## Locally connected recursion categories (2006)

Citations: | 1 - 0 self |

### BibTeX

@TECHREPORT{Lengyel06locallyconnected,

author = {Florian Lengyel},

title = {Locally connected recursion categories},

institution = {},

year = {2006}

}

### OpenURL

### Abstract

Abstract. A recursion category is locally connected if connected domains are jointly epimorphic. New proofs of the existence of non-complemented and recursively inseparable domains are given in a locally connected category. The use of local connectedness to produce categorical analogs of undecidable problems is new; the approach allows us to relax the hypotheses under which the results were originally proved. The results are generalized to non-locally

### Citations

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29 | Restriction categories i: Categories of partial maps
- Cockett, Lack
(Show Context)
Citation Context ...t and Lack observe that P -categories are “symmetric monoidal categories in which each object has a a monoidal natural cocommutative coassociative comultiplication (and possibly an unnatural counit)” =-=[CL02]-=-. Dually, one may speak of a category with a binary coproduct +, together with natural transformations i0, i1, ∇, ass+, tr+ satisfying the duals of conditions i), ii) and iii), where the injections i0... |

15 |
Dominical categories: recursion theory without elements
- Paola, Heller
- 1987
(Show Context)
Citation Context ...H87, Proposition 7.3] Complements and quasicomplements coincide. Rosolini has written in [Ros88a, page 313] that in view of his representation Theorem 2.8 (loc. cit.), all the results of section 5 of =-=[DPH87]-=- “...can be generalized to P -categories with ranges stable under products by checking that they hold in a P -category of the form M–Ptl(A).” However, the preceding proposition shows this is false for... |

10 |
An existence theorem for recursive categories
- Heller
- 1990
(Show Context)
Citation Context ...ants, and connected domains [DPH87, Hel90, DPM91, Len04]. Connected domains arose in connection with Heller’s existence theorem for recursion categories, which applied to locally connected categories =-=[Hel90]-=-. Di Paola and Montagna produced examples of non-locally connected recursion categories [DPM91]; they raised the question of generalizing Heller’s existence theorem to handle this case; this question ... |

5 |
More existence theorems for recursion categories, Annals of Pure and Applied Logic 125
- Lengyel
- 2004
(Show Context)
Citation Context ...ntagna produced examples of non-locally connected recursion categories [DPM91]; they raised the question of generalizing Heller’s existence theorem to handle this case; this question was addressed in =-=[Len04]-=-. Here we follow the approach of [Len04], in which range functors were used. The dominical categories of Di Paola and Heller were the original categorical setting for recursion categories; the setting... |

3 |
Some properties of the syntactic p-recursion categories generated by consistent, recursively enumerable extensions of Peano arithmetic
- Paola, Montagna
- 1991
(Show Context)
Citation Context ... with Heller’s existence theorem for recursion categories, which applied to locally connected categories [Hel90]. Di Paola and Montagna produced examples of non-locally connected recursion categories =-=[DPM91]-=-; they raised the question of generalizing Heller’s existence theorem to handle this case; this question was addressed in [Len04]. Here we follow the approach of [Len04], in which range functors were ... |

3 | Pathologies” in two syntactic categories of partial maps - Montagna - 1989 |

2 |
Continuity and effectiveness in topoi, D.phil
- Rosolini
- 1986
(Show Context)
Citation Context ...ural transformation ∆ : 1C → × ◦ ∆C, and for each object X of C, natural transformations p 0( ),X : ( ) × X → 1C, p 1X,( ) : X × ( ) → 1C such that the conditions i), ii) and iii) below are satisfied =-=[Ros86]-=-. i) The following equations hold. p0X,X∆X = 1X = p1X,X∆X, (p0X,Y × p1X,Y )∆X×Y = 1X×Y , p0X,Y (1X × p0Y,Z) = p0X,Y ×Z, p0X,Z(1X × p1Y,Z) = p0X,Y ×Z, p1X,Z(p0X,Y × 1Z) = p1X×Y,Z, p1Y,Z(p1X,Y × 1Z) = p... |

2 | Representation theorems for special p-categories - Rosolini - 1988 |

2 | A relativization mechanism in recursion categories - Stefani - 1993 |

1 |
Representation theorems for p-categories, Categorical algebra and its applications (Louvain-La-Neuve
- Rosolini
- 1987
(Show Context)
Citation Context ...ns in a dominical recursion category C under the assumption that C satisfies the axiom of choice [DPH87, Theorem 8.15]. Rosolini later generalized this to a P -category satisfying the axiom of choice =-=[Ros88a]-=-. Here we produce a pair of recursively inseparable domains in a locally connected P -recursion category C which satisfies the weak axiom of choice, as well as a criterion, valid in classical recursio... |

1 | An algebraic approach to categories of partial morphisms - Stefani |