This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirty-five years. The first three were “Set-theoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have

We give an overview of issues surrounding computerverified theorem proving in the standard pure-mathematical context.

The pragmatic goal of this internship was to establish the theoretical soundness of a particular feature of the Agda proof assistant, the so-called 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 well-known 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

Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical set-theoretical 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 tar-attachment to the source file the revised and expanded version of the proof development which had been attached to math.HO/0311260.