language and constraints on
how they can be connected. Using typed open-graphs, we can construct free
symmetric monoidal categories, PROPs, and more general monoidal theories. Thus
open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting
review
On string topology of classifying spaces
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of
the space of free loops in the classifying space of G is known to be the value
on the circle in a homological conformal field theory. This means in particular
that it admits operations parameterized by homology classes of classifying
spaces of diffeomorphism groups of surfaces. Here we present a radical
extension of this result, giving a new construction in which diffeomorphisms
are replaced with homotopy equivalences, and surfaces with boundary are
replaced with arbitrary spaces homotopy equivalent to finite graphs. The result
is a novel kind of field theory which is related to both the diffeomorphism
groups of surfaces and the automorphism groups of free groups with boundaries.
Our work shows that the algebraic structures in string topology of classifying
spaces can be brought into line with, and in fact far exceed, those available
in string topology of manifolds. For simplicity, we restrict to the
characteristic 2 case. The generalization to arbitrary characteristic will be
addressed in a subsequent paper.Comment: 93 pages; v4: minor changes; to appear in Advances in Mathematic
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
Topological conformal field theories and gauge theories
This paper gives a construction, using heat kernels, of differential forms on
the moduli space of metrised ribbon graphs, or equivalently on the moduli space
of Riemann surfaces with boundary. The construction depends on a manifold with
a bundle of Frobenius algebras, satisfying various conditions. These forms
satisfy gluing conditions which mean they form an open topological conformal
field theory, i.e. a kind of open string theory.
If the integral of these forms converged, it would yield the purely quantum
part of the partition function of a Chern-Simons type gauge theory. Yang-Mills
theory on a four manifold arises as one of these Chern-Simons type gauge
theories.Comment: A few more typos correcte
Edge contraction on dual ribbon graphs and 2D TQFT
We present a new set of axioms for 2D TQFT formulated on the category of cell
graphs with edge-contraction operations as morphisms. We construct a functor
from this category to the endofunctor category consisting of Frobenius
algebras. Edge-contraction operations correspond to natural transformations of
endofunctors, which are compatible with the Frobenius algebra structure. Given
a Frobenius algebra A, every cell graph determines an element of the symmetric
tensor algebra defined over the dual space A*. We show that the
edge-contraction axioms make this assignment depending only on the topological
type of the cell graph, but not on the graph itself. Thus the functor generates
the TQFT corresponding to A.Comment: accepted in Journal of Algebra (22 pages, 13 figures
Feynman Categories
In this paper we give a new foundational, categorical formulation for
operations and relations and objects parameterizing them. This generalizes and
unifies the theory of operads and all their cousins including but not limited
to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads,
their colored versions, as well as algebras over operads and an abundance of
other related structures, such as crossed simplicial groups, the augmented
simplicial category or FI--modules.
The usefulness of this approach is that it allows us to handle all the
classical as well as more esoteric structures under a common framework and we
can treat all the situations simultaneously. Many of the known constructions
simply become Kan extensions.
In this common framework, we also derive universal operations, such as those
underlying Deligne's conjecture, construct Hopf algebras as well as perform
resolutions, (co)bar transforms and Feynman transforms which are related to
master equations. For these applications, we construct the relevant model
category structures. This produces many new examples.Comment: Expanded version. New introduction, new arrangement of text, more
details on several constructions. New figure
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