18 research outputs found
Involutive Categories and Monoids, with a GNS-correspondence
This paper develops the basics of the theory of involutive categories and
shows that such categories provide the natural setting in which to describe
involutive monoids. It is shown how categories of Eilenberg-Moore algebras of
involutive monads are involutive, with conjugation for modules and vector
spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS)
construction is identified as a bijective correspondence between states on
involutive monoids and inner products. This correspondence exists in arbritrary
involutive categories
Category Algebras and States on Categories
The purpose of this paper is to build a new bridge between category theory
and a generalized probability theory known as noncommutative probability or
quantum probability, which was originated as a mathematical framework for
quantum theory, in terms of states as linear functional defined on category
algebras. We clarify that category algebras can be considered as generalized
matrix algebras and that the notions of state on category as linear functional
defined on category algebra turns out to be a conceptual generalization of
probability measures on sets as discrete categories. Moreover, by establishing
a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain
a representation of category algebras of -categories on certain
generalized Hilbert spaces which we call semi-Hilbert modules over rigs.Comment: 16 page
Relating Operator Spaces via Adjunctions
This chapter uses categorical techniques to describe relations between
various sets of operators on a Hilbert space, such as self-adjoint, positive,
density, effect and projection operators. These relations, including various
Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual
adjunctions, and maps between them. Of particular interest is the connection
with quantum structures, via a dual adjunction between convex sets and effect
modules. The approach systematically uses categories of modules, via their
description as Eilenberg-Moore algebras of a monad
Proof Orders for Decreasing Diagrams
We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders differ with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo
Inversion, Iteration, and the Art of Dual Wielding
The humble ("dagger") is used to denote two different operations in
category theory: Taking the adjoint of a morphism (in dagger categories) and
finding the least fixed point of a functional (in categories enriched in
domains). While these two operations are usually considered separately from one
another, the emergence of reversible notions of computation shows the need to
consider how the two ought to interact. In the present paper, we wield both of
these daggers at once and consider dagger categories enriched in domains. We
develop a notion of a monotone dagger structure as a dagger structure that is
well behaved with respect to the enrichment, and show that such a structure
leads to pleasant inversion properties of the fixed points that arise as a
result. Notably, such a structure guarantees the existence of fixed point
adjoints, which we show are intimately related to the conjugates arising from a
canonical involutive monoidal structure in the enrichment. Finally, we relate
the results to applications in the design and semantics of reversible
programming languages.Comment: Accepted for RC 201
Reversible Monadic Computing
AbstractWe extend categorical semantics of monadic programming to reversible computing, by considering monoidal closed dagger categories: the dagger gives reversibility, whereas closure gives higher-order expressivity. We demonstrate that Frobenius monads model the appropriate notion of coherence between the dagger and closure by reinforcing Cayley's theorem; by proving that effectful computations (Kleisli morphisms) are reversible precisely when the monad is Frobenius; by characterizing the largest reversible subcategory of Eilenberg–Moore algebras; and by identifying the latter algebras as measurements in our leading example of quantum computing. Strong Frobenius monads are characterized internally by Frobenius monoids
Monads on Dagger Categories
The theory of monads on categories equipped with a dagger (a contravariant
identity-on-objects involutive endofunctor) works best when everything respects
the dagger: the monad and adjunctions should preserve the dagger, and the monad
and its algebras should satisfy the so-called Frobenius law. Then any monad
resolves as an adjunction, with extremal solutions given by the categories of
Kleisli and Frobenius-Eilenberg-Moore algebras, which again have a dagger. We
characterize the Frobenius law as a coherence property between dagger and
closure, and characterize strong such monads as being induced by Frobenius
monoids.Comment: 28 page
Dagger Categories of Tame Relations
Within the context of an involutive monoidal category the notion of a
comparison relation is identified. Instances are equality on sets, inequality
on posets, orthogonality on orthomodular lattices, non-empty intersection on
powersets, and inner product on vector or Hilbert spaces. Associated with a
collection of such (symmetric) comparison relations a dagger category is
defined with "tame" relations as morphisms. Examples include familiar
categories in the foundations of quantum mechanics, such as sets with partial
injections, or with locally bifinite relations, or with formal distributions
between them, or Hilbert spaces with bounded (continuous) linear maps. Of one
particular example of such a dagger category of tame relations, involving sets
and bifinite multirelations between them, the categorical structure is
investigated in some detail. It turns out to involve symmetric monoidal dagger
structure, with biproducts, and dagger kernels. This category may form an
appropriate universe for discrete quantum computations, just like Hilbert
spaces form a universe for continuous computation
Algebraic Rieffel Induction, Formal Morita Equivalence, and Applications to Deformation Quantization
In this paper we consider algebras with involution over a ring C which is
given by the quadratic extension by i of an ordered ring R. We discuss the
*-representation theory of such *-algebras on pre-Hilbert spaces over C and
develop the notions of Rieffel induction and formal Morita equivalence for this
category analogously to the situation for C^*-algebras. Throughout this paper
the notion of positive functionals and positive algebra elements will be
crucial for all constructions. As in the case of C^*-algebras, we show that the
GNS construction of *-representations can be understood as Rieffel induction
and, moreover, that formal Morita equivalence of two *-algebras, which is
defined by the existence of a bimodule with certain additional structures,
implies the equivalence of the categories of strongly non-degenerate
*-representations of the two *-algebras. We discuss various examples like
finite rank operators on pre-Hilbert spaces and matrix algebras over
*-algebras. Formal Morita equivalence is shown to imply Morita equivalence in
the ring-theoretic framework. Finally we apply our considerations to
deformation theory and in particular to deformation quantization and discuss
the classical limit and the deformation of equivalence bimodules.Comment: LaTeX2e, 51pages, minor typos corrected and Note/references adde