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A Coalgebraic Foundation for Linear Time Semantics

John Power, Daniele Turi
1999 Electronical Notes in Theoretical Computer Science  
In Set, this gives a category with ordinary transition systems as objects and with morphisms characterised in terms of a linear notion of bisimulation.  ...  The final object in this category is the canonical abstract model for trace equivalence and can be obtained by extending the final coalgebra of the deterministic action behaviour to the Kleisli category  ...  Background and Notation A coalgebra X, h of an endofunctor H on a category C is given by an object X and a map h : X / / H(X) of C; we shall often refer to a coalgebra X, h simply by its carrier X or by  ... 
doi:10.1016/s1571-0661(05)80319-6 fatcat:dinawi2t75h7radvzp7ql3j7uy

Polynomial functors and trees [article]

Joachim Kock
2009 arXiv   pre-print
Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees.  ...  Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets.  ...  This work was prompted by Ieke Moerdijk's talk on dendroidal sets at the Mac Lane Memorial Conference in April 2006 -I am grateful for the input.  ... 
arXiv:0807.2874v3 fatcat:mcwqwovzkbaszfn35hxipvzklu

Coalgebras for modelling observable behaviour of programs

William Steingartner, Valerie Novitzká
2017 Journal of Applied Mathematics and Computational Mechanics  
Coalgebras enable one to model execution of programs step by step using categorical structures and polynomial endofunctors.  ...  In our paper, we start with a short introduction of basic notions and we use this method for modelling structural operational semantics of a simple imperative language.  ...  Acknowledgment This work has been supported by Grant No. 002TUKE-4/2017: Innovative didactic methods of education process at university and their importance in increasing education mastership of teachers  ... 
doi:10.17512/jamcm.2017.2.12 fatcat:zd5vhwnu6bfwhfx6eozsgyhb5y

Endofunctors modelling higher-order behaviours [article]

Marco Peressotti
2016 arXiv   pre-print
To achieve this property, we shift from term passing to behaviour passing: in the former higher-order is expressed by passing around syntactic objects (such as terms or processes) as representations of  ...  These functors have the challenging peculiarity to be circularly defined with their own final coalgebra.  ...  When C is Set, this category is the well-known topos of trees [5] .  ... 
arXiv:1602.06221v2 fatcat:fudx2s3ckrbvpmupf22wsc4fke

Polynomial Functors and Trees

J. Kock
2010 International mathematics research notices  
Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees.  ...  Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets.  ...  A map of polynomial endofunctors S → T is completely determined by its value on the edge set. Proof. Let R ⊂ S be an element of sub(S).  ... 
doi:10.1093/imrn/rnq068 fatcat:vm6daubnavegbe3ok67gohv3o4

Higher-order Algebras and Coalgebras from Parameterized Endofunctors

Jiho Kim
2010 Electronical Notes in Theoretical Computer Science  
The study of algebras and coalgebras involve parametric description of a family of endofunctors. Such descriptions can often be packaged as parameterized endofunctors.  ...  A parameterized endofunctor generates a higher-order endofunctor on a functor category.  ...  Acknowledgement The author gratefully acknowledges the helpful discussions with Jan-Li Lin who made him aware of the work on the arrow category [5] .  ... 
doi:10.1016/j.entcs.2010.07.018 fatcat:lru3gm5ghbesflf4x6gmff3fp4

Category theory for operational semantics

Marina Lenisa, John Power, Hiroshi Watanabe
2004 Theoretical Computer Science  
We use the concept of a distributive law of a monad over a copointed endofunctor to define and develop a reformulation and mild generalisation of Turi and Plotkin's notion of an abstract operational rule  ...  Following Turi and Plotkin, our definition, suitably restricted, agrees with the notion of a set of GSOS-rules, allowing one to construct both an operational model and a canonical, internally fully abstract  ...  The functor T is given by the free monad on the endofunctor , i.e., it is determined by the left adjoint to the forgetful functor from -Alg to Set.  ... 
doi:10.1016/j.tcs.2004.07.024 fatcat:cyptihavlzdgnlmeoi6gu3psvu

Dualising initial algebras

NEIL GHANI, CHRISTOPH LTH, FEDERICO DE MARCHI, JOHN POWER
2003 Mathematical Structures in Computer Science  
We attempt to emulate this result for the coalgebraic case by first defining a notion of cosignature and coequation and then proving that the models of such coalgebraic presentations are precisely the  ...  of the representing monad.  ...  to figure out the structure of the final coalgebra in Example 5.10, and the anonymous referees for their constructive criticism.  ... 
doi:10.1017/s0960129502003912 fatcat:t5qb2brau5ayhfh2jfchqq5uze

