Author
Martti Karvonen
Other affiliations: Aalto University, University of Ottawa
Bio: Martti Karvonen is an academic researcher from University of Edinburgh. The author has contributed to research in topics: Morphism & Monad (functional programming). The author has an hindex of 7, co-authored 17 publications receiving 108 citations. Previous affiliations of Martti Karvonen include Aalto University & University of Ottawa.
Papers
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TL;DR: 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 monads and its algebras should satisfy the Frobenius law.
Abstract: 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.
24 citations
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TL;DR: 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 monads and its algebras should satisfy the Frobenius law.
Abstract: 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.
18 citations
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24 Jun 2019TL;DR: In this paper, the authors study simulation and quantum resources in the setting of the sheaf-theoretic approach to contextuality and nonlocality, where resources are viewed behaviourally, as empirical models.
Abstract: We study simulation and quantum resources in the setting of the sheaf-theoretic approach to contextuality and nonlocality. 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.
15 citations
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TL;DR: In this paper, a notion of a dagger limit is proposed, which is suitable for a wide class of dagger categories up to unitary isomorphism, and can be expressed as dagger adjoints to a diagonal functor.
Abstract: 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 e.g. in categorical quantum mechanics and algebraic field theory. 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.
12 citations
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TL;DR: Frobenius monads model the appropriate notion of coherence between the dagger and closure by reinforcing Cayley's theorem and proving that effectful computations are reversible precisely when the monad is Frobenius.
10 citations
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521 citations
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TL;DR: In this paper, the authors consider functions f: I → ℝ, where I ⊆ ℩ is a special kind of subset, called an interval, and of one of the forms.
Abstract: We shall be particularly concerned with functions f: I → ℝ, where I ⊆ ℝ is a special kind of subset, called an interval, and of one of the forms.
156 citations
01 Jan 2016
TL;DR: The the theory of categories is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can download it instantly.
Abstract: Thank you very much for downloading the theory of categories. Maybe you have knowledge that, people have look hundreds times for their favorite readings like this the theory of categories, but end up in infectious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they cope with some harmful bugs inside their computer. the theory of categories is available in our book collection an online access to it is set as public so you can download it instantly. Our book servers spans in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the the theory of categories is universally compatible with any devices to read.
87 citations
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TL;DR: In this article, a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions is introduced, and dualizable and invertible 1-morphisms in these 2-categories are analyzed.
Abstract: We introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple. We analyze dualisable and invertible 1-morphisms in these 2-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.
48 citations