Kock and Reyes 1977 Anders Kock and Gonzalo Reyes, "Doctrines in categorical logic," in J. Barwise, editor, "Handbook of Mathematical Logic," Studies in Logic and Foundations of Mathematics 90, North-Holland (1977), 283--313.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....cubical sites can be interpreted as classifying categories for these theories. This allows to recover the universal property of the cubical sites and to exhibit them as presentation free versions of the theories. The exposition is modelled on the case of algebraic theories in cartesian categories [9, 20, 25] with the necessary generalisations; some of the ideas behind the analysis can be found in [2] and [21] For conciseness, we restrict here to the framework needed to discuss the cubical sites. Thus, the signatures are single sorted and the languages only allow weakening and exchange as structural ....

A. Kock, G. Reyes, Doctrines in categorical logic, In: J. Barwise (editor), Handbook of mathematical logic, North Holland 1977.

....is a C indexed family of natural transformations : 0 ; Q( 1) subject to some coherence conditions. A detailed treatment of indexed categories and brations can be found in [16] 2 Syntax In the following, we introduce several kinds of indexed categories called doctrines [19]. We abuse terminology, since a doctrine is generally understood to be an indexed category where reindexing functors have left adjoints, and this property does not always holds for our doctrines. We have chosen this terminology to emphasize the relation between indexed categories used for the ....

A. Kock and G. E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283-313. North Holland, 1977.

....the arrows V i V j in B may be regarded as j tuples of type expressions, each having all type variables among t 1 ; t i . Composition in B corresponds to substitution of types for type variables. This relatively standard interpretation of an algebraic language is explained in [KR77], for example. It is helpful to note that we may 13 regard an arrow V i V j as either a j tuple of types, or a context Gamma = fx 1 : 1 ; x j : j g of length j, with all type expressions over the same set of i type variables. We now consider the interpretation of terms. For each ....

A. Kock, G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283--313. North-Holland, 1977.

.... together with a family of suitable natural transformations, usually denoted as diagonals and projections (related papers range from [37,70] to the more recent [41,48] Then, our definition of algebraic theory can be proved equivalent to the classical one, dating back to the early work of Lawvere [47,50]. The following, classical result states the equivalence between these theories and the usual term algebra construction for ordinary signatures. Proposition 2.1 (algebraic theories and term algebras) Given an ordinary signature Sigma, for all n; m 2 N c there exists a one to one ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283--313. North Holland, 1977.

.... ordinals, n] and the morphisms [n] m] to be m tuples of elements of the polynomial Heyting algebra in n indeterminates, H[X 1 ; X n ] Composition is given by substitution and the identity on [n] is (X 1 ; X n ) Lawvere s categorical treatment of algebraic theories (see [7], for example) tells us that C (has finite products and) contains the generic model of the algebraic theory of Heyting algebras equipped with a morphism from H . In particular, C does contain a Heyting algebra object, namely U = 1] its top and bottom elements are ( 0] 1] and its ....

A. Kock and G. E. Reyes, Doctrines in categorical logic. In: J. Barwise (ed.), Handbook of Mathematical Logic (North-Holland, Amsterdam, 1977), Chapter A.8.

....and its use in providing a functorial view of algebras. As Kock and Reyes summarize, the right way of conceiving the totality of operations for an equational theory was found by Lawvere, who realized that substitution should be viewed as the composition of arrows on a certain kind of category [27]. Definition 3.1 (algebraic theories) An algebraic theory C is a category whose objects are (underlined) natural numbers, and where for each n there is an n tuple of distinguished morphisms (or projections) f n i : n 1 j i = 1 : ng, making n the n fold categorical product of 1, that is, ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283-313. North Holland, 1989.

....a presentation biased towards the process algebra framework we deal with in Sections 3 and 4. For a comprehensive introduction we refer the reader to [12] 2. 1 Building States We open this section recalling some definitions from graph theory, that will be used to introduce algebraic theories [21,18]. Developed in the early Sixties, these theories received a lot of attention during the Seventies from computer scientists as a suitable characterization of the ordinary notion of term algebras. Definition 2.1 (graphs) A graph G is a 4 tuple hOG ; AG ; ffi 0 ; ffi 1 i: OG , AG are sets whose ....

A. Kock, G.E. Reyes, Doctrines in Categorical Logic, in Handbook of Mathematical Logic, ed. John Bairwise, North Holland, 1977, pp. 283-313.

.... with a family of suitable natural transformations, usually denoted as diagonals and projections (related papers range from [20, 39] to the more recent [22, 28] Then, our notion of algebraic theory can be proved equivalent to the classical definition, dating back to the early work of Lawvere [27, 31]: Hence, a classical result states the equivalence of these theories with the usual term algebras. Proposition 1 (algebraic theories and term algebras) Let Sigma be a signature. Then for all n; m 2 IlN c there exists a one to one correspondence between the set of arrows from n to m of A( Sigma) ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283--313. North Holland, 1977.