Towards Synthetic Descriptive Set Theory: An instantiation with represented spaces [article]

Arno Pauly, Matthew de Brecht
2014 arXiv   pre-print
Presently, we mainly focus on developing the ideas in the category of represented spaces.  ...  Synthetic descriptive set theory promises elegant explanations for various phenomena in both classic and effective descriptive set theory.  ...  Acknowledgements The first author would like to thank Vasco Brattka, Vassilis Gregoriades, Takayuki Kihara, Luca Motto-Ros, Matthias Schröder and Victor Selivanov for fruitful dicussions on the subject of  ... 
arXiv:1307.1850v2 fatcat:6ac5k2xk2fdpneydzvx7cchafi

An abstract view on syntax with sharing [article]

Richard Garner
2011 arXiv   pre-print
Inductively generated syntax has an abstract expression in terms of initial algebras for certain endofunctors on the category of sets, which permits one to go beyond the set-based case, and speak of inductively  ...  In this paper we give a similar abstract expression to the notion of term graph. Aspects of the concrete theory are redeveloped in this setting, and applications beyond the realm of sets discussed.  ...  However, whilst in principle this approach allows one to move beyond endofunctors of the category of sets, as the authors of [8] note in their Section 4.2, there are aspects of their development that  ... 
arXiv:1009.3682v2 fatcat:pnae2wagrrhfne6zd76gdmggqe

An abstract view on syntax with sharing

R. Garner
2011 Journal of Logic and Computation  
Inductively generated syntax has an abstract expression in terms of initial algebras for certain endofunctors on the category of sets, which permits one to go beyond the set-based case, and speak of inductively  ...  Aspects of the concrete theory are redeveloped in this setting, and applications beyond the realm of sets discussed. Vol. 22 No. 6,  ...  However, while in principle this approach allows one to move beyond endofunctors of the category of sets, as the authors of [9] note in their Section 4.2, there are aspects of their development that  ... 
doi:10.1093/logcom/exr021 fatcat:i5xfi7227vednl6dba6x5uu2zq

On Finitary Functors and Their Presentations [chapter]

Jiří Adámek, Stefan Milius, Lawrence S. Moss
2012 Lecture Notes in Computer Science  
Finitary endofunctors of locally presentable categories are proved to have equational presentations.  ...  Special attention is paid to the Hausdorff functor of non-empty compact subsets of a complete metric space.  ...  Power [9]: given objects A and B of A , let A, B be the endofunctor of A assigning to X the power of B to the set A (X, A): A, B X = B A (X,A ) .  ... 
doi:10.1007/978-3-642-32784-1_4 fatcat:u7xpe22frnesxlf463nqwnhkba

Relating coalgebraic notions of bisimulation

Sam Staton, Alexander Kurz
2011 Logical Methods in Computer Science  
The theory of coalgebras, for an endofunctor on a category, has been proposed as a general theory of transition systems.  ...  We investigate and relate four generalizations of bisimulation to this setting, providing conditions under which the four different generalizations coincide.  ...  In this setting, the object X of states is no longer a set, but a presheaf.  ... 
doi:10.2168/lmcs-7(1:13)2011 fatcat:khbpalgotbb6jatpya6dgsle3a

Modularity of Behaviours for Mathematical Operational Semantics

Marco Kick, John Power
2004 Electronical Notes in Theoretical Computer Science  
From such a distributive law, one can readily induce a distributive law of the monad over the cofree comonad on the behaviour endofunctor, and much of their analysis can be carried out in the latter terms  ...  Our leading examples are given by combining action and timing, with a modular account of the operational semantics for the combination induced by that of each of the two components.  ...  There is a range of obvious examples, leading ones given by the set of natural numbers and by the set of non-negative real numbers, each with monoid structure given by sum.  ... 
doi:10.1016/j.entcs.2004.02.030 fatcat:yomj2rtcdbdtzedw6mkmt5ytsm

Data Base Mappings and Monads: (Co)Induction [article]

Zoran Majkic
2011 arXiv   pre-print
The objects in this category are the database-instances (a database-instance is a set of n-ary relations, i.e., a set of relational tables as in standard RDBs).  ...  Thus DB category, as a base category for the semantics of databases and mappings between them, is different from the Set category used dominantly for such issues, and needs the full investigation of its  ...  Hence, if the action of a monad preserves this particular kind of colimits, its action on any object will be determined by its action on the fp objects; such a monad is called finitary.  ... 
arXiv:1102.4769v1 fatcat:dfdpjl2mhja6tbsvfj3btwelsi
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