D\'{e}vissage for Waldhausen K-theory

A d\'evissage-type theorem in algebraic K-theory is a statement that identifies the K-theory of a Waldhausen category $\mathscr{C}$ in terms of the K-theories of a collection of Waldhausen subcategories of $\mathscr{C}$ when a d\'evissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between d\'evissage theorems of single type and of multiple type depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original d\'evissage theorem for abelian categories (single type) and Waldhausen's theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of d\'evissage-type theorems and prove a general d\'evissage theorem of single type and a general d\'evissage theorem of multiple type.Comment: 28 page

Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references polished) Submitted to the International Journal of Theoretical Physic