Gerard Huet and Amokrane Sabi. Constructive category theory. In In honor of Robin Milner. Cambridge University Press, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....enterprise. We decided to develop Universal Algebra as a general tool to de ne algebraic structures. Previous work on Algebra in Type Theory was done by Paul Jackson using the proof system Nuprl (see [14] by Peter Aczel on Galois Theory (see [1] and by Huet and Sa bi on Category Theory (see [13]) A large class of algebraic structures has been developed in Coq by Lo c Pottier. Another aim is the use of a two level approach to the derivation of propositions about algebraic objects (see [4] In this approach, statements about objects are lifted to a syntactic level where they can be ....

....[6] but the version of (intensional) type theory implemented in Coq does not. Nevertheless a model of extensional type theory inside intensional type theory has been constructed by Martin Hofmann (see [10] We use a variant of this model, which has already been implemented by Huet and Sa bi in [13] and used by Pottier. The elements of a type are build up using some constructors, and elements of a type are said to be equal when they are convertible. Thus a type cannot be de ned by a predicate over an other type (subtyping) or by rede ning the equality (quotienting) We allow ourselves to be ....

Gerard Huet and Amokrane Sabi. Constructive category theory. In In honor of Robin Milner. Cambridge University Press, 1997.

....partial setoids, but there are at least four possible de nitions for morphisms of partial setoids. Below we give these possible de nitions of setoids. None of them is original. The rst de nition has been used e.g. in the formalisation of Galois theory [4, 12] and of constructive category theory [29]. The second de nition has been used e.g. in the formalisation of polynomials [7] The other de nitions do not seem to appear in the context of formal proofs but have been used by M. Hofmann to interpret extensional concepts in intensional type theory [26, 27] 2.1 Total setoids A total setoid ....

G. Huet and A. Sabi. Constructive category theory. In G. Plotkin, C. Stirling, and M. Tofte, editors, Proof, Language and Interaction|Essays in honour of Robin Milner, pages 239-275. MIT Press, 2000. Setoids in type theory 27

....It is, however, not our intention to provide a complete formalization of category theory in type theory; we only define certain categorical notions that are necessary to express the subsequent polytypic developments. For a more elaborated account of category theory within type theory see e.g. [8]. Functors are twofold mappings: they map source objects to target objects and they map morphisms of the source category to morphisms of the target category with the requirement that identity arrows and composition are preserved. Here, we restrict the notion of functors to the category of types ....

G. Huet and A. Saibi. Constructive Category Theory. In Proceedings of the joint CLICS-TYPES Workshop on Categories and Type Theory, January 1995.

....1 Introduction Constructive type theory has been shown to be adequate for representing categorical reasoning. In this work we use Calculus of Inductive Constructions as implemented in Coq V6.1 to formalize a segment of category theory. We follow the axiomatization proposed by Huet and Saibi (see [HS95]) where objects are modeled as types and hom sets as hom setoids. We start this work by first presenting the axiomatization of the notion of category proposed by Huet and Saibi. Afterwards we define some examples to illustrate the previous axiomatization. Finally we present basic concepts and ....

....Build Equivalence. End BinRel. 2.2 Setoids To define a category as general as possible the objects and the morphisms can not be sets, or else we are only defining small categories. One possible axiomatization of category theory in Coq that solves this problem was proposed by Huet and Saibi (see [HS95]) In this work we adopt this solution and so we start by defining the structure Setoid. 2.2.1 The Setoid Structure Setoids are triples composed of a type Carrier, a relation Equal over Carrier and a proof that Equal is an equivalence relation. It is usual in mathematics to overload the notation ....

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G'erard Huet and Amokrane Saibi. Constructive category theory. Technical report, INRIA Rocquencourt, 1995.

....Formalisation of general logics 2.1 Syntax The syntax of a logical language is parametrised by a collection of signatures with morphisms between some of these signatures. The natural framework for de ning such a collection is category theory (for a formalisation of category theory in CoQ, see [8]) However, we do not adopt this framework and, in the following, we suppose this collection is countable and we de ne the type SIGN as follows: Record SIGN : Type : mkSign S : IN Set ; S : 8i; j : IN (S(n i ) S(j) Type . Hence, in order to de ne a object of type SIGN, we have to ....

G. Huet and A. Sabi. Constructive Category Theory. MIT Press, To appear 1998. preliminary version presented at CLICS-TYPES BRA meeting, 95.

