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Presenting Morphisms of Distributive Laws

Bartek Klin, Beata Nachyla, Marc Herbstritt
2015 Conference on Algebra and Coalgebra in Computer Science  
A format for well-behaved translations between structural operational specifications is derived from a notion of distributive law morphism, previously studied by Power and Watanabe.  ...  A simple distributive law of an endofunctor Σ over an endofunctor B is a natural transformation λ : ΣB =⇒ BΣ.  ...  So far in this paper we only treated simple SOS, to explain the general idea of presenting distributive law morphisms in a relatively basic setting.  ... 
doi:10.4230/lipics.calco.2015.190 dblp:conf/calco/KlinN15 fatcat:aabse6lr5bcyziargok7lytbai

A unified treatment of structural definitions on syntax for capture-avoiding substitution, context application, named substitution, partial differentiation, and so on [article]

Tom Hirschowitz, Ambroise Lafont
2022 arXiv   pre-print
We introduce a category-theoreticabstraction of a syntax with auxiliary functions, called an admissiblemonad morphism.  ...  Relying on an abstract form of structural recursion,we then design generic tools to construct admissible monad morphismsfrom basic data.  ...  From distributive laws Let us now show how to construct admissible monad morphisms from distributive laws.  ... 
arXiv:2204.03870v1 fatcat:fkldci7cnjdmlicsnqmsscugai

Presenting Distributive Laws

Marcello Bonsangue, Helle Hansen, Alexander Kurz, Jurriaan Rot, Stefan Milius
2015 Logical Methods in Computer Science  
In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad.  ...  Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction.  ...  Thus, from Proposition 4.10 and Corollary 4.6 we obtain a distributive law κ of P ω ((−) * ) over Id × F such that i • q : λ ⇒ κ is a morphism of distributive laws, i.e., κ is presented by λ and the equations  ... 
doi:10.2168/lmcs-11(3:2)2015 fatcat:s4rlgr4wkrhsjde2hzwxnzicva

Presenting Distributive Laws [chapter]

Marcello M. Bonsangue, Helle Hvid Hansen, Alexander Kurz, Jurriaan Rot
2013 Lecture Notes in Computer Science  
In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad.  ...  Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction.  ...  Thus, from Proposition 3 and Corollary 1 we obtain a distributive law κ of P ω (Id * ) over 2 × Id A such that i • q : λ ⇒ κ is a morphism of distributive laws, i.e., κ is presented by λ and the equations  ... 
doi:10.1007/978-3-642-40206-7_9 fatcat:i3bxe3y3n5cctkdzbv2fzquffu

Structural Operational Semantics and Modal Logic, Revisited

Bartek Klin
2010 Electronical Notes in Theoretical Computer Science  
Secondly, a more concrete understanding of the assumptions of the theorem is provided, where proving compositionality amounts to finding a syntactic distributive law between two collections of predicate  ...  Firstly, a more abstract, conceptual proof of the main compositionality theorem is given, based on an understanding of modal logic as a study of coalgebras in slice categories of adjunctions.  ...  Connections ρ and ζ are simply morphisms of endofunctors, just as λ and χ are endomorphisms on them; also sliced distributive laws χ λ are distributive law morphisms in the sense of [24] .  ... 
doi:10.1016/j.entcs.2010.07.019 fatcat:5ndmofo7l5h6nfnod5ilzfxgdu

Factorisations of distributive laws

Ulrich Krähmer, Paul Slevin
2016 Journal of Pure and Applied Algebra  
Here we define the category of factorisations of a distributive law, show that it acts on this construction, and give some explicit examples.  ...  Kassel MSC: 16T05; 18C15; 18C20; 19D55 Recently, Böhm and Ştefan constructed duplicial (paracyclic) objects from distributive laws between (co)monads.  ...  The aim of the present paper is to extend this procedure to any duplicial object defined by a distributive law.  ... 
doi:10.1016/j.jpaa.2015.09.008 fatcat:sf7n7yitwnccpg6dd4574epfjq

The Tensor Product of Triples as Multilinear Product

M.A. López López, NISTAL J.L. Freire
1993 Revista Matemática Complutense  
In this paper we introduce a notion of multilinear product ¡br triples in Set, which if it is given by a distributive law Ihen coincides with the one given by Hunge.  ...  Wc also demonstrate that the tensor product of two triples, if there exist, is an initial object jo a suitable category of multilinear products.  ...  The aim of the present paper is to introduce a notion of multilinear product for triples in Set, which if it is given by a distributive law then coincides with the one given by Bunge, ami to demonstrate  ... 
doi:10.5209/rev_rema.1993.v6.n1.17839 fatcat:2tbdajli5nfkrp3r7qiknewxxe

