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53
Galois Theory of Second Order Covering Maps of Simplicial Sets
 J. Pure Appl. Algebra
, 1995
"... this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but ..."
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this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but
Higher order symmetry of graphs
 In Lecture given at the September Meeting of the Irish Mathematical Society, available on the Author's web
, 1994
"... It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein’s famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. ..."
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It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein’s famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. The structure of group alone may not give all the expression one needs of the intuitive idea of symmetry. One often needs structured groups (for example topological, Lie, algebraic, order,...). Here we consider groups with the additional structure of directed graph, which we abbreviate to graph. This type of structure appears in [18, 14]. We shall associate with a graph A a group AUT(A) which is also a graph. The vertices of AUT(A) are the automorphism of the graph A and the edges between automorphisms give an expression of “adjacency ” of automorphisms. The vertices of this graph form a group, and so also do the edges. The automorphisms of A adjacent to the identity will be called the inner automorphisms of the graph A. One aspect of the problem is to describe these inner automorphisms in terms of the internal structure of the graph A. The second theme is that of regarding the usual category of sets and mappings as but one environment for doing mathematics, and one which may be replaced by others. We use the word “environment ” here rather than “foundation”, because the former word implies a more relativistic approach.
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
Factorization Systems For Symmetric CatGroups
 THEORY AND APPLICATIONS OF CATEGORIES, PREPRINT
, 2000
"... This paper is a first step in the study of symmetric catgroups as the 2dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective fu ..."
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This paper is a first step in the study of symmetric catgroups as the 2dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective functor followed by a faithful one. Both these factorizations give rise to a factorization system, in a suitable 2categorical sense, in the 2category of symmetric catgroups. An application to exact sequences is given.
Normal Functors And Strong Protomodularity
 APPLICATIONS OF CATEGORIES
, 2000
"... The notion of normal subobject having an intrinsic meaning in any protomodular category, we introduce the notion of normal functor, namely left exact conservative functor which reflects normal subobjects. The point is that for the category Gp of groups the change of base functors, with respect t ..."
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The notion of normal subobject having an intrinsic meaning in any protomodular category, we introduce the notion of normal functor, namely left exact conservative functor which reflects normal subobjects. The point is that for the category Gp of groups the change of base functors, with respect to the fibration of pointed objects, are not only conservative (this is the definition of a protomodular category), but also normal. This leads to the notion of strongly protomodular category. Some of their properties are given, the main one being that this notion is inherited by the slice categories.
NOTES ON 2GROUPOIDS, 2GROUPS AND CrossedModules
, 2005
"... This paper contains some basic results on 2groupoids, with special emphasis on computing derived mapping 2groupoids between 2groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s k ..."
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This paper contains some basic results on 2groupoids, with special emphasis on computing derived mapping 2groupoids between 2groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2groupoids introduced by Moerdijk and Svensson.
ON WEAK MAPS BETWEEN 2GROUPS
, 2008
"... We give an explicit handy cocyclefree description of the groupoid of weak maps between two crossedmodules using what we call a butterfly (Theorem 8.4). We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2category of pointed homotopy 2types. ..."
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We give an explicit handy cocyclefree description of the groupoid of weak maps between two crossedmodules using what we call a butterfly (Theorem 8.4). We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2category of pointed homotopy 2types. This has applications in the study of 2group actions (say, on stacks), and in the theory of gerbes bound by crossedmodules and principal2bundles).
SEMISTRICT TAMSAMANI NGROUPOIDS AND CONNECTED NTYPES
, 2007
"... Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of co ..."
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Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of connected ntypes.