....objects. It may also be the appropriate way to develop category theory inside a constructive framework such as Martin Lof type theory. As such it provides an alternative to E category theory (category theory where each hom set is equipped with an equivalence relation) as studied by (Aczel 1993) (Huet and Saibi 1995), and (Duval and Reynaud 1994) E categories were also studied abstractly by (Lack 1995) who shows them to be bicategories enriched over a monoidal bicategory. The fact that the P category theory we use can be formalized (programmed) in MartinL of type theory is one way of ensuring that our ....

G. Huet and A. Saibi , Constructive Category Theory, in: Proceedings of the Joint CLICSTYPES Workshop on Categories and Type Theory (Goteborg), Jan. 1995.

....style, which are mechanically checked by the system. The expressive power of this formalism allows to formalize and prove constructive mathematics, and some non trivial mathematical results have been checked with the Coq system (the most important one is a formalization of Category Theory [10]) Proofs in the Coq system are constructive, contrary to other systems like Mizar, HOL or PVS. The main interest of intuitionnistic logic is to have a computational interpretation. We will explain this point in details later on, but let use give the idea: to prove the proposition A ) B is to ....

G. Huet and A. Saibi. Constructive category theory. To appear, 1996.

....theory. A stronger representation is presented in Section 4.2. Note that we use the type variables o and a for objects and arrows, respectively. This means that any HOL type can be used to represent objects and arrows. Also note that we use HOL equality to compare arrows. In ALF [4] Coq [7] and LEGO [1] the equality is part of the category structure; so each category has its own equality on arrows. It is convenient to treat a category as one entity CC (corresponding to C) instead of as a 5 tuple. Therefore we shall make use of projections to extract the components of a category. ....

G. Huet and A. Sa ibi. Constructive category theory. Draft, 1995.

....monoidal category is then obtained by introducing an equivalence relation on such proofs. We can also view the present formalization as part of the larger enterprise of developing a formal constructive category theory inside intuitionistic type theory, see for example Aczel [1] Huet and Saibi [15], and Dybjer and Gaspes [8] We can compare this category theory to Rydeheard and Burstall s [18] Computational Category Theory. An essential difference is that the programming language of type theory, with dependent types and Curry Howard interpretation of the logical constants, is much more ....

G. Huet and A. Saibi. Constructive category theory. In Proceedings of the Joint CLICS-TYPES Workshop on Categories and Type Theory, Goteborg, January 1995.

.... theory which can be formalised in a constructive setting, for definiteness in extensional Martin Lof type theory with subset types as described in [NPS90] It seems worthwhile to carry out the formalization of category theory in a type theoretic setting, this is a topic of current research, e.g. [HS95]. 1.2 An intuitionistic completeness proof C. Coquand, T. Coquand and Dybjer have observed that there is a close analogy between intuitionistic completeness proofs and normalization. Indeed, there is an intriguing relationship between an intuitionistic completeness proof for Kripkestyle semantics ....

G'erard Huet and Amokrane Saibi. Constructive category theory. In Peter Dybjer and Randy Pollack, editors, Informal proceedings of the joint CLICSTYPES workshop on categories and type theory, 1995.

....are needed too. Here we show that almost a whole proof of this coherence theorem is hidden in a Curry Howard interpretation of a proof of normalization for monoids. The second point of the paper is to contribute to the development of constructive category theory in the sense of Huet and Saibi [16], who implemented part of elementary category theory in the proof assistant Coq. Here we extend the scope of constructive category theory to the area of coherence theorems (cf. also [9] We have formalized our proof in Martin Lof type theory and implemented it in the proof assistant ALF. An ....

....point is that we define N as the set of lists with elements in X and introduce an explicit injection J : N M . There are no true subsets in type theory. Hence the normalization function is defined as follows: Nf (a) J( a] e) 4. 4 Monoidal Categories We follow Aczel [1] Huet and Saibi [16], and Dybjer and Gaspes [10] and formalize a notion of category in intuitionistic type which does not have equality of objects as part of the structure. A category thus consists of a set of objects, but setoids of arrows indexed by pairs of objects. There is a family of identity arrows indexed by ....

G. Huet and A. Saibi. Constructive category theory. In Proceedings of the Joint CLICS-TYPES Workshop on Categories and Type Theory, Goteborg, January 1995.

