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Enriched stratified systems for the foundations of category
"... This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much ..."
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This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have
Universe Subtyping in MartinLöf Type Theory Internship Report
"... The pragmatic goal of this internship was to establish the theoretical soundness of a particular feature of the Agda proof assistant, the socalled universe polymorphism. This is part of a series of work by my supervisor Andreas Abel [ACD07] aiming to provide a coherent metatheory for the different ..."
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The pragmatic goal of this internship was to establish the theoretical soundness of a particular feature of the Agda proof assistant, the socalled universe polymorphism. This is part of a series of work by my supervisor Andreas Abel [ACD07] aiming to provide a coherent metatheory for the different features implemented or desired in this proof assistant. More precisely, we first considered universe cumulativity, a wellknown feature for universe hierarchy, then universe polymorphism, which is a less explored area, and finally attempted to add irrelevance to the type system, to ease universe levels manipulations. All those notions will be introduced in more detail in the first section. I will highlight the main goals and constraints of the desired theory, in comparison to related works. I will also have to discuss some of the technical choices and issues we have faced; those technical decisions have large consequences – often unforeseen – on the technical development and its structure. The second section will describe the formal type system – or type systems, as we will consider variants – used throughout the report. To this declarative type
Computer theorem proving in math
, 2004
"... Abstract—We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. This is based on my talk at the PQR conference (Brussels, June 2003). ..."
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Abstract—We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. This is based on my talk at the PQR conference (Brussels, June 2003).
unknown title
, 2004
"... Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tara ..."
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Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tarattachment to the source file the revised and expanded version of the proof development which had been attached to math.HO/0311260.
unknown title
"... Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tara ..."
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Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tarattachment to the source file the revised and expanded version of the proof development which had been attached to math.HO/0311260.
Settheoretical mathematics in Coq
"... Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tara ..."
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Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tarattachment to the source file the revised and expanded version of the proof development which had been attached to math.HO/0311260.
FOUNDATIONS OF UNLIMITED CATEGORY THEORY: WHAT REMAINS TO BE DONE
"... Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unli ..."
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Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited ” or “naive ” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution. From the very beginnings of the subject of category theory as introduced by Eilenberg & Mac Lane (1945) it was recognized that the notion of category lends itself naturally to