.... together with a family of suitable natural transformations, usually denoted as diagonals and projections (related papers range from [37,70] to the more recent [41,48] Then, our definition of algebraic theory can be proved equivalent to the classical one, dating back to the early work of Lawvere [47,50]. The following, classical result states the equivalence between these theories and the the usual term algebra construction for ordinary signatures. Proposition 2.1 (algebraic theories and term algebras) Given an ordinary signature Sigma, for all n; m 2 N c there exists a one to one ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283--313. North Holland, 1977.

....rewriting . Before explaining concisely the main contribution of the paper, it is worth recalling that the terms over a given signature Sigma can be regarded as the arrows of a cartesian category (called the algebraic theory of Sigma ) freely generated (in a suitable way) by Sigma (see e.g. [21, 18]) Such a category has (underlined) natural numbers as objects, and its generators are arrows like g : n 1, where g is an operator of arity n in Sigma ; in this category arrows from n to m are in oneto one correspondence with m tuples of terms over n variables. Furthermore, a term rewrite rule ....

A. Kock and G.E. Reyes. Doctrines in Categorical Logic. In J. Bairwise, editor, Handbook of Mathematical Logic, pages 283--313. North Holland, 1977.

....Section 6. 4) Before explaining concisely the main contribution of the paper, it is worth recalling that the terms over a given signature Sigma can be regarded as the arrows of a Cartesian category (called the algebraic theory of Sigma) freely generated (in a suitable way) by Sigma (see, e.g. [40, 43]) Such a category has (underlined) natural numbers as objects, and its generators are arrows like g : n 1, where g is an operator of rank n in Sigma. It 1 Here gs stands for graph substitution , an acronym whose explanation we defer to the end of Section 5. definitivo.tex; 30 11 1999; ....

....for both the fields of mathematics and computer science. As Kock and Reyes summarised, the right way of conceiving the totality of operations for an equational theory was found by Lawvere, who realized that substitution should be viewed as the composition of arrows on a certain kind of category [40]. DEFINITION 27. Algebraic theories) An algebraic theory C is a category whose objects are underlined natural numbers, and which for definitivo.tex; 30 11 1999; 16:52; p.25 26 each n is equipped with an n tuple of maps f n i : n 1 j i = 1 : ng, making n the n fold Cartesian product of ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283--313. North Holland, 1977.

....and its use in providing a functorial view of algebras. As Kock and Reyes summarize, the right way of conceiving the totality of operations for an equational theory was found by Lawvere, who realized that substitution should be viewed as the composition of arrows on a certain kind of category [11]. Definition 3 (algebraic theories) An algebraic theory C is a category whose objects are (underlined) natural numbers, such that for each n it is equipped with an n tuple of distinguished morphisms (or projections) f n i : n 1 j i = 1 : ng, making n the n fold categorical product of 1, ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283--313. North Holland, 1977.

....the different classes of abstract sequents. All the relevant categorical notions are given in the appendix, but note however that this section can be safely skipped at a first reading, without affecting the understanding of the rest of the paper. Introduced in the early Sixties, algebraic theories [Law63, KR77] received a lot of attention during the Seventies from computer scientists: they were used to characterize in an alternative way the usual definition of term algebra over an ordinary signature. The following definition provides, for a given signature Sigma , the categorical description of the ....

A. Kock, G.E. Reyes, Doctrines in Categorical Logic, in Handbook of Mathematical Logic, ed. John Bairwise, North Holland, 1977, pp. 283-313.

....( x(c] 8x: c x ) 3. c: T (hx(ci ) 9x: c x ) 4. c: T ; v: c v) hx(ci(x = v) 3 Categorical semantics Given a category C with nite products, the general pattern for interpreting a typed calculus according to Lawvere s functorial semantics goes as follows (see [KR77, Law63]) a context and a type type are interpreted by objects of C, by abuse of notation we will indicate these objects with and respectively; in particular, the empty context ; is interpreted by the terminal object 1 and the context ; x: is interpreted by ; a term e: is ....

.... (e.g. see [Pit88, Pit89, Mog91] In categorical logic there are two approaches which extend Lawvere s functorial semantics to typed predicate logics: the internal approach interprets formulas as subobjects, while the external approach interprets formulas in the bers of a C indexed category (see [KR77, Osi73, PS78]) The obvious trade o between these two approaches is that the rst is closer to models, while the second is closer to theories. In this section we recall the categorical structures proposed in the literature for interpreting computational types (strong monads) and logical constants. 11 3.1 ....

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A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic. North Holland, 1977.

....a few features) Syntax independent view. The rst ingredient of such a methodology should be to abstract as much as possible from the concrete presentation of a language, so that one can focus only on the underlying mathematical structures . This is standard practice in Categorical Logic (see [KR77, Pitar]) where theories are identi ed with categories having certain additional structure. We follow a similar paradigm for programming languages. In particular, we propose to identify a programming language where type expressions are evaluated independently from program expressions with an indexed ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic. North Holland, 1977.

