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This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on Realisability [8] and the other on Sheaf Models in Algebraic Set Theory [9]. Sheaf theory and realisability have been effective methods for constructing models of various constructive and intuitionistic(More)
1 Introduction This paper is the second in a series on the relation between algebraic set theory [19] and predicative formal systems. The purpose of the present paper is to show how realizability models of constructive set theories fit into the framework of algebraic set theory. It can be read independently from the first part [5]; however, we recommend(More)
We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occurring in topology such as Segal's category , or the total category of a crossed simplicial group such as Con-nes' cyclic category. For any generalized Reedy category R and any cofibrantly generated model category E,(More)