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Preservation of Equations by Monoidal Monads

Louis Parlant, Jurriaan Rot, Alexandra Silva, Bas Westerbaan, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
It has already been shown that any monoidal monad preserves linear equations; affine monads preserve drop equations (where some variable appears only on one side, such as x⋅ y = y) and relevant monads  ...  Finally, we identify a subclass of equations such that their preservation is equivalent to relevance.  ...  Later, Manes and Mulry show in [14] and [15] that a correspondence between categorical properties of one monad and algebraic features of another may lead to a distributive law.  ... 
doi:10.4230/lipics.mfcs.2020.77 dblp:conf/mfcs/ParlantR0W20 fatcat:d4mclyzjwvgbzfgfz7fb3syuf4

Quotients in monadic programming: Projective algebras are equivalent to coalgebras [article]

Dusko Pavlovic, Peter-Michael Seidel
2017 arXiv   pre-print
of projective algebras is equivalent with the category of coalgebras for the comonad induced by any monad resolution.  ...  Both equivalences also entail several general corollaries concerning monadicity and comonadicity.  ...  1 Introduction The story Background: Monadic programming. Monads are one of functional programmers' favorite tools, and possibly one of the most popular categorical concepts [14, 13] .  ... 
arXiv:1701.07601v3 fatcat:cp6zezf4zzferertdbaw4qx3sm

Homotopy Theory of T-algebras over Top-Cat ? [article]

Ilias Amrani
2012 arXiv   pre-print
We discuss some properties and consequences of such path object. We also explain the construction of a 2-monad which algebras are (symmetric) monoidal topological categories.  ...  In this article, we interconnect two different aspects of higher category theory, in one hand the theory of infinity categories and on an other hand the theory of 2-categories.We construct an explicit  ...  Actually, this equality holds only in the case where r = s and z = v. Eventual model structure on T -algebras. This section si purely conjectural.  ... 
arXiv:1205.5376v1 fatcat:hm5yszqzb5hqromcteb2etsnc4

Polynomial monads and delooping of mapping spaces [article]

Michael Batanin, Florian De Leger
2020 arXiv   pre-print
small categories to polynomial monads and their algebras.  ...  As an application we give a categorical proof of the Dwyer-Hess and Turchin results concerning the explicit double delooping of spaces of long knots.  ...  The first author also gratefully acknowledges the financial support of Max Planck Institut für Mathematik. Bibliography.  ... 
arXiv:1712.00904v4 fatcat:7lfs2uccvfajpefphdt7654ep4

Instances of Computational Effects: An Algebraic Perspective

Sam Staton
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science  
We investigate the connections between computational effects, algebraic theories, and monads on functor categories.  ...  We use our framework to give a general account of several notions of computation that had previously been analyzed in terms of monads on presheaf categories: the analysis of local store by Plotkin and  ...  The development of the framework is guided by the categorical foundation of enriched algebraic theories [30] , building on recent developments by Lack and Rosický [32] , Melliès et al. ([37] , [38]  ... 
doi:10.1109/lics.2013.58 dblp:conf/lics/Staton13 fatcat:gpnz4fipongvbkmnrd3mtjknpi

Preservation of Equations by Monoidal Monads [article]

Louis Parlant, Jurriaan Rot, Alexandra Silva, Bas Westerbaan
2020 arXiv   pre-print
It has already been shown that any monoidal monad preserves linear equations; affine monads preserve drop equations (where some variable appears only on one side, such as x· y = y) and relevant monads  ...  Finally, we identify the subclass of equations such that their preservation is equivalent to relevance.  ...  Later, Manes and Mulry show in [16] and [17] that a correspondence between categorical properties of one monad and algebraic features of another may lead to a distributive law.  ... 
arXiv:2001.06348v2 fatcat:nfvoeeuusbfmlafhz5jrb72pg4

Duality Theory and Categorical Universal Logic: With Emphasis on Quantum Structures

Yoshihiro Maruyama
2014 Electronic Proceedings in Theoretical Computer Science  
Categorical Universal Logic is a theory of monad-relativised hyperdoctrines (or fibred universal algebras), which in particular encompasses categorical forms of both first-order and higher-order quantum  ...  We finally consider how to reconcile Birkhoff-von Neumann's quantum logic and Abramsky-Coecke's categorical quantum mechanics (which is modernised quantum logic as an antithesis to the traditional one)  ...  Both OML and EA are algebraic categories, i.e., can be described as categories of algebras of monads on Set.  ... 
doi:10.4204/eptcs.171.10 fatcat:cpjr7g6yh5b77pkow4gzgzts2e

