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18
The identity type weak factorisation system
 U.U.D.M. REPORT 2008:20
, 2008
"... ... theory T with axioms for identity types admits a nontrivial weak factorisation system. After characterising this weak factorisation system explicitly, we relate it to the homotopy theory of groupoids. ..."
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... theory T with axioms for identity types admits a nontrivial weak factorisation system. After characterising this weak factorisation system explicitly, we relate it to the homotopy theory of groupoids.
A minimalist twolevel foundation for constructive mathematics
, 2008
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms” paradigm and acts as a programming language.
General structural operational semantics through categorical logic (Extended Abstract)
, 2008
"... Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rulebased definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formul ..."
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Cited by 9 (6 self)
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Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rulebased definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formulation we recover precise results for particular language idioms by interpreting the logic in particular categories. For instance, results for firstorder calculi, such as CCS, arise from considering the general results in the category of sets. Results for languages involving substitution and name generation, such as the πcalculus, arise from considering the general results in categories of sheaves and group actions. As an extended example, we develop a tyft/tyxtlike rule format for open bisimulation in the πcalculus.
Constructivist and Structuralist Foundations: Bishop’s and Lawvere’s Theories of Sets
, 2009
"... Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of MartinLöf. The theory, CETCS, provides a str ..."
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Cited by 8 (1 self)
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Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of MartinLöf. The theory, CETCS, provides a structuralist foundation for constructive mathematics in the style of Bishop.
JOYAL’S ARITHMETIC UNIVERSE AS LISTARITHMETIC PRETOPOS
"... Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three ..."
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Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three reasons: first, Joyal’s arithmetic universes are listarithmetic pretopoi; second, the initial arithmetic universe among Joyal’s constructions is equivalent to the initial listarithmetic pretopos; third, any listarithmetic pretopos enjoys the existence of free internal categories and diagrams as required to prove Gödel’s incompleteness. In doing our proofs we make an extensive use of the internal type theory of the categorical structures involved in Joyal’s constructions. The definition of listarithmetic pretopos is equivalent to the general one that I came to know in a recent talk by André Joyal. 1.
Consistency of the minimalist foundation with Church thesis and Bar Induction. submitted
, 2010
"... We consider a version of the minimalist foundation previously introduced to formalize predicative constructive mathematics. This foundation is equipped with two levels to meet the usual informal practice of developing mathematics in an extensional set theory (its extensional level) with the possibil ..."
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We consider a version of the minimalist foundation previously introduced to formalize predicative constructive mathematics. This foundation is equipped with two levels to meet the usual informal practice of developing mathematics in an extensional set theory (its extensional level) with the possibility of formalizing it in an intensional theory enjoying a proofs as programs semantics (its intensional level). For the intensional level we show a realizability interpretation validating Bar Induction and formal Church thesis for typetheoretic functions. This is possible because in our foundation the wellknown result by Kleene that Brouwer’s principle of Bar Induction is inconsistent with the formal Church thesis for choice sequences can be decomposed as follows: Brouwer’s Bar Induction, where choice sequences are functional relations, is inconsistent with the formal Church thesis for typetheoretic functions (from natural numbers to natural numbers) and the axiom of unique choice transforming a functional relation between natural numbers into a typetheoretic function. As a consequence this model disproves the validity of the axiom of unique choice in our foundation. This model can serve to interpret the whole foundation in a classical predicative set theory by keeping the computational interpretation of predicative sets as data types and their typetheoretic functions as programs. Moreover it shows that choice sequences of Cantor space, those of Baire space, and real numbers both as Dedekind cuts or Cauchy sequences, do not form a set in the minimalist foundation.
Why topology in the minimalist foundation must be pointfree
, 2013
"... We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our twolevel minimalist foundation, the choice for a pointfree approach to topology is not just a matter of convenience or mathematical elegance, but becomes compulsory. The main ..."
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We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our twolevel minimalist foundation, the choice for a pointfree approach to topology is not just a matter of convenience or mathematical elegance, but becomes compulsory. The main reason is that in our foundation real numbers, either as Dedekind cuts or as Cauchy sequences, do not form a set.
Quotients over Minimal Type Theory
 In Computation and Logic in the Real World CiE 2007, Siena, volume 4497 of LNCS
, 2007
"... Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid a ̀ la Bishop we build ..."
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Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid a ̀ la Bishop we build a model of qmTT over mTT. The design of an extensional type theory with quotients and its interpretation in mTT is a key technical step in order to build a two level system to serve as a minimal foundation for constructive mathematics as advocated in the mentioned paper about mTT.
CONSTRUCTING CATEGORIES AND SETOIDS OF SETOIDS IN TYPE THEORY
, 2013
"... Vol. 10(3:25)2014, pp. 1–14 www.lmcsonline.org ..."
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