## Unique Factorisation Lifting Functors and Categories of Linearly-Controlled Processes (1999)

Venue: | Mathematical Structures in Computer Science |

Citations: | 7 - 2 self |

### BibTeX

@INPROCEEDINGS{Bunge99uniquefactorisation,

author = {Marta Bunge and Marcelo P. Fiore},

title = {Unique Factorisation Lifting Functors and Categories of Linearly-Controlled Processes},

booktitle = {Mathematical Structures in Computer Science},

year = {1999},

pages = {200--0}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearly-controlled ones. To this end, we introduce and study a notion of path-linearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearly-controlled processes (viz., processes controlled by path-linearisable categories) are sheaf models. Introduction This work is an investigation into the mathematical structure of processes. The processes to be considered embody a notion of state space varying according to a control. This we formalise as a category of states (and their inter-relations) Xequipped with a control functor X C f . There are different ways in which the control category C may be required to control t...

### Citations

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Citation Context ...wing adjoint situation c B + # j ! // ? j oo c B # // ? ?s( ) # oo UFL=B which for B = M(A) yields (17). Finally we concentrate on process equivalence. A standard such notion is given by bisimilaritys=-=[Mil89]-=-. We recall the definition. A (strong) bisimulation between transition graphs X and Y is a relation R between their respective state sets such that whenever x R y, then if x a // x 0 then y a // y 0 a... |

998 |
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Citation Context ... subcategory of Cat of ufl functors. We show in Section 2 that UFL=C may be embedded (as a full reflective subcategory) in a presheaf topos; viz., c C # where C # is Mac Lane's twisted arrow category =-=[ML71]-=-. On the one hand, this result establishes connections with presheaf models; thus opening the possibility of reusing the techniques developed in that setting [CW97]. On the other hand, it highlights i... |

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275 | A Classification of Models for Concurrency
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Citation Context ...dels of linearly-controlled processes. Some concluding remarks are provided in Section 6. 1 Background and motivation Labelled transition systems are a model of computation widely used in concurrency =-=[WN95]-=-. We recall the basic definitions. A transition system over a set of actions A consists of a set of states S and a transition relation // ` S \Theta A \Theta S. A tuple (s; a; s 0 ) 2 // is usually de... |

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Citation Context ...o property (1) of synchronisation trees. We will show below that, by broadening the notion of fibration, one can develop a theory for transition systems with a treatment of bisimulation via open-maps =-=[JNW96]-=- in the same spirit of Winskel's et al. presheaf models. The required fibration condition amounts to a unique factorisation lifting property [Law86b, Str96, BN98]. We illustrate this concept in the co... |

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Citation Context ...ing either of the following equivalent conditions: for ? an initial object in P, (R--i) The set T (?) is terminal in Set. (R--ii) There exists a unique map ? // T in b P(that is, 0 // ? is orthogonal =-=[FK72]-=- to T // 1 in b P). In the definition of the category of synchronisation trees, we have implicitly identified the rooted presheaf T : P(A) op // Set with the synchronisation tree with initial state gi... |

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Citation Context ...ified by the labelling functor t. More concretely, a morphism s e // s 0 in Twitnesses the capability of the system to evolve from the state s to the state s 0 producing the observable behaviour t(e) =-=[WN97]-=-. In this light, the property (DF) has the following computational interpretation: a state s 0 occurring after an observation a, as indicated in (4) above, comes from a unique state s and transition s... |

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Citation Context ...s Mac Lane's twisted arrow category [ML71]. On the one hand, this result establishes connections with presheaf models; thus opening the possibility of reusing the techniques developed in that setting =-=[CW97]-=-. On the other hand, it highlights important differences. First, the path category C # is obtained from the control category C and this passage equips the former with extra structure; notably notions ... |

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Citation Context ...roperty is better defined as the conjunction of a "cancellation" and a "fill-in" property (see Definition 4.1). The cancellation property has been considered by Lawvere in the cont=-=ext of processes in [Law89]-=- and, as argued there, seems to be inherent in the notion of a given evolution of a process. The further imposition of the fill-in property has important conceptual and mathematical consequences. Conc... |

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Citation Context ...maps such that (x; y) 2 W (that is, (x; y) factors through (w 1 ; w 2 )). The notion of bisimulation provided by the path functor u : M(A)# // TGA corresponds to back-and-forth bisimulation on states =-=[DNMV90]-=-. 4 Path-linearisable categories In the previous section we have regarded and analysed the category of transition graphs UFL=M(A) as an interleaving model of concurrent computation. In this section co... |

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Citation Context ... jargon of computer science. From the mathematical perspective, as shown in Section 4, the cancellation and fill-in properties on a category C are equivalent to an "interval glueing" propert=-=y used in [BN98]-=- to show that UFL=C is a (model-generated) topos. We give another proof of this result in Section 5 where we show that, for C a path-linearisable category, UFL=C is a topos (of sheaves on C # ). We re... |

8 | Categories of spaces may not be generalized spaces as exemplified by directed graphs - Lawvere - 2005 |

6 |
A note on discrete conduché fibrations. Theory and Applications of Categories
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Citation Context ...pos (of sheaves on C # ). We remark that Johnstone showed that for C the commutative square, the simplest example of failure of fill-in, the category UFL=C is not a topos. This result can be found in =-=[Joh99]-=-, where it is also shown that, for a category C satisfying a factorisation preordered (our cancellation) and a factorisation strongly 2 connected (our fill-in) condition, UFL=C is a sheaf topos. Organ... |

1 | Handbook of Categorical Algebra. Number 50--52 in Encyclopedia of Mathematics and its Applications - Borceux - 1994 |

1 |
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Citation Context ... ? ? ? ? ' X 0 fl 0sX fls? ? ? ? ?s// B 0 (13) commutes, identities given by id (';X;fl) = id X , and composition given as in B ; the ufl functor u(fi) maps a morphism as in (13) to X X 0 . Following =-=[Law86c]-=- we will refer to the categories [[B fi // B 0 ]] as interval categories ; they range from the initial object (id B ; B; fi) to the terminal object (fi; B 0 ; id B 0 ). Associated to every functor f :... |

1 |
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Citation Context ...serve that TGA ' UFL=M(A) (15) 13 and argue that this representation embodies the dynamics of transition graphs. The equivalence (15) above for the case A = 1 was first noticed by Schanuel and Street =-=[SS88]. To -=-see (15) observe that the adjunction j; " : F a U : Cat // Graph induces the following adjoint situation Graph=`(A) \Sigma j // ? j oo Graph=UF(`A) ? Cat=F(`A) (16) where the left adjoint lands i... |