J. Cartmell. Generalised Algebraic Theories and Contextual Categories. PhD thesis, University of Oxford, July 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 jAj (8U 2 jAj:X U implies a 2 U) The easy details are omitted. 2 132 = Logical System Our goal is to define a logic in which we are able to solve recursive domain equations with the aid of the fixpoint type. The approach we adopt is to set up a logic in which there is a universal type [Car86]. The elements of this type act as codes for the external or observable types. Thus a recursive type can be realised by considering the corresponding fixpoint of the universal type. In order to make things precise, we shall define a dependently typed equational logic called FIX = This is ....

....claim is immediate from the FIX = rules. 2 139 140 Categorical Semantics of the FIX 9.1 Categories for Modelling Dependent Type Theories We review a categorical structure which can be used to model dependent type theories. Some of the earliest work in this area was undertaken by Cartmell [Car86] with additional work by Taylor [Tay86] Here we shall give a presentation of categories with attributes which is based on on Pitts account in [Pit95] Further useful information can be found in [Str89] HP89] CGW89] and [Ben85] Categories with Attributes Definition 9.1.1 A category with ....

J. Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209--243, 86.

....This is given by Kripke resource models, which are monoid indexed sets of functorial Kripke models. The indexing element can be seen as the resource able to realize the structure it indexes. We work with indexed categories rather than, for example, with Cartmell s contextual categories [4], as the indexed approach allows a better separation of the evident conceptual issues. Kripke resource models generalize, as we might expect, the functorial Kripke models of the lP calculus [18] These consist of a functor J: W ; C ; Cat] where W is a Kripke world structure, C is a category ....

....More specifically, in the case of the type theory, the judgement G S M:A is modelled as the arrow 1 JAK in the fibre over JGK in the strict indexed category E :C Cat. Alternative approaches to the semantics of (intuitionistic) dependent types are presented in, for example, Cartmell [4], Pitts [16] These presentations lack the conceptual distinction provided by indexed categories. We need the base category C to account for the structural features of the type theory and its internal logic, hence the following definition: Definition 3. A doubly monoidal category is a category C ....

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J Cartmell. Generalised algebraic theories and contextual categories. Ann. Pure Appl. Logic, 32:209--243, 1986.

....monoidindexed sets of functorial Kripke models, fJ r : W ; C op ; Cat] j r 2 Rg. The indexing element r 2 R can be seen as the resource able to realize the functorial Kripke structure it indexes. We work with indexed category theory, rather than, for example, Cartmell s contextual categories [5], as the indexed approach allows us to separate certain conceptual issues and, hence, allows us to recognize the extra structure needed for studying the model theory of structurally weaker logics and type theories than the intuitionistic ones. We will see this later, in Section 3.2, when we ....

....category theory [24] More speci cally, in the case of the type theory, the judgement M :A is modelled as the arrow 1 JMK JAK in the bre over J K in the strict indexed category E :C op Cat. We remark that this is not the only technique for modelling a typing judgement; Cartmell [5], Pitts [25] and several other authors use a more onedimensional structure which relies on the properties of certain classes of maps to model the intuitionistic fragment of the calculus. These are formally equivalent to the indexed approach but the latter is appealing for one main reason: it ....

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J Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209-243, 1986.

....basic ones as auxiliary. This may be preferred because the three auxiliary judgment forms correspond directly to the categorical notions of object, morphism, and equality between morphisms, or merely because it is formally neat. One can think of the logical framework as the (generalised algebraic [13]) theory of a categorical structure in which the key component is a category of contexts. The intuition for a context is that it is a coordinate space in which the points are ordered sequences or vectors of values. A morphism between contexts is a translation of points which defines each ....

J. Cartmell. Generalised Algebraic Theories and Contextual Categories. PhD thesis, University of Oxford, July 1990.

....Cambridge Univ. Press, 1994) This talk will summarize the recent progress on the problem of whether each pure morphism in a accessible category is a regular monomorphism. JANELIDZE, G. Higher Dimensional Central Extensions: A Categorical Approach to Homology Theory of Groups As shown in [2], the so called double central extensions of groups can be described as the 2 dimensional coverings with respect to the standard adjunction between groups and abelian groups. Now we extend this result to higher dimensions. Using the generalized Hopf formula [1] we show that all homology groups of ....

....is needed. The construction Circ (defined in [5] is discussed. This provides a fundamental example of a bicategory of circuits (which is also called Circ) and it comes equipped with a tensor product and a feedback operation. In fact, this bicategory is an example of a (weak) equipment (see [2]) How the theory of the latter may be used as a tool for handling some constructions important to Computer Science is indicated. The Grothendieck construction is shown to exhibit circuits as a class of spans of categories (which are characterized) relating the theory to [6] As a result of this ....

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J. Cartmell, Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 (1986), 209-243.

....cally, in the case of the type theory, the judgement M :A is modelled as the arrow 1 JMK JAK in the bre over J K in the strict indexed category E :D op V , where V is a category of values . We remark that this is not the only technique for modelling a typing judgement; Cartmell [28], Pitts [138] and several other authors use a more one dimensional structure which relies on the properties of certain classes of maps to model type dependency and dependent function spaces. These are essentially equivalent to the indexed approach but the latter is appealing for the main reason ....

....conceptually separate issues. Moreover, indexed techniques seem better suited to generalizations concerned with weaker type theories [86] For instance, at a logical level, the base and bres deal, respectively, with terms and propositions. Nevertheless, these ideas owe much to work of Cartmell [28], Pitts [138] Seely [160] Streicher [167] and others. We take a Kripke prestructure to be a functor J : W; D op ; V] where W is a small category (of worlds) D op = W D op W , where W ranges over the objects of W, and each DW (the base at W ) is small; V, a subcategory of the ....

J. Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209-243, 1994.

....of arrows is associative with the unit arrows acting as left and right unit, that is, such that (f Delta g) Delta h = f Delta (g Delta h) and 1 Delta f = f Delta 1 = f for arbitrary arrows f , g and h of the appropriate sorts. The generalized algebraic theories (GAT) of Cartmell [5] may be used to present the axiom system for an abstract category. In doing so there is a choice concerning the treatment of arrow equality. One alternative is to express the arrow equations, such as, for instance, associativity of arrow composition, in terms of the built in equality that goes ....

J. Cartmell. Generalised Algebraic Theories and Contextual Categories. Annals of Pure and Applied Logic, 32:209--243, 1986.

.... Gamma Gamma f R R Gamma Gamma Gamma Gamma p(f; A 00 ) We require a canonical choice of p(f; A 00 ) and f A 00 ) which is functorial in f , i.e. p(f ; f 0 ; A 00 ) p(f; A 00 ) p(f 0 ; f A 00 ) dually to contextual categories, cf. Car86] This will be needed for Prop. 8.3 only. As an example, we define the logical system associated with many sorted equational logic. Definition 2.11 Let Sig EQ be the category of many sorted algebraic signatures having: Objects: Pairs Sigma = S; Omega Gamma consisting of a set S of type ....

J. Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209--243, 1986.

....monoid indexed sets of functorial Kripke models, fJ r : W ; C op ; Cat] j r 2 Rg. The indexing element r 2 R can be seen as the resource able to realize the functorial Kripke structure it indexes. We work with indexed category theory, rather than, for example, Cartmell s contextual categories [5], as the indexed approach allows us to separate certain conceptual issues and, hence, allows us to recognize the extra structure needed for studying the model theory of structurally weaker logics and type theories than the intuitionistic ones. We will see this later, in Section 3.2, when we ....

....[21] More specifically, in the case of the type theory, the judgement Gamma Sigma M :A is modelled as the arrow 1 JMK JAK in the fibre over J GammaK in the strict indexed category E :C op Cat. We remark that this is not the only technique for modelling a typing judgement; Cartmell [5], Pitts [22] and several other authors use a more one dimensional structure which relies on the properties of certain classes of maps to model the intuitionistic fragment of the calculus. These are formally equivalent to the indexed approach but the latter is appealing for one main reason: it ....

[Article contains additional citation context not shown here]

J Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209--243, 1986.

....case of the type theory, the judgement Gamma Sigma M :A is modelled as the arrow 1 JMK JAK in the fibre over J GammaK in the strict indexed category E :D op V , where V is a category of values . We remark that this is not the only technique for modelling a typing judgement; Cartmell [28], Pitts [138] and several other authors use a more one dimensional structure which relies on the properties of certain classes of maps to model type dependency and dependent function spaces. These are essentially equivalent to the indexed approach but the latter is appealing for the main reason ....

....separate issues. Moreover, indexed techniques seem better suited to generalizations concerned with weaker type theories [86] For instance, at a logical level, the base and fibres deal, respectively, with terms and propositions. Nevertheless, these ideas owe much to work of Cartmell [28], Pitts [138] Seely [160] Streicher [167] and others. We take a Kripke prestructure to be a functor J : W; D op ; V] where W is a small category (of worlds) D op = W D op W , where W ranges over the objects of W, and each DW (the base at W ) is small; V, a subcategory of the ....

J. Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209--243, 1994.

....types in Section 3. We will just consider a single example of this, namely the categorical semantics of dependent products in Section 6.5. Several different, but interconnected, categorical structures have been proposed for interpreting the basic framework of dependent types by Seely [ 1984 ] Cartmell [ 1986 ] Taylor [ 1986 ] Ehrhard [ 1988 ] Streicher [ 1989, 1991 ] Hyland and Pitts [ 1989 ] Obtu lowicz [ 1989 ] Curien [ 1989 ] and Jacobs [ 1991 ] This reflects the fact that the categorical interpretation of dependent types is undoubtedly more complicated than the other varieties of ....

....of categorical logic explained in this chapter. This is due to the structural complications implicit in the logic. One complication is that in general one must consider the satisfaction of equations not only between terms (of the same type) but also between types. Another complication is that 1 Cartmell [ 1986 ] calls such theories generalized algebraic . Categorical Logic 63 proofs of well formedness of expressions cannot be separated from proofs of equality, and are by no means unique. If one assigns meanings to expressions by induction on the derivation of their well formedness, one therefore has ....

[Article contains additional citation context not shown here]

J. Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209--243, 1986.

.... calculus. This is given by Kripke resource models, which are monoid indexed sets of functorial Kripke models. The indexing element can be seen as the resource able to realize the structure it indexes. We work with indexed categories rather than, for example, with Cartmell s contextual categories [4], as the indexed approach allows us to separate certain conceptual issues and, hence, allows us to recognize the extra structure needed for studying the model theory of structurally weaker logics and type theories than intuitionistic ones. Kripke resource models generalize, as we might expect, the ....

....theory. More specifically, in the case of the type theory, the judgement Gamma Sigma M :A is modelled as the arrow 1 JMK JAK in the fibre over J GammaK in the strict indexed category E :C op Cat. We remark that this is not the only technique for modelling a typing judgement; Cartmell [4], Pitts [15] and several other authors use a more onedimensional structure which relies on the properties of certain classes of maps to model the intuitionistic fragment of the calculus. These are formally equivalent to the indexed approach but the latter is appealing for the main reason that ....

[Article contains additional citation context not shown here]

J Cartmell. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32:209--243, 1986.

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J. Cartmell. Generalised Algebraic Theories and Contextual Categories. PhD thesis, University of Oxford, July 1990.

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