A Category Theoretic View of Nondeterministic Recursive Program Schemes

Daniel Schwencke, Marc Herbstritt
2011 Annual Conference for Computer Science Logic  
The Kleisli category A M of a monad (M, η, µ) on a category A is given as follows: the objects of A M are the same objects as the ones of A; the morphisms of A M between X and Y are all morphisms X → M  ...  and by Jf = η Y · f : X → M Y on morphisms f : X → Y .  ...  Proposition 3. 5 . 5 For the canonical distributive law λ of a polynomial Set-functor H over the monad Definition 4. 1 . 1 Given a distributive law δ : N M → M N of monads, a δ-distributive law of an  ... 
doi:10.4230/lipics.csl.2011.496 dblp:conf/csl/Schwencke11 fatcat:eemeg2qyybahzj5p2cx2uebusu

Interacting Frobenius Algebras are Hopf

Ross Duncan, Kevin Dunne
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
However the more general theory does not arise from a distributive law.  ...  Bonchi, Sobocinski, and Zanasi [9] have shown that, given a suitable distribution law, a pair of Hopf algebras forms two Frobenius algebras.  ...  The distributive laws λF , ρ and σ form a distributive series of monads [12] , and hence determine a PROP FG presented by the equations of M + PG + C and equations (P1), (P2) and (F).  ... 
doi:10.1145/2933575.2934550 dblp:conf/lics/DuncanD16 fatcat:nfolwrzdqvcjtpjj2iz63t7d3u

String diagrams for free monads (functional pearl)

Maciej Piróg, Nicolas Wu
2016 SIGPLAN notices  
Full terms of use are available: Abstract We show how one can reason about free monads using their universal properties rather than any concrete implementation.  ...  Acknowledgments Diagrams similar to the ones shown in Section 3 first appeared in an unpublished appendix to a paper by the present authors and Jeremy Gibbons (2015) .  ...  The authors would like to thank Tom Schrijvers and Alexander Vandenbroucke for their notes on an early draft of this paper, and the anonymous reviewers for their constructive remarks.  ... 
doi:10.1145/3022670.2951947 fatcat:oxckyyqapnc7vmhnuatu2nyv34

String diagrams for free monads (functional pearl)

Maciej Piróg, Nicolas Wu
2016 Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming - ICFP 2016  
Full terms of use are available: Abstract We show how one can reason about free monads using their universal properties rather than any concrete implementation.  ...  Acknowledgments Diagrams similar to the ones shown in Section 3 first appeared in an unpublished appendix to a paper by the present authors and Jeremy Gibbons (2015) .  ...  The authors would like to thank Tom Schrijvers and Alexander Vandenbroucke for their notes on an early draft of this paper, and the anonymous reviewers for their constructive remarks.  ... 
doi:10.1145/2951913.2951947 dblp:conf/icfp/PirogW16 fatcat:mces5ufbjfdadarzmqx7fmyyka

Adding recursive constructs to bialgebraic semantics

Bartek Klin
2004 The Journal of Logic and Algebraic Programming  
For a class of recursive equations called regular unfolding rules, this construction yields distributive laws as analyzed by Turi and Plotkin.  ...  Rather than interpreting recursive constructs by means of operational rules, separate recursive equations are added to semantic descriptions of languages.  ...  Gordon Plotkin's insightful remarks motivated the developments presented in Section 8. Thanks are  ... 
doi:10.1016/j.jlap.2004.03.005 fatcat:xi7bctaqjrfyjgyilipeuqvb4q

Interacting Frobenius Algebras are Hopf [article]

Ross Duncan, Kevin Dunne
2016 arXiv   pre-print
Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras.  ...  We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.  ...  be composed via a distributive law ρ : C; PG → PG; C, yielding a PROP presented by the equations of C + PG and equation (P2).  ... 
arXiv:1601.04964v1 fatcat:jbxzvcqsgjey7bhlhmzg2lkwoa

Weakening and Iterating Laws using String Diagrams [article]

Alexandre Goy
2022 arXiv   pre-print
We also provide examples of new weak distributive laws arising from iteration.  ...  The main contribution of this paper is to generalise the associativity of iterated composition to weak distributive laws.  ...  The presentation of the present paper -including the string diagrammatic approach -is greatly inspired by the one of Street-weak distributive laws in [31] .  ... 
arXiv:2205.03640v1 fatcat:iohqvnvw4veq7ooo6ixvwubhqe

Factorisations of distributive laws [article]

Ulrich Kraehmer, Paul Slevin
2014 arXiv   pre-print
Here we define the category of factorisations of a distributive law, show that it acts on this construction, and give some explicit examples.  ...  Recently, Boehm and Stefan constructed duplicial (paracyclic) objects from distributive laws between (co)monads.  ...  The aim of the present paper is to extend this procedure to any duplicial object defined by a distributive law.  ... 
arXiv:1409.7521v1 fatcat:vj4es5bfivc5lemg6ewtqx45lm
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