On semiflexible, flexible and pie algebras

John Bourke, Richard Garner
2013 Journal of Pure and Applied Algebra  
We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones.  ...  Clearly, to speak of an algebra for a 2-monad's being "free at the level of objects" is vague; and even if made precise, it would be insufficiently general, limiting us to 2-monads on Cat and similarly  ...  In the case that K is cocomplete, we may view [J, K] as the algebras for a 2-monad on C = [ob J, K], and now the pointwise equivalences therein are equally well the algebra maps which become equivalences  ... 
doi:10.1016/j.jpaa.2012.06.002 fatcat:uvxqgschlfegbj2moja63gqwae

On semiflexible, flexible and pie algebras [article]

John Bourke, Richard Garner
2011 arXiv   pre-print
We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones.  ...  Pie algebras are contrasted with flexible and semiflexible algebras via a series of characterisations of each class; particular attention is paid to the case of pie, flexible and semiflexible weights,  ...  In the case that K is cocomplete, we may view [J, K] as the algebras for a 2-monad on C = [ob J, K], and now the pointwise equivalences therein are equally well the algebra maps which become equivalences  ... 
arXiv:1112.1448v1 fatcat:x7cikjianzhgxbl3hunk7h6qf4

A Categorical Construction of the Real Unit Interval [article]

John van de Wetering
2021 arXiv   pre-print
The algebras of this monad carry an order, multiplication, addition and complement, and as such model much of the operations we need to do on probabilities.  ...  We show that the real unit interval is the unique non-initial, non-final irreducible algebra of a particular monad on the category of bounded posets.  ...  Acknowledgements The author would like to thank Bas and Bram Westerbaan for insightful discussions. The author is supported by a Rubicon fellowship financed by the Dutch Research Council (NWO).  ... 
arXiv:2106.10094v1 fatcat:siiwkgvlsfggjm7s4gknvscuta

Heterogeneous substitution systems revisited [article]

Benedikt Ahrens, Ralph Matthes
2016 arXiv   pre-print
we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover.  ...  Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is  ...  Thanks to Paige North for discussion of the subject matter, and to Anders Mörtberg for providing feedback to a draft of this article.  ... 
arXiv:1601.04299v1 fatcat:7avpaxbofrb47pblvajndvimuy

Reversible Monadic Computing

Chris Heunen, Martti Karvonen
2015 Electronical Notes in Theoretical Computer Science  
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  ...  We 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  ...  Each side of the Frobenius law (1) equals ( ) † • ; one of these equations is equivalent to (1) .  ... 
doi:10.1016/j.entcs.2015.12.014 fatcat:7djszc3rrzep5d3l4q2mjdfmv4

Extensional Universal Types for Call-by-Value [chapter]

Kazuyuki Asada
2008 Lecture Notes in Computer Science  
., β-equality and ηequality hold for not only values but all terms. We give monadic style categorical semantics, so that the results can be applied also to languages like Haskell.  ...  We give syntax and its sound and complete categorical semantics, and describe how we can construct concrete models with a variety of computational effects.  ...  Acknowledgements I would like to thank Masahito Hasegawa for many helpful discussions and comments on earlier drafts.  ... 
doi:10.1007/978-3-540-89330-1_9 fatcat:icvmldilfzgm7jmgpo3thtwxv4

Kleisli category and database mappings

Zoran Majkic, Bhanu Prasad
2010 International Journal of Intelligent Information and Database Systems  
This paper presents the semantics of database mappings in the relational database (DB) category, based on the power-view monad T and monadic algebras.  ...  Consequently, the semantics of database mappings in this DB category are defined based on the power-view monad T and the Kleisli category for databases, which can be "internalized" in this basic DB category  ...  Categorial symmetry and behavioral equivalence Let us consider the problem of how to define equivalent (categorically isomorphic) objects (database instances) from a behavioral point of view based on observations  ... 
doi:10.1504/ijiids.2010.035774 fatcat:zo3uprm35jhrhesqcwwtuuket4

Healthiness from Duality

Wataru Hino, Hiroki Kobayashi, Ichiro Hasuo, Bart Jacobs
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
This framework is based on a dual adjunction induced by a dualizing object and on our notion of relative Eilenberg-Moore algebra.  ...  Building upon our previous works on socalled state-and-effect triangles, we contribute a unified categorical framework for investigating healthiness conditions.  ...  Special thanks are due to John Power for the lectures he gave on the occasion of his visit to Tokyo; the notion of relative algebra is inspired by them.  ... 
doi:10.1145/2933575.2935319 dblp:conf/lics/HinoKH016 fatcat:odnri772tnhkpfcneqj23idskq
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