A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found application in the theory of distributed systems.

products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the non-transverse intersection of submanifolds is

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to

In this paper the 2-category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2-group G on (some version of) Kapranov and Voevodsky’s 2-category of (complex) 2-vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants π0(G), π1(G) and [α]∈H 3 (π0(G), π1(G)) classifying G. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain that the monoidal category

In this paper we unfold the 2-category structure of the representations of a (strict) 2-group on (a suitable version of) Kapranov and Voevodsky’s 2-category of finite dimensional 2-vector spaces and we discuss the relationship with classical representation theory of groups on finite dimensional vector spaces. In particular, we prove that the monoidal category of representations of any group G appears as a full subcategory of the category of endomorphisms of a particular object in the 2-category of representations of G when G is thought of as a 2-group with only identity arrows. As an easy consequence of the unfolding process, we also see that every 2-group with a compact Lie group as base group has a rank one representation faithful with respect to the base group, contrary to a claim by Barrett and Mackaay (unpublished work). 1

Abstract. In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorial properties are well behaved under enlargements. Contents

Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “p-typical” Witt vectors of p-adic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two different ways. We allow not just p-typical Witt vectors but also, for example, those taken with respect to any set of primes in any ring of integers in any global field. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of the Greenberg transform. We investigate whether many standard geometric properties of spaces and maps are preserved by Witt vector functors.

Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves

Abstract. This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author’s existing results for orbifolds. It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie algebra of vector fields. 1.

This thesis investigates relations over a category C relative to an (E; M)-factori-