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51
Relations in Concurrency
"... The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the seman ..."
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Cited by 304 (36 self)
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The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higherorder processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higherorder processes. For this it is useful to generalise event structures to allow events which “persist.”
Containers  Constructing Strictly Positive Types
, 2004
"... ... with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are t ..."
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Cited by 85 (28 self)
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... with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are the notions of containers and container functors, introduced in Abbott, Altenkirch, and Ghani (2003a). These provide a new conceptual analysis of data structures and polymorphic functions by exploiting dependent type theory as a convenient way to define constructions in MartinLöf categories. We also show that morphisms between containers can be full and faithfully interpreted as polymorphic functions (i.e. natural transformations) and that, in the presence of Wtypes, all strictly positive types (including nested inductive and coinductive types) give rise to containers.
On the Interpretation of Type Theory in Locally Cartesian Closed Categories
 Proceedings of Computer Science Logic, Lecture Notes in Computer Science
, 1994
"... . We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of exten ..."
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Cited by 61 (1 self)
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. We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory. 1 Introduction and Motivation Interpreting dependent type theory in locally cartesian closed categories (lcccs) and more generally in (non split) fibrational models like the ones described in [7] is an intricate problem. The reason is that in order to interpret terms associated with substitution like pairing for \Sigma types or application for \Pitypes one needs a semantical equivalent to syntactic substitution. To clarify the issue let us have a look at the "naive" approach described in Seely's seminal paper [14] which contains a subtle inaccuracy. Assume some dependently typed calculus like the one defined in [10] and an lccc C (a category ...
Internal Type Theory
 Lecture Notes in Computer Science
, 1996
"... . We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with f ..."
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Cited by 55 (8 self)
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. We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in MartinLof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families. 1 Introduction In a previous paper [8] I introduced a general notion of simultaneous inductiverecursive definition in intuitionistic type theory. This notion subsumes various reflection principles and seems to pave the way for a natural development of what could be called "internal type theory", that is, the construction of models of (fragments of) type theory in type theory, and more generally, the formalization of the metatheory of type theory in type theory. The present paper is a first investigation of such an internal type theor...
Wellfounded Trees in Categories
, 1999
"... this paper, we give an abstract 2 categorical characterization of Wtypes. We calculate these Wtypes explicitly in some categories of presheaves and sheaves on a site, and in the gluing category or Freyd cover. (We also have an explicit description in the case of Hyland's realizability topos, ..."
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Cited by 54 (10 self)
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this paper, we give an abstract 2 categorical characterization of Wtypes. We calculate these Wtypes explicitly in some categories of presheaves and sheaves on a site, and in the gluing category or Freyd cover. (We also have an explicit description in the case of Hyland's realizability topos, which will be presented in [17].) These explicit calculations can be formalized in a weak predicative metatheory, and lead to the result that if E is any suitably filtered pretopos with dependent products and Wtypes, then so is the category of internal sheaves on a site in E (Remark 5.9). Our paper is organized as follows. In Section 2 we review some standard definitions concerning pretoposes and dependent products. In Section 3 we present the categorical definition of the Wconstruction, and in Section 4 we prove some of its basic functoriality properties; e.g., that it turns coequalizers into equalizers. In Section 5, a construction is presented which to each map between (pre)sheaves of sets associates a sheaf of wellfounded trees, and it is proved that this is in fact the Wtype in the category (pre)sheaves of sets (Theorem 5.6). In Section 6, we discuss the Wconstruction for the Freyd cover. Finally, in Section 7 it is shown how these categorical constructions are not only analogous to but explicitly related to MartinLof type theory. 2 Pretoposes and dependent products
Presheaf Models for Concurrency
, 1999
"... In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their wo ..."
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Cited by 49 (19 self)
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In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a builtin notion of bisimulation. We show how
Interpolation in Grothendieck Institutions
 THEORETICAL COMPUTER SCIENCE
, 2003
"... It is well known that interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which ..."
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Cited by 39 (3 self)
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It is well known that interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which have recently emerged as an important mathematical structure underlying heterogenous multilogic specification. Our main result can be used in the applications in several different ways. It can be used to establish interpolation properties for multilogic Grothendieck institutions, but also to lift interpolation properties from unsorted logics to their many sorted variants. The importance of the latter resides in the fact that, unlike other structural properties of logics, many sorted interpolation is a nontrivial generalisation of unsorted interpolation. The concepts, results, and the applications discussed in this paper are illustrated with several examples from conventional logic and algebraic specification theory.
The Discrete Objects in the Effective Topos
 Proc. London Math. Soc
, 1990
"... The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subc ..."
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Cited by 34 (7 self)
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The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subcategories arises from