Tripos theory

Hyland, J. M. E.; Johnstone, Peter; Pitts, Andrew M.

doi:10.1017/s0305004100057534

Cited by 129 publications

(150 citation statements)

References 6 publications

(10 reference statements)

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“…In contrast, full induction principles, for example Ind, hold for arbitrary formulas, or equivalently for arbitrary subclasses of the domain. Such principles are expressed 15 I am grateful to Peter Aczel for pointing this out to me. 16 Various simpler formulations of Infinity are possible in the presence of the Exp, Pow or Sep axioms.…”

Section: Infinity and Inductionmentioning

confidence: 89%

“…15 Because of this, we follow Lawvere and turn primitive recursion into the defining property of the natural numbers: 16 Infinity (Inf ) N is a set, 0 ∈ N , s ∈ N N and, for every set A, element z ∈ A, and function f ∈ A A , there exists a unique function h ∈ A N satisfying: h(0) = z and h(s(x)) = f (h(x)) for all x ∈ N .…”

Section: Infinity and Inductionmentioning

confidence: 99%

“…28 There are many interesting examples of lcc pretoposes that are not elementary toposes. For example, if one performs the usual construction that builds a realizability topos from a partial combinatory algebra (pca) [15], but does so using a typed pca in the sense of Longley then the resulting category is an lcc pretopos, but not necessarily a topos, see [19]. A particularly natural mathematical example of an lcc pretopos that is not an elementary topos is the "exact completion" of the category of topological spaces [8].…”

Section: Categories Of Setsmentioning

confidence: 99%

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Constructive Set Theories and Their Category-Theoretic Models

Simpson¹

2005

From Sets and Types to Topology and Analysis

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We advocate a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of category-theoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar category-theoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.

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Section: Infinity and Inductionmentioning

confidence: 89%

Section: Infinity and Inductionmentioning

confidence: 99%

Section: Categories Of Setsmentioning

confidence: 99%

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Constructive Set Theories and Their Category-Theoretic Models

Simpson¹

2005

From Sets and Types to Topology and Analysis

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“…The main examples of this construction are localic toposes and realizability toposes obtained from a tripos, see [HJP80,Pit02,vO08].…”

Section: Proposition For An Elementary Doctrinementioning

confidence: 99%

“…Hyland, P.T. Johnstone and A.M. Pitts, see [HJP80] and which made apparent the abstract construction behind Higg's complete Heyting valued toposes and toposes obtained from Kleene's realizability like the effective topos, see [Hyl82]. In [MR15,Pas15b] it was shown that the tripos-to-topos construction T P of a given tripos P : C op G G Heyt can be obtained as the exact completion of the doctrine P cx : Prd P op G G Heyt obtained by freely completing the original tripos with full comprehensions and comprehensive diagonals.…”

Section: Introductionmentioning

confidence: 99%

Triposes, exact completions, and Hilbert's ε-operator

Maietti¹,

Pasquali²,

Rosolini³

2017

Tbilisi Math. J.

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Triposes were introduced as presentations of toposes by J.M.E. Hyland, P.T. Johnstone and A.M. Pitts. They introduced a construction that, from a tripos P : C op G G Pos, produces an elementary topos TP in such a way that the fibration of the subobjects of the topos TP is freely obtained from P . One can also construct the "smallest" elementary doctrine made of subobjects which fully extends P , more precisely the free full comprehensive doctrine with comprehensive diagonals P cx : PrdP op G G Pos on P . The base category has finite limits and embeds into the topos TP via a functor K: PrdP G G TP determined by the universal property of P cx and which preserves finite limits. Hence it extends to an exact functor K ex : (PrdP ) ex/lex G G TP from the exact completion of PrdP .We characterize the triposes P for which the functor K ex is an equivalence as those P equipped with a so-called ε-operator. We also show that the tripos-to-topos construction need not preserve ε-operators by producing counterexamples from localic triposes constructed from wellordered sets. A characterization of the tripos-to-topos construction as a completion to an exact category is instrumental for the results in the paper and we derived it as a consequence of a more general characterization of an exact completion related to Lawvere's hyperdoctrines.

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Hyperdoctrines, Natural Deduction and the Beck Condition

Seely

1983

Mathematical Logic Qtrly

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IntroductionI n the late sixties F. W. LAWVERE showed that the logical connectives and quantifiers were examples of the categorical notion of adjointness. In [9] and [lo] he amplified this notion by a more thorough discussion of the structure of a hyperdoctrine, which had much of the flavour of intuitionistic logic with equality. In this eontext it was natural to "stratify" formulae and proofs according to the free variables occurring in them, a procedure later to become standard in categorical logic. (See MAKKAI- REYES [12], FOURMAN [l], KOCK-REYES [7], for example.) In this paper, we make the relationship between hyperdoctrines and logic precise, showing that hyperdoctrines are naturally equivalent to first order intuitionistic theories with equality, where here "theory" is intended to include some proof theoretic structure, and not merely the notion of entailment. Moreover, we will show that this equivalence restricts to one giving a natural logical interpretation to the BECK (or CHEVALLEY) condition: in a given hyperdoctrine, the Beck condition for a pullback diagram is just the condition that the corresponding theory "recognizes " the pull back.This work has an obvious relationship to LAMBEK [S], and to SZABO [Zl], but, apart from the evident difference in using natural deduction rather than the sequent calculus, one important variant must be noted. SZABO treats the quantifiers as infinite conjunctions and disjunctions, whereas here (following LAWVERE [9]) they are operations adjoint to substitution. This avoids any need to refer to infinitary logic, and more closely reflects their nature: the adjunctions are explicit in the rules for the quantifiers (in either Gentzen system).There are also connections between hyperdoctrines and Dialectia interpretations (see P. SCOTT [17]) and realizability (see HYLAND, JOHNSTONE, PITTS [5]; note: a "tripos" is a po-hyperdoctrine with a generic predicate). We plan to explore these connections further in a sequel, particularly with respect to GIRARD'S type theory (GIRARD [3]).Basics of category theory may be found in MAC LANE [ l l ] or GOLDBLATT [4]. First Order LogicWe base our logic, LPCE, on a natural deduction formulation of intuitionistic, multisorted, first order predicate calculus with equality. The main modification we must introduce (essentially to be able to allow interpretations with uninhabited sorts) is l ) These results are contained in the author's Ph. D. thesis, SEELY [18].

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Tripos theory

Cited by 129 publications

References 6 publications

Constructive Set Theories and Their Category-Theoretic Models

Constructive Set Theories and Their Category-Theoretic Models

Triposes, exact completions, and Hilbert's ε-operator

Hyperdoctrines, Natural Deduction and the Beck Condition

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