Papers by thomas streicher
Oxford University Press eBooks, Oct 6, 2005
We discuss a notion of universe in toposes which from a logical point of view gives rise to an ex... more We discuss a notion of universe in toposes which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH) that allows one to construct families of types in such a universe by structural recursion and to quantify over such families. Further, we show that (hierarchies of) such universes do exist in all sheaf and realizability toposes but neither in the free topos nor in the V ω+ω model of Zermelo set theory. Though universes in Set are necessarily of strongly inaccessible cardinality it remains an open question whether toposes with a universe allow one to construct internal models of Intuitionistic Zermelo Fraenkel set theory (IZF). The background information about toposes and fibred categories as needed for our discussion in this paper can be found e.g. in the fairly accessible sources [MM, Jac, Str2].
arXiv (Cornell University), Jan 9, 2018
The notion of fibered category was introduced by A. Grothendieck for purely geometric reasons. Th... more The notion of fibered category was introduced by A. Grothendieck for purely geometric reasons. The "logical" aspect of fibered categories and, in particular, their relevance for category theory over an arbitrary base category with pullbacks has been investigated and worked out in detail by Jean Bénabou. The aim of these notes is to explain Bénabou's approach to fibered categories which is mostly unpublished but intrinsic to most fields of category theory, in particular to topos theory and categorical logic. There is no claim for originality by the author of these notes. On the contrary I want to express my gratitude to Jean Bénabou for his lectures and many personal tutorials where he explained to me various aspects of his work on fibered categories. I also want to thank J.-R. Roisin for making me available his handwritten notes [Ben2] of Des Catégories Fibrées, a course by Jean Bénabou given at the University of Louvain-la-Neuve back in 1980. The current notes are based essentially on [Ben2] and quite a few other insights of J. Bénabou that I learnt from him personally. The last four sections are based on results of J.-L. Moens's Thése [Moe] from 1982 which itself was strongly influenced by [Ben2].
Archive for Mathematical Logic, Jun 7, 2006
arXiv (Cornell University), May 2, 2023
We prove that cloven Grothendieck fibrations over a fixed base B are the pseudo-coalgebras for a ... more We prove that cloven Grothendieck fibrations over a fixed base B are the pseudo-coalgebras for a lax idempotent 2-comonad on Cat /B. We show this via an original observation that the known colax idempotent 2-monad for fibrations over a fixed base has a right 2-adjoint. As an important consequence, we obtain an original cofree construction of a fibration on a functor. We also give a new, conceptual proof of the fact that the forgetful 2-functor from split fibrations to cloven fibrations over a fixed base has both a left 2-adjoint and a right 2-adjoint, in terms of coherence phenomena of strictification of pseudo-(co)algebras. The 2-monad for fibrations yields the left splitting and the 2-comonad yields the right splitting. Moreover, we show that the constructions induced by these coherence theorems recover Giraud's explicit constructions of the left and the right splittings.
arXiv (Cornell University), May 12, 2020
In [HJP80] Hyland, Johnstone and Pitts introduced the notion of tripos for the purpose of organiz... more In [HJP80] Hyland, Johnstone and Pitts introduced the notion of tripos for the purpose of organizing the construction of realizability toposes in a way that generalizes the construction of localic toposes from complete Heyting algebras. In [Pit02] one finds a generalization of this notion eliminating an unnecessary assumption of [HJP80]. The aim of this paper is to characterize triposes over a base topos S in terms of so-called constant objects functors from S to some elementary topos. Our characterization is slightly different from the one in Pitts's PhD Thesis [Pit81] and motivated by the fibered view of geometric morphisms as described in [Str20]. In particular, we discuss the question whether triposes over Set giving rise to equivalent toposes are already equivalent as triposes.
Semantics of Type Theory
Birkhäuser Boston eBooks, 1991
Logical Methods in Computer Science, Sep 11, 2014
In this paper we revise and simplify the notion of observationally induced algebra introduced by ... more In this paper we revise and simplify the notion of observationally induced algebra introduced by Simpson and Schröder for the purpose of modelling computational effects in the particular case where the ambient category is given by classical domain theory. As examples of the general framework we consider the various powerdomains. For the particular case of the Plotkin powerdomain the general recipe leads to a somewhat unexpected result which, however, makes sense from a Computer Science perspective. We analyze this "deviation" and show how to reobtain the original Plotkin powerdomain by imposing further conditions previously considered by R. Heckmann and J. Goubault-Larrecq.
