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18
Fibrations of Graphs
 DISCRETE MATH
, 1996
"... A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, 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 ..."
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Cited by 25 (6 self)
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A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, 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.
Quasismooth Derived Manifolds
"... products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is ..."
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Cited by 14 (0 self)
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products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes 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 ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes 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
Representation theory of 2groups on Kapranov and Voevodsky’s 2vector spaces
 Adv. Math
"... In this paper the 2category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2group G on (some version of) Kapranov and Voevodsky’s 2category of (complex) 2vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invaria ..."
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Cited by 10 (1 self)
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In this paper the 2category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2group G on (some version of) Kapranov and Voevodsky’s 2category of (complex) 2vector 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
A categorification of quantum sl(2
 Adv. Math
"... We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs betwee ..."
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Cited by 10 (4 self)
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We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs between 1morphisms are graded lifts of a semilinear form defined on ˙U. Graded lifts of various homomorphisms and antihomomorphisms of U ˙ arise naturally in the context of our graphical calculus. For each positive integer N a representation of U˙ is constructed using iterated flag varieties that categorifies the irreducible (N + 1)dimensional representation of ˙ U.
The basic geometry of Witt vectors
"... Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “ptypical” Witt vectors of padic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two di ..."
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Cited by 8 (2 self)
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Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “ptypical” Witt vectors of padic 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 ptypical 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 padic 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.
Categorified quantum sl(2) and equivariant cohomology of iterated flag variaties
, 2008
"... A 2category was introduced in math.QA/0803.3652 that categorifies Lusztig’s version of quantum sl(2). Here we construct for each positive integer N a representation of this 2category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N + ..."
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Cited by 6 (3 self)
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A 2category was introduced in math.QA/0803.3652 that categorifies Lusztig’s version of quantum sl(2). Here we construct for each positive integer N a representation of this 2category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N + 1)dimensional representation of quantum sl(2). 1
Representation theory of 2groups on finite dimensional 2vector spaces, in preparation
, 2004
"... In this paper we unfold the 2category structure of the representations of a (strict) 2group on (a suitable version of) Kapranov and Voevodsky’s 2category of finite dimensional 2vector spaces and we discuss the relationship with classical representation theory of groups on finite dimensional vect ..."
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Cited by 4 (1 self)
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In this paper we unfold the 2category structure of the representations of a (strict) 2group on (a suitable version of) Kapranov and Voevodsky’s 2category of finite dimensional 2vector 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 2category of representations of G when G is thought of as a 2group with only identity arrows. As an easy consequence of the unfolding process, we also see that every 2group 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
Enlargements of categories
 Theory Appl. Categ
"... 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 ..."
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Cited by 3 (0 self)
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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
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... 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 ..."
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Cited by 2 (2 self)
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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 2monad. 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