## Restriction categories I: Categories of partial maps (2001)

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Venue: | Theoretical Computer Science |

Citations: | 31 - 7 self |

### BibTeX

@ARTICLE{Lack01restrictioncategories,

author = {Stephen Lack},

title = {Restriction categories I: Categories of partial maps},

journal = {Theoretical Computer Science},

year = {2001},

volume = {270},

pages = {223--259}

}

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### Abstract

### Citations

225 |
Locally presentable and accessible categories
- Adámek, Rosický
- 1994
(Show Context)
Citation Context ...e; and so complete and cocomplete. Proof. rCat0 is the category of algebras for a nitary monad on the locally nitely presentable category Cat0. It follows, for example by the nal remark of Chap. 2 in =-=[1]-=-, that rCat0 is locally nitely presentable, and so in particular complete and cocomplete. Remark2.5. Since U is monadic, limits in rCat0 are constructed as in Cat0. For general colimits in rCat0, howe... |

117 | Weakly distributive categories
- Cockett, Seely
- 1997
(Show Context)
Citation Context ...ral colimits in rCat0, however, we have no explicit description, although it is easy to see that coproducts are formed in rCat0 as in Cat0, whence we conclude that rCat0 is extensive, in the sense of =-=[5, 6]-=-. The rest of this section is devoted to giving an explicit description of the left adjoint F to U : rCat0 → Cat0. To motivate the construction, we rst recall from Lemma 2.1 that if f =ug then gf =f; ... |

99 |
Introduction to extensive and distributive categories
- Carboni, Lack, et al.
- 1993
(Show Context)
Citation Context ...ral colimits in rCat0, however, we have no explicit description, although it is easy to see that coproducts are formed in rCat0 as in Cat0, whence we conclude that rCat0 is extensive, in the sense of =-=[5, 6]-=-. The rest of this section is devoted to giving an explicit description of the left adjoint F to U : rCat0 → Cat0. To motivate the construction, we rst recall from Lemma 2.1 that if f =ug then gf =f; ... |

78 |
Two-dimensional monad theory
- Blackwell, Kelly, et al.
- 1989
(Show Context)
Citation Context ... set about adding 2-cells to rCat0, the monad described in this section cannot be enriched to become a 2-monad. It does admit enrichment over invertible 2-cells (for the importance of such monads see =-=[2]-=-) but we shall not make any use of this fact. 2.2.2. The 2-category rCat A natural transformation between restriction functors is called a restriction transformation if all of its components are total... |

63 |
Axiomatic Domain Theory in Categories of Partial Maps
- Fiore
- 1994
(Show Context)
Citation Context ...gory with a formal stable system of monics. The key results here are Theorem 4.2, 3 A stable system of monics is (essentially) what Roslini called a dominion in [24] (this name was also used by Fiore =-=[11]-=-), Robinson and Rosolini called an admissible class of subobjects in [22], and Mulry called a domain structure in [18].s226 J.R.B. Cockett, S. Lack / Theoretical Computer Science 270 (2002) 223–259 wh... |

63 |
Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads
- Kelly, Power
- 1993
(Show Context)
Citation Context ...ic, and then give an explicit description of the monad. In provingthe monadicity of rCat0, we observe that Cat0 is a locally nitely presentable category, and then give a presentation, in the sense of =-=[16]-=-, for a nitary monad on Cat0, whose category of algebras is rCat0. These presentations involve operations and equations, each havingan arity which is a nitely presentable category. We shall also give ... |

59 |
Categories of partial maps
- Robinson, Rosolini
- 1988
(Show Context)
Citation Context ...ty through “zero morphisms” (maps which are nowhere de ned) and the presence of “near products”, which are tensor products which behave like products with respect to total maps. Robinson and Rosolini =-=[22]-=- quickly pointed out that the zero structure was not really necessary to obtain a theory of partiality. They observed that it could be obtained through the “near-product” structure alone. Accordingly ... |

40 | Syntactic control of interference revisited
- O’Hearn, Power, et al.
- 1999
(Show Context)
Citation Context ...o which a map with that restriction idempotent can quotient its domain. Thus, we should regard the maps in FSet as beingequipped with generalized apartness relations. It is interestingto note that in =-=[21]-=- a very similar idea (there called the “category of worlds”) was used to provide a semantics for modeling non-interference in a programming language. Suppose now that we have a functor H : X → UY wher... |

28 | An Introduction to the Structure Theory - Grillet, Semigroups - 1995 |

25 |
Fibrations in Bicategories, Cahiers de Topologie et Géométrie Différentielle 21
- Street
- 1980
(Show Context)
Citation Context ...(X) → X is faithful. We call @ : R(X) → X the restriction bration associated to the restriction category X. It turns out that @ : R(X) → X is actually a bration in the bicategory rCat in the sense of =-=[25]-=-, but we shall not pursue this point of view here. The posets RIdX(X ) are actually meet-semilattices, with binary meets given by e1 ∧e2 =e1e2; and the identity 1X as top element. Also we have RIdX(f)... |

