## A categorical version of the Brouwer-Heyting-Kolmogorov interpretation (2002)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Palmgren02acategorical,

author = {Erik Palmgren},

title = {A categorical version of the Brouwer-Heyting-Kolmogorov interpretation},

year = {2002}

}

### OpenURL

### Abstract

In this paper we interpret (fragments of) intuitionistic logic in categories with weak closure properties, such as quasi left exact categories and locally cartesian closed categories (LCCC) with sums. We also interpret the full choice scheme in an LCCC. The interpretation can be seen as a categorical form of the usual Brouwer-Heyting-Kolmogorov (BHK) interpretation. The standard interpretation of geometric logic in a pretopos is obtained by applying the image functor to the BHK-interpretation The standard interpretation of...

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Citation Context ... but this means that f has a section. Since f was arbitrary, AC holds. This argument holds more generally in any topos (Awodey 1995). 8. Exact Completions In the practice of constructive mathematics (=-=Bishop and Bridges 198-=-5), and as well as in computer checked formalisations in Martin-Lof type theories, there is a well-established notion of set. A set is here a type, or preset, X together with an equivalence relation R... |

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Citation Context ...objects and morphisms. The foundational diculties of such notions in relation to set theory have been recognised and addressed using Grothendieck universes orsbered category theory, see for instance (=-=Borceux 1994).) A-=-nother possibility is to interpret the relation as a morphism f : M(R) ! M(S 1 ) M(Sn ). In this case there may, loosely speaking, be several \reasons" or \proofs" for a tuple to belong t... |

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Citation Context ...logic is naturally dierent so that e.g. internal surjections correspond to external split epimorphisms. Nevertheless, it can be shown for pretoposes (or more generally lextensive regular categories (C=-=arboni et al. 1993-=-)) that the standard interpretation is obtained by applying the image functor to the BHK-interpretation (see Section 4). In a locally cartesian closed category (LCCC), withsnite sums, the BHKinterpret... |

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(Show Context)
Citation Context ...ithsnite sums, the BHKinterpretation veries not onlysrst-order logic but also the full choice scheme (Awodey 1995); see Section 6 and 7. This is of course expected in view of Seely's interpretation (S=-=eely 19-=-84) of Martin-Lof type theory (Martin-Lof 1984). (However (Seely 1984) is slightlysawed by some subtle coherence problems relating to equality of types. These defects were corrected later by Curien (1... |

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(Show Context)
Citation Context ...)) ! (m) such that p 2s = hm 1 ; ^ ci. By composition with p 1 we obtain a map p q (r) ! p1 (m) over U as desired. Example 7.4 The category of types with arbitrary equivalence relations (Moerdijk=-= and Palmgren 200-=-0) in Martin-Lof type theory is an LCCC withsnite sums (and a pretopos as well). Remark 7.5 Awodey (1995) proves a version of Theorem 7.3. (See also Remark 2.6 in (Moerdijk and Palmgren 2000).) Remark... |

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Citation Context ...theory (see Example 2.4 below) appears to lack strong pullbacks. From a categorical perspective a more natural notion is that of a weak left exact category in which products may be weak as well; see (=-=Carboni and Vitale 1998-=-). E. Palmgren 4 Example 2.3 Let Grpd be the category having small groupoids as objects, equivalence classes of functors (between groupoids) as morphisms, and where two functors are identi ed if they ... |

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Citation Context ... M(T ). We shall assume that the category C hassnite products. For the interpretation of relations there seems to be at least two natural choices. In the standard interpretation of categorical logic (=-=Makka-=-i and Reyes 1977), a relation symbol R on S 1 Sn is interpreted as a subobject of M(S 1 ) M(Sn ), represented by a monomorphism M(R) M(S 1 ) M(Sn ). An interpretation of a formula ' with free... |

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Citation Context ...nd the topos-theoretic standard semantics on the other side. First we explain the propositions-as-types interpretation in categories, and then its relation to standard semantics in exact completions (=-=Carboni 1995-=-), and how the standard semantics, for important fragments of intuitionistic logic, can be derived from this interpretation. Most of the results are well-known to the specialists in categorical logic,... |

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Citation Context ...umes classical set theory as a meta-theory.) In thesnal section, we see that categories withsnite limits, or even lextensive LCCCs, can be extended using the exact completion procedure (Carboni 1995; =-=Carboni and Rosolini 2000-=-). The standard interpretation in the completed category is then shown to be equivalent to the present BHK-interpretation of the original category. It seems that the BHK-interpretation has, among topo... |

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Citation Context ...new results, e.g. the interpretation of fragments of logic in quasi left exact categories and weak hyperdoctrines. The idea behind this interpretation is the not so well-known Curry-Lauchli adjoint (L=-=awvere 1996-=-). The interpretation of a many-sortedsrst-order language L in a category C involves assignment of categorical entities to sorts (or types), constants, functions and relations. A sort S is naturally i... |

3 | Axiom of choice and excluded middle in categorical logic
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(Show Context)
Citation Context ...nctor to the BHK-interpretation (see Section 4). In a locally cartesian closed category (LCCC), withsnite sums, the BHKinterpretation veries not onlysrst-order logic but also the full choice scheme (A=-=wodey 19-=-95); see Section 6 and 7. This is of course expected in view of Seely's interpretation (Seely 1984) of Martin-Lof type theory (Martin-Lof 1984). (However (Seely 1984) is slightlysawed by some subtle c... |