In this paper we define a notion of relativization for higher order logic. We then show that there is a higher order theory of Grothendieck topoi such that all Grothendieck topoi relativizes to all models of set theory with choice.

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares zigzags. This result makes questions about associated cohomology groups spectral sequences easy to understand. describe notion ‘universal’ quasi-isomorphism behaviour decomposition under tensor product compute Grothendieck ring category bounded complexes with...

The Weil conjecture is a delightful theorem for algebraic varieties on finite fields and an important model for dynamical zeta functions. In this paper, we prove a functional equation of Lefschetz zeta functions for infinite cyclic coverings which is analogous to the Weil conjecture. Applying this functional equation to knot theory, we obtain a new view point on the reciprocity of the Alexander...

Of the 18 chapters in the book, we will only cover the 5 listed below and thus allow for a more leisurely pace. There is also some supplementary material in the bibliography to expand upon if needed. The selected chapters are thematically divided into two groups. The first one is by far the larger and is concerned with the interaction between ideas from (simplicial) geometry and topology on the...

We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.

Article history: Received 28 June 2011 Available online 31 January 2012 Communicated by Michel Van den Bergh