## Restriction categories III: colimits, partial limits, and extensivity (2006)

Citations: | 4 - 3 self |

### BibTeX

@MISC{Cockett06restrictioncategories,

author = {J. R. B. Cockett and Stephen Lack},

title = {Restriction categories III: colimits, partial limits, and extensivity},

year = {2006}

}

### OpenURL

### Abstract

### Citations

960 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ... p-category, while the symmetry is the τ. The unit is the restriction terminal object, and the unit constraint X × T ∼ = X is the projection. In light of the coherence results for monoidal categories =-=[15]-=-, we shall allow ourselves to omit explicit mention of the associativity isomorphisms, and write as if the tensor product were strictly associative. As observed by Carboni [2], for each object X the d... |

115 | Weakly distributive categories
- Cockett, Seely
- 1997
(Show Context)
Citation Context ... coproducts and the left adjoint I : D → D+1 is bijective on objects, D+1 has coproducts; the injections are in the image of I and so total. Thus D+1 has restriction coproducts. It is well known (see =-=[4, 5]-=- for example) that the free completion under (finite) coproducts of a category C can be formed as the category Fam(C ) of finite families of objects of C . Explicitly, an object of Fam(C ) is a finite... |

97 |
Introduction to extensive and distributive categories
- Carboni, Lack, et al.
- 1993
(Show Context)
Citation Context ... coproducts and the left adjoint I : D → D+1 is bijective on objects, D+1 has coproducts; the injections are in the image of I and so total. Thus D+1 has restriction coproducts. It is well known (see =-=[4, 5]-=- for example) that the free completion under (finite) coproducts of a category C can be formed as the category Fam(C ) of finite families of objects of C . Explicitly, an object of Fam(C ) is a finite... |

49 |
Strong functors and monoidal monads
- Kock
- 1972
(Show Context)
Citation Context ...ds of [1]. Let C be a symmetric monoidal category, with tensor product ⊗, unit I, and symmetry τ. The associativity and unit isomorphisms will be suppressed where possible. A symmetric monoidal monad =-=[14]-=- on C is a monad T = (T, η, µ) equipped with a natural transformation ϕA,B : T A ⊗ T B → T (A ⊗ B) satisfying the equations T A ⊗ T B ϕA,B �� T (A ⊗ B) A ⊗ B τ τ �� �� ηA⊗ηB� � � T A ⊗ T B ����� � ϕA,... |

30 | Restriction categories I: categories of partial maps
- Cockett, Lack
- 2002
(Show Context)
Citation Context ...partial map f : A → A which is defined whenever f is, in which it acts as the identity. This operation is taken as fundamental in the notion of restriction category studied in the earlier instalments =-=[6, 7]-=- of this sequence of papers and again here. The maps of the form f are always idempotents, and are called restriction idempotents. This time the trivial case is characterized by the fact that the rest... |

17 |
Dominical categories: recursion theory without elements
- Paola, Heller
- 1987
(Show Context)
Citation Context ...object, and the unique map out of the initial object is in M , then ParM (C ) has zero maps, given by the “nowhere defined” partial maps. These were fundamental in the approach of di Paola and Heller =-=[11]-=-. The presence of these zero maps means that only when the category itself is trivial can the partiality be trivial. D. To every partial map f : A → B we can associate the partial map f : A → A which ... |

13 |
Restriction categories II: partial map classification, Theoretical Computer Science 294
- Cockett, Lack
- 2003
(Show Context)
Citation Context ...either the restriction category structure or the symmetric monoidal structure. 5.2 Classified restriction categories and equational lifting categories In this brief section we revisit the analysis in =-=[7]-=- of classified restriction categories, in particular its connection with the equational lifting monads of [1]. Let C be a symmetric monoidal category, with tensor product ⊗, unit I, and symmetry τ. Th... |

8 |
group representations
- Matrices
- 1991
(Show Context)
Citation Context ...It is well known that in the category of sets and binary relations the disjoint union (of a finite family of sets) serves both as coproduct and product, so that there is a “calculus of matrices”: see =-=[3]-=-, for example. In this section we consider the extent to which this can be adapted to deal not with relations but with partial functions. We then consider when such a calculus is available in an abstr... |

6 | The extensive completion of a distributive category
- Cockett, Lack
(Show Context)
Citation Context ...s are extensive we obtain as the total category a lextensive category. This provides, in particular, an alternative description of the extensive completion of a distributive category to that given in =-=[9]-=-. This is described in Section 5.4. But what is the importance of being extensive? Section 2 answers this question for partial map categories very concretely: extensivity means that there is a “calcul... |

4 |
Freyd and Andre Scedrov, Categories, allegories
- Peter
- 1990
(Show Context)
Citation Context ...f ≤ g if g is defined whenever f is, and they then agree. This makes ParM (C ) into a bicategory, and is the approach taken by Carboni in [2]; it is also closely related to Freyd’s notion of allegory =-=[12]-=-. The trivial case is characterized by the fact that the partial order is discrete, in the sense that f ≤ g only if f = g. B. If the category C of total maps has finite products, then this induces a s... |

3 | Equational lifting monads
- Bucalo, Führmann, et al.
(Show Context)
Citation Context ...egories and equational lifting categories In this brief section we revisit the analysis in [7] of classified restriction categories, in particular its connection with the equational lifting monads of =-=[1]-=-. Let C be a symmetric monoidal category, with tensor product ⊗, unit I, and symmetry τ. The associativity and unit isomorphisms will be suppressed where possible. A symmetric monoidal monad [14] on C... |

2 |
Bicategories of partial maps
- Carboni
- 1987
(Show Context)
Citation Context ... be identified. A. Given partial maps f, g : A → B, we define f ≤ g if g is defined whenever f is, and they then agree. This makes ParM (C ) into a bicategory, and is the approach taken by Carboni in =-=[2]-=-; it is also closely related to Freyd’s notion of allegory [12]. The trivial case is characterized by the fact that the partial order is discrete, in the sense that f ≤ g only if f = g. B. If the cate... |

1 | Restriction categories IV: Enriched restriction categories, in preparation - Cockett, Lack |

1 |
On partial algebras, Colloq
- Hoehnke
- 1977
(Show Context)
Citation Context ...triction categories with restriction products are exactly the partial cartesian categories in the sense of Curien and Obtulowicz [10], and or alternatively the pre-dht-symmetric categories of Hoehnke =-=[13]-=-, and they are a special case of the bicategories of partial maps of Carboni [2]. For more details on these correspondences, see [16]. 4.3 Categories with products and a restriction Before leaving our... |

1 |
Categories of partial maps, Information and computation 79:94–130
- Robinson, Rosolini
- 1988
(Show Context)
Citation Context ...d by the fact that this symmetric monoidal structure on the category of partial maps is in fact cartesian (that is, given by the categorical product). This approach was taken by Robinson and Rosolini =-=[16]-=- and by Curien and Obtulowicz [10]. C. If C has a strict initial object, and the unique map out of the initial object is in M , then ParM (C ) has zero maps, given by the “nowhere defined” partial map... |