We show that a certain simple call-by-name cointinuation semantics of Parigot's A, -calculus is c... more We show that a certain simple call-by-name cointinuation semantics of Parigot's A, -calculus is cornplete. More precisely, for every A,-theory we coinstruct a Cartesian closed category such that the ensuing continuation-style interpretation of A,, which maps terms to functions sending abstract continuations to responses, is full and faithful. Thus, any &-category in the sense of is isomorphic to a continuation model derived from a cartesiairlclosed category of continuations.
Categorical reconstruction of a reduction free normalization proof
Lecture Notes in Computer Science, 1995
RST-Order Set Theories, Toposes and Categories of Classes
Theory and Applications of Categories, Jul 20, 2021
We give examples of essential local geometric morphisms which are not locally connected although ... more We give examples of essential local geometric morphisms which are not locally connected although their inverse image parts give rise to exponential ideals.
Reduction-Free Normalisation for a Polymorphic System
The Scott Model of PCF
WORLD SCIENTIFIC eBooks, Dec 1, 2006
Computability in Domains
WORLD SCIENTIFIC eBooks, Dec 1, 2006
Model Theory of Denotational Semantics
Informatik-Fachberichte, 1985
The category of complete partial orders (cpo-s) has been suggested as the category where to inter... more The category of complete partial orders (cpo-s) has been suggested as the category where to interpret programs, expressions, declarations etc. One usually restricts oneself to certain subcategories such as ;-algebraic cpo-s, Scott domains or Plotkins SFP objects (sequences of finite partial orders). A quite different category is the category of sequential algorithms on concrete data structures introduced by Berry and Curien. Thus we can conclude that the model we choose to interpret a programming language depends on certain additional assumptions. There does not exist a unique model and so it is worthwhile to find a general definition of a notion of model for the metalanguage of denotational semantics. 1. The method of denotational semantics
Correctness of the Interpretation of the Calculus of Constructions in Doctrines of Constructions
Birkhäuser Boston eBooks, 1991
According to Gerard Huet and Thierry Coquand the Calculus of Constructions is a typed λ-calculus ... more According to Gerard Huet and Thierry Coquand the Calculus of Constructions is a typed λ-calculus where type expressions themselves can contain λ-expressions as subterms. The original presentation in [Col] is characterized by a lot of overloading which has puzzled many readers as there products of types, functional abstraction and universal quantification were represented by one and the same syntactic construct, namely [x:A]B. Another instance of overloading was their systematic mixing of two aspects of propositions, namely propositions as objects of type Prop and propositions as types. This ambiguity we avoid by the type-forming construct Proof, which when applied to an object p of type Prop gives the type Proof(p) of proofs of the proposition p.
Specification and design of shared resource arbitration
International Journal of Parallel Programming, Feb 1, 1991
The specification, modular design and verification of distributed communicating systems is demons... more The specification, modular design and verification of distributed communicating systems is demonstrated by an example. The scheduling of the access to a common shared resource by a finite number of devices with priorities is a well known problem from hardware and operating systems design. Using the concepts of stream processing continuous function a variety of formal requirement and design specifications for this problem are given and the design specifications are proven correct w.r.t the requirement specifications. From the design specifications one can immediately read off applicative programs implementing the requirement specifications. Special attention is paid to the aspect of modelling time.
Independence results for calculi of dependent types
Springer eBooks, Nov 22, 2005
Based on a categorical semantics for impredicative calculi of dependent types we prove several in... more Based on a categorical semantics for impredicative calculi of dependent types we prove several independence results. Especially we prove that there exists a model where all syntactical concepts can be interpreted with one exception: in the model the strong sum of a family of propositions indexed over a proposition need not be a proposition again. The method of proof consists of restricting the set of propositions in the well-known ω-Set model due to E. Moggi.
Contextual Categories and Categorical Semantics of Dependent Types
By 1978, John Cartmell has introduced the notion of a contextual category in his Ph.D.Thesis on G... more By 1978, John Cartmell has introduced the notion of a contextual category in his Ph.D.Thesis on Generalised Algebraic Theories and Contextual Categories [Cartl], part of which has been published as [Cart2]. We give a detailed exposition of his work on contextual categories in order to be able to explain our notion of categorical model for the Calculus of Constructions which is based on Cartmell’s notion of contextual category.
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Papers by thomas streicher