This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of many-sorted predicate logic.

This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...

this paper, we give an abstract 2 categorical characterization of W-types. We calculate these W-types 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 W-types, 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 W-construction, 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 W-type in the category (pre)sheaves of sets (Theorem 5.6). In Section 6, we discuss the W-construction for the Freyd cover. Finally, in Section 7 it is shown how these categorical constructions are not only analogous to but explicitly related to Martin-Lof type theory. 2 Pretoposes and dependent products

It is well known that one can build models of full higher-order dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 -spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this category|in contradistinction to Top 0 |is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene-Kreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally cartesian closed categories. We also show how to interpret rst-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic rst-order logic. As a consequence, a modied double-negation translation into type theory (without bracket types) is complete for all of classical rst-order logic.

TinkerType is a pragmatic framework for compact and modular description of formal systems (type systems, operational semantics, logics, etc.). A family of related systems is broken down into a set of clauses – individual inference rules – and a set of features controlling the inclusion of clauses in particular systems. Simple static checks are used to help maintain consistency of the generated systems. We present TinkerType and its implementation and describe its application to two substantial repositories of typed lambda-calculi. The first repository covers a broad range of typing features, including subtyping, polymorphism, type operators and kinding, computational effects, and dependent types. It describes both declarative and algorithmic aspects of the systems, and can be used with our tool, the TinkerType Assembler,to generate calculi either in the form of typeset collections of inference rules or as executable ML typecheckers. The second repository addresses a smaller collection of systems, and provides modularized proofs of basic safety properties.

Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects

We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on Fraenkel-Mostowski (FM) set theory how to address this through first-class names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier N , name-binding, and unique choice of fresh names. The Schanuel topos --- the category underlying FM set theory --- is an instance of this axiomatisation.

. Data types like trees which are finitely branching and of (possibly) infinite depth are described by iterating initial algebras and terminal coalgebras. We study proof principles for such data types in the context of categorical logic, following and extending the approach of [14, 15]. The technical contribution of this paper involves a description of initial algebras and terminal coalgebras in total categories of fibrations for lifted "datafunctors". These lifted functors are used to formulate our proof principles. We test these principles by proving some elementary results for four kinds of trees (with finite or infinite breadth or depth) using the proof tool pvs. 1 Introduction Algebras and coalgebras are of well-established importance in computer science, notably in the theory of datatypes, where especially initial algebras and terminal coalgebras play a distinguished role. Over the past decade there is more and more interest in the logic associated with initial algebras and ter...

The Theory of Contexts is a type-theoretic axiomatization which has been recently proposed by some of the authors for giving a metalogical account of the fundamental notions of variable and context as they appear in Higher Order Abstract Syntax. In this paper, we prove that this theory is consistent by building a model based on functor categories. By means of a suitable notion of forcing, we prove that this model validates Classical Higher Order Logic, the Theory of Contexts, and also (parametrised) structural induction and recursion principles over contexts. The approach we present in full detail should be useful also for reasoning on other models based on functor categories. Moreover, the construction could be adopted, and possibly generalized, also for validating other theories of names and binders. Contents 1 The object language 4 2 The metalanguage (Framework System #) 6 2.1 Syntax 6 2.2 Typing and logical judgements 7 2.3 Adequacy of the encoding 8 2.4 Remarks on the design of # 9 3 Category-theoretic preliminaries 11 4.1 The ambient categories 4.2 Interpreting types 16 4.3 Interpreting environments 18 4.4 Interpreting the typing judgement of terms 19 4.5 Interpreting logical judgements 21 is a model of # 22 5.1 Forcing 22 5.2 Characterisation of Leibniz equality 23 models logical axioms and rules 26 models the Theory of Contexts 27 6 Recursion 28 6.1 First-order recursion 28 6.2 Higher-order recursion 31 7 Induction 33 7.1 First-order induction 34 7.2 Higher-order induction 37 8 Connections with tripos theory 38 9 Related work 41 9.1 Semantics based on functor categories 41 9.2 Logics for nominal calculi 44 10 Conclusions 45 A Proofs 46 A.1 Proof of Proposition 4.2 46 A.2 Proof of Proposition 4.3 47 A.3 Proof of Theorem 5.1 48 A.4 Proof of...