....not the metalanguage. In Categorical Logic it is common practice to identify a theory T with a category F(T ) with additional structure such that there is a one one correspondence between models of T in a category C with additional structure and structure preserving functors from F(T ) to C (see [KR77]) 3 . This identi cation was originally proposed by Lawvere, who also showed that algebraic theories can be viewed as categories with nite products. In Section 2.2 we give a class of theories that can be viewed as categories with a monad, so that any category with a monad is, up to equivalence ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic. North Holland, 1977.

....ttl has been accepted for publication [7] 2 Background In tile logic, configurations and observations form two monoidal categories sharing the same class of objects. Category theory offers a convenient characterization of both configurations and observations also in terms of algebraic theories [18,17], described in Appendix A. The free algebraic theory associated to a (one sorted) signature Sigma is called the Lawvere theory for Sigma. A Lawvere theory is just an alternative presentation of a signature, because the additional term structure is generated in a completely free way. We refer to ....

A. Kock and G.E. Reyes, Doctrines in categorical logic, in: John Barwise, Ed., Handbook of Mathematical Logic, North Holland, 283--313 (1977).

.... with a family of suitable natural transformations, usually denoted as diagonals and projections (related papers range from [19, 37] to the more recent [21, 27] Then, our definition of algebraic theory can be proved equivalent to the classical one, dating back to the early work of Lawvere [30, 26]: Hence, a classical result states the equivalence of these theories with the usual term algebra. Proposition6 (algebraic theories and term algebras) Let Sigma be a signature. Then for all n; m 2 IlN c there exists a one to one correspondence between the set of arrows from n to m of A( Sigma) ....

A. Kock and G.E. Reyes. Doctrines in Categorical Logic. In J. Bairwise, editor, Handbook of Mathematical Logic, pages 283--313. North Holland, 1977.

....term tile systems (tTS) 2.2 Term Tile Systems In what follows we consider one sorted signature only. The many sorted case can be handled very easily in a similar way, but requires a more complex notation that is not necessary for our case study and therefore avoided. An algebraic theory [27, 28, 24] is just a cartesian category having underlined natural numbers as objects. The free algebraic theory associated to a (one sorted) signature Sigma is called the Lawvere theory for Sigma , and is denoted by Th[ Sigma] the arrows from m to n are in a one to one correspondence with n tuples of ....

A. Kock and G.E. Reyes. Doctrines in Categorical Logic. In: John Bairwise, Ed., Handbook of Mathematical Logic. North Holland, 283--313 (1977).

....view, we can generalize the notion of signature to consist of functional symbols which are arrows from IlN to IlN, where IlN is the set of underlined natural numbers. Thus we have arrows of the form f : n m. Starting from a given signature Sigma, we can define an associated algebraic theory [24, 21], written A( Sigma) which have been shown to be an alternative characterization of term algebras T Sigma (X) over a set X of variables. In fact, we will use algebraic theories to describe the terms representing sub parts of a system and also (but not in their generality) the synchronization ....

A. Kock, G.E. Reyes, Doctrines in Categorical Logic, in Handbook of Mathematical Logic, ed. John Bairwise, North Holland, 1977, pp. 283-313.

....the translations defined in Examples 2.10 and 2.12 are indeed interpretations between suitable HML theories. 3 Categorical view The idea of functorial semantics is that certain fragments of logical theories correspond to certain choices of categorical properties P , sometimes called doctrines ([Law75, KR77]) so that theories of kind P can be identified with categories with P structure. In this setting, interpretations of a theory T in a category C with P structure correspond to P preserving functors from T to C, that is, models of T in C are objects in the functor category [T; C] P . According to ....

A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of mathematical logic, pages 283--313. North-Holland, 1977.

....and the functoriality axiom (of tensor product Omega ) expresses a basic fact about the true concurrency of the model. A second example, showing that the use of categories offer a general and convenient characterization also of configurations, is given by Lawvere theories. An algebraic theory [44, 45, 40] is just a cartesian category having natural numbers as objects. The free algebraic theory associated to a (one sorted) signature Sigma is called the Lawvere theory for Sigma, and is denoted by Th[ Sigma] also L Sigma ) the arrows from m to n are in a one to one correspondence with n tuples ....

A. Kock and G.E. Reyes. Doctrines in Categorical Logic. In: John Bairwise, Ed., Handbook of Mathematical Logic. North Holland. 1977. pp. 283--313.

....3.4 Cartesian 2 Categories and Term Rewriting Systems We open this section lifting the definition of cartesian product and terminal object to cells, then showing the relationship between 2 categories with such an additional structure and trs s. In particular, we will introduce Lawvere Theories [Law63, KR77] as a well suited category for representing terms of an algebra: their structure is exactly matched by the cartesian one of these theories. For a quick reference to universal algebras and trs s, we refer the reader to Appendix A. We recall that a category C is cartesian if it has a terminal object ....

A. Kock, G.E. Reyes, Doctrines in Categorical Logic, in Handbook of Mathematical Logic, ed. John Bairwise, North Holland, 1977, pp. 283-313.

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Kock and Reyes 1977 Anders Kock and Gonzalo Reyes, "Doctrines in categorical logic," in J. Barwise, editor, "Handbook of Mathematical Logic," Studies in Logic and Foundations of Mathematics 90, North-Holland (1977), 283--313.

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