19 | Lifting theorems for Kleisli categories, in - Mulry - 1994 |

17 |
Dominical categories: recursion theory without elements
- Paola, Heller
- 1987
(Show Context)
Citation Context ...esults in computability and algebraic geometry. Because partial maps are central to so many issues in computer science there has been a considerable e ort to develop their theory. Di Paola and Heller =-=[10]-=- introduced the notion of “dominical categories” as an algebraic setting in which one could study partial maps (and computability theory). They approached partiality through “zero morphisms” (maps whi... |

13 |
Restriction categories II: partial map classification, Theoretical Computer Science 294
- Cockett, Lack
- 2003
(Show Context)
Citation Context ...ects as actual subobjects. Some of the main theorems of the paper involve the existence or construction of certain adjunctions. These are summarized in the diagram appearing in Section 5. In a sequel =-=[8]-=- to this paper we shall consider the relationship of partial map classiers to restriction categories. This is closely related to more recent work on partiality in which the monad arisingfrom the parti... |

10 |
The algebra of distributive and extensive categories
- Lack
- 1995
(Show Context)
Citation Context ...ude Mulry’s work [18–20]). Secondly, one of the main motivatingresults, the description of the extensive completion of a distributive category, had also been proved independently by the second author =-=[17]-=- at roughly the same time. Thus, it had been resolved that we should try to pool resources and publish the results jointly ::: and that event had to wait for a time when we could get together physical... |

5 |
Restriction categories III: Partial structures, to appear. [9] T. Coquand, Categories of embedding
- Cockett, Lack
- 1988
(Show Context)
Citation Context ...siers to restriction categories. This is closely related to more recent work on partiality in which the monad arisingfrom the partial map classi er is taken as primitive; see [3]. In a further sequel =-=[9]-=-, we shall consider restriction categories arising as categories of partial maps in a category with extra structure such as products, coproducts, distributivity, extensivity, and so on; and it is in t... |

4 | Partial functions, ordered categories, limits and cartesian closure
- Jay
- 1990
(Show Context)
Citation Context ...l core ective sub-2-category. A category with a restriction has a natural poset enrichment using the de nition f6g ⇔f =gf which allows one to regard it as a 2-category=bicategory of partial maps (see =-=[4, 15]-=-). We do not pursue this here beyond the observation that it allows us to construct a left 2-adjoint to the inclusion Triv : Cat → rCat. For as each X in rCat can be regarded as a 2-category, we can u... |

3 | Monads and algebras in the semantics of partial data types, Theoret - Mulry |

3 | Domains and dominical categories, Riv - Rosolini - 1985 |

2 |
Bicategories of partial maps
- Carboni
- 1987
(Show Context)
Citation Context ...tative coassociative comultiplication (and possibly an unnatural counit). These categories were considered by Jacobs [14] as the semantics of weakening. A quite di erent approach was taken by Carboni =-=[4]-=-, where the bicategory structure was taken as primitive. These P-categories are also, essentially, what the rst author called “copy categories” in a manuscript [7] which started circulatingin about 19... |

2 |
Copy categories, unpublished manuscript
- Cockett
- 1995
(Show Context)
Citation Context ... erent approach was taken by Carboni [4], where the bicategory structure was taken as primitive. These P-categories are also, essentially, what the rst author called “copy categories” in a manuscript =-=[7]-=- which started circulatingin about 1995 but was never published. The manuscript never reached publication for two main reasons. First, much of the material on partiality was already available in the a... |

2 |
Semantics of weakeningand contraction
- Jacobs
- 1994
(Show Context)
Citation Context ...e symmetric monoidal categories in which each object has a monoidal natural cocommutative coassociative comultiplication (and possibly an unnatural counit). These categories were considered by Jacobs =-=[14]-=- as the semantics of weakening. A quite di erent approach was taken by Carboni [4], where the bicategory structure was taken as primitive. These P-categories are also, essentially, what the rst author... |

1 |
Lifting in Category theory and computer science (Santa Margherita Ligure
- Bucalo, Rosolini
- 1997
(Show Context)
Citation Context ...nship of partial map classiers to restriction categories. This is closely related to more recent work on partiality in which the monad arisingfrom the partial map classi er is taken as primitive; see =-=[3]-=-. In a further sequel [9], we shall consider restriction categories arising as categories of partial maps in a category with extra structure such as products, coproducts, distributivity, extensivity, ... |

1 |
Partial map classi ers and partial cartesian closed categories, Theoret
- Mulry
- 1992
(Show Context)
Citation Context ...ntially) what Roslini called a dominion in [24] (this name was also used by Fiore [11]), Robinson and Rosolini called an admissible class of subobjects in [22], and Mulry called a domain structure in =-=[18]-=-.s226 J.R.B. Cockett, S. Lack / Theoretical Computer Science 270 (2002) 223–259 which establishes an adjunction between rCat and the 2-category sLatFib of bred semilattices; and Theorem 4.3, which des... |

1 |
Continuity and e ectiveness in topi, D.Phil Thesis
- Rosolini
- 1986
(Show Context)
Citation Context ... structure alone. Accordingly they introduced P-categories – categories with a “near-product” structure – as the basis for a theory of partiality (these ideas were also presented in Rosolini’s thesis =-=[24]-=-). In more modern terms these are symmetric monoidal categories in which each object has a monoidal natural cocommutative coassociative comultiplication (and possibly an unnatural counit). These categ... |