38 research outputs found
Biproducts without pointedness
We show how to define biproducts up to isomorphism in an arbitrary category
without assuming any enrichment. The resulting notion coincides with the usual
definitions whenever all binary biproducts exist or the category is suitably
enriched, resulting in a modest yet strict generalization otherwise. We also
characterize when a category has all binary biproducts in terms of an
ambidextrous adjunction. Finally, we give some new examples of biproducts that
our definition recognizes.Comment: 6 page
A comonadic view of simulation and quantum resources
We study simulation and quantum resources in the setting of the
sheaf-theoretic approach to contextuality and non-locality. Resources are
viewed behaviourally, as empirical models. In earlier work, a notion of
morphism for these empirical models was proposed and studied. We generalize and
simplify the earlier approach, by starting with a very simple notion of
morphism, and then extending it to a more useful one by passing to a co-Kleisli
category with respect to a comonad of measurement protocols. We show that these
morphisms capture notions of simulation between empirical models obtained via
`free' operations in a resource theory of contextuality, including the type of
classical control used in measurement-based quantum computation schemes.Comment: To appear in Proceedings of LiCS 201
Uncloneable Quantum Advice
The famous no-cloning principle has been shown recently to enable a number of
uncloneable functionalities. Here we address for the first time unkeyed quantum
uncloneablity, via the study of a complexity-theoretic tool that enables a
computation, but that is natively unkeyed: quantum advice. Remarkably, this is
an application of the no-cloning principle in a context where the quantum
states of interest are not chosen by a random process. We show the
unconditional existence of promise problems admitting uncloneable quantum
advice, and the existence of languages with uncloneable advice, assuming the
feasibility of quantum copy-protecting certain functions. Along the way, we
note that state complexity classes, introduced by Rosenthal and Yuen (ITCS
2022) - which concern the computational difficulty of synthesizing sequences of
quantum states - can be naturally generalized to obtain state cloning
complexity classes. We make initial observations on these classes, notably
obtaining a result analogous to the existence of undecidable problems.
Our proof technique establishes the existence of ingenerable sequences of
finite bit strings - essentially meaning that they cannot be generated by any
uniform circuit family. We then prove a generic result showing that the
difficulty of accomplishing a computational task on uniformly random inputs
implies its difficulty on any fixed, ingenerable sequence. We use this result
to derandomize quantum cryptographic games that relate to cloning, and then
incorporate a result of Kundu and Tan (arXiv 2022) to obtain uncloneable
advice. Applying this two-step process to a monogamy-of-entanglement game
yields a promise problem with uncloneable advice, and applying it to the
quantum copy-protection of pseudorandom functions with super-logarithmic output
lengths yields a language with uncloneable advice.Comment: 58 pages, 6 figure
Reversible Effects as Inverse Arrows
Reversible computing models settings in which all processes can be reversed.
Applications include low-power computing, quantum computing, and robotics. It
is unclear how to represent side-effects in this setting, because conventional
methods need not respect reversibility. We model reversible effects by adapting
Hughes' arrows to dagger arrows and inverse arrows. This captures several
fundamental reversible effects, including serialization and mutable store
computations. Whereas arrows are monoids in the category of profunctors, dagger
arrows are involutive monoids in the category of profunctors, and inverse
arrows satisfy certain additional properties. These semantics inform the design
of functional reversible programs supporting side-effects.Comment: 15 pages; corrected Example 3.
Way of the dagger
A dagger category is a category equipped with a functorial way of reversing morphisms,
i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories
with additional structure have been studied under different names in categorical
quantum mechanics, algebraic field theory and homological algebra, amongst others.
In this thesis we study the dagger in its own right and show how basic category theory
adapts to dagger categories.
We develop a notion of a dagger limit that we show is suitable in the following
ways: it subsumes special cases known from the literature; dagger limits are unique
up to unitary isomorphism; a wide class of dagger limits can be built from a small
selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to
a diagonal functor; dagger limits can be built from ordinary limits in the presence of
polar decomposition; dagger limits commute with dagger colimits in many cases.
Using cofree dagger categories, the theory of dagger limits can be leveraged to
provide an enrichment-free understanding of limit-colimit coincidences in ordinary
category theory. We formalize the concept of an ambilimit, and show that it captures
known cases. As a special case, we show how to define biproducts up to isomorphism
in an arbitrary category without assuming any enrichment. Moreover, the limit-colimit
coincidence from domain theory can be generalized to the unenriched setting and we
show that, under suitable assumptions, a wide class of endofunctors has canonical fixed
points.
The theory of monads on dagger categories works best when all structure 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 use dagger categories to study reversible computing. Specifically, we model reversible
effects by adapting Hughes’ arrows to dagger arrows and inverse arrows. This
captures several fundamental reversible effects, including serialization and mutable
store computations. Whereas arrows are monoids in the category of profunctors, dagger
arrows are involutive monoids in the category of profunctors, and inverse arrows
satisfy certain additional properties. These semantics inform the design of functional
reversible programs supporting side-effects
Limits in dagger categories
We develop a notion of limit for dagger categories, that we show is suitable
in the following ways: it subsumes special cases known from the literature;
dagger limits are unique up to unitary isomorphism; a wide class of dagger
limits can be built from a small selection of them; dagger limits of a fixed
shape can be phrased as dagger adjoints to a diagonal functor; dagger limits
can be built from ordinary limits in the presence of polar decomposition;
dagger limits commute with dagger colimits in many cases
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
Metronidazole and ursodeoxycholic acid for primary sclerosing cholangitis : A randomized placebo-controlled trial
No effective medical therapy is currently available for primary sclerosing cholangitis (PSC). Ursodeoxycholic acid (UDCA) improves liver enzymes, but its effect on liver histology is controversial. Metronidazole (MTZ) prevents PSC-like liver damage in animal models and reduces intestinal permeability. We recruited 80 patients with PSC into a randomized placebo-controlled study to evaluate the effect of UDCA and MTZ (UDCA/MTZ) compared with UDCA/placebo on the progression of PSC. Patients (41 UDCA/placebo and 39 UDCA/ MTZ) were followed every third month. Assessment of liver function test, histological stage and grade, and cholangiography (via ERCP) at baseline showed no differences between the groups. After 36 months, serum aminotransferases gamma-glutamyltransferase, and alkaline phosphatase (ALP) decreased markedly in both groups, serum ALP more significantly in the UDCA/MTZ group (-337 +/- 54 U/L, PPeer reviewe
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