....proof of normalization for monoids almost directly yields a coherence proof for monoidal categories. It is an interesting example of an application of intuitionistic type theory and can be viewed as part of the larger enterprise of constructive category theory in the sense of Huet and Saibi [10]. The paper shows how the ALF formalization makes essential use of the CurryHoward interpretation while the quite parallel HOL development does not. In this case study we make a systematic comparison of the two systems, both with respect to their very different logic bases (ALF is based on ....

G'erard Huet and Amokrane Saibi. Constructive category theory. In Proceedings of the Joint CLICS-TYPES Workshop on Categories and Type Theory, Goteborg, January 1995.

....stand for a setoid, the corresponding italic letter will stand for its carrier, and ref, trans, sym (with an italic letter as a subscript) stand for the proofs of reflexivity, transitivity, and symmetry, respectively. Categories and Functors in Type Theory. We follow Aczel [1] and Huet and Saibi [15] and define a category to have a set of objects, but hom setoids. We shall not need to refer to equality of objects. The object part of a functor is a function between the object sets and the morphism part is a family of maps between the hom setoids, such that the functor laws are satisfied with ....

....relation in the hom setoid. Setoids and maps under extensional equality ext form a category in type theory which plays the role of the category of sets in ordinary category theory. For the description of an implementation (in Coq) of category theory along these lines we refer to Huet and Saibi [15]. Setoid Indexed Families of Setoids in Type Theory. Definition4. Let A be a setoid. An A indexed family of setoids consists of a family B of setoids indexed by the set A; a reindexing map (P ) B(x 0 ) B(x) whenever P : x A x 0 . This family is coherent provided (ref A ) ....

G. Huet and A. Saibi. Constructive category theory. In Proceedings of the Joint CLICS-TYPES Workshop on Categories and Type Theory, Goteborg, January 1995.

.... (x A)R(op(x, e) x) Monoid One can use this type for proving some universal properties of monoids, or to check that a particular structure forms a monoid. 2.3 Categories Categories have been already considered in the context of intuitionistic type theory by P. Aczel[1] G. Huet and A. Sa#bi[19], P. Dybjer and V. Gaspes[13] All these de nitions basically follow the same pattern: a category consists of an object set and a family of Hom setoids. We 2.3 Categories 13 have not invented anything better and use the same de nition in this work. What seems interesting is that none of the ....

....de nition unfolding impressively shorted. See [3] for a comparison of the two formalisations. The work can be viewed as a part of a larger enterprise of formalising categorical notions in type theory. The most extensive work in this direction, to our knowledge, has been performed by Huet and Saibi [19, 29]. They implemented a large segment of category theory in Coq, a powerful system based on the Calculus of Constructions. Other examples include Aczel [1] Hedberg [17] Dybjer and Gaspes [13] However, none of these works considered coherence theorems. To our knowledge, they did not chase large ....

G#rard Huet and Amokrane Saibi. Constructive category theory. In Proceedings of the Joint CLICS-TYPES Workshop on Categories and Type Theory, G#teborg, January 1995.

.... there is a constructive proof of the Intermediate Value Theorem from Bishop s book [11] The most up to date library is by Forester [31] building on work of Chirimar and Howe [20] The Coq and Alf research groups have also formalized a number of interesting results in constructive mathematics [44]. Pollock [64] proved a strong normalization theorem for Lego. Werner and PaulinMohring [61] have proved a decidability theorem for propositional logic. A topological completeness proof for predicate logic was proved in Alf by Persson [63] and the Hahn Banach theorem in Coq [19] Anthony Bailey is ....

G. Huet and A. Sa ¨ ibi. Constructive category theory. In Gordon Plotkin, Colin Stirling and Mads Tofte, editors, Proof, Language and Interaction: Essays in Honour of Robin Milner. MIT Press, 1998. Presented at CLICS-TYPES BRA '95.

....C D) NTsetoid C D) CompCatFunct IdCatFunct) Having checked that we have all categorical properties, we may now define the functor category. Definition CatFunct : BuildCategory (Functor C D) NTsetoid C D) CompCatFunct IdCatFunct AssocCatFunct IdlCatFunct IdrCatFunct) End catfunctor. In [19] we give a few examples of applications of the above notions; for instance, we prove the Interchange Lemma. 3.5 Conclusion The development shown in this paper is but a tiny initial fragment of the theory of categories. However, it is quite promising, in that the power of dependent types and ....

G. Huet and A. Saibi. "Constructive Category Theory". Submitted for publication.

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G. Huet and A. Saibi. Constructive category theory. Technical report, INRIA Rocquencourt, 1995.

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