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Higher fundamental functors for simplicial sets, Cahiers Topologie Géom
 Diff. Catég
"... Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how ..."
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Cited by 11 (8 self)
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Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (nonreversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy ncategory functor ↑Πn: Smp = nCat, left adjoint to a nerve Nn = nCat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.
Higher Hopf formulae for homology via Galois Theory, preprint math.AT/0701815
, 2007
"... and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case ..."
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Cited by 10 (3 self)
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and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A
Finite Sets And Symmetric Simplicial Sets
 Theory Appl. Categ
, 2001
"... The category of finite cardinals (or, equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the augmented symmetric simplicial sets. We prove here that this ground category has characterisations simil ..."
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Cited by 4 (2 self)
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The category of finite cardinals (or, equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the augmented symmetric simplicial sets. We prove here that this ground category has characterisations similar to the classical ones for the category of finite ordinals, by the existence of a universal symmetric monoid, or by generators and relations. The latter provides a definition of symmetric simplicial sets by faces, degeneracies and transpositions, under suitable relations.
Higher central extensions and Hopf formulae
, 902
"... Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained. 0 ..."
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Cited by 3 (1 self)
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Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained. 0
BAER INVARIANTS IN SEMIABELIAN CATEGORIES II: HOMOLOGY
 THEORY AND APPLICATIONS OF CATEGORIES
, 2004
"... This article treats the problem of deriving the reflector of a semiabelian ..."
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Cited by 2 (1 self)
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This article treats the problem of deriving the reflector of a semiabelian
Extension Theories for Categories
, 1994
"... This paper is a slight revision of a paper I wrote in 1980 and never submitted for publication. I would greatly appreciate any information about more recent work on this topic. ..."
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This paper is a slight revision of a paper I wrote in 1980 and never submitted for publication. I would greatly appreciate any information about more recent work on this topic.
and Theories Version 1.1, Reprinted by Theory and Applications of Categories
"... The first author gratefully acknowledges the support he has received from the NSERC of Canada for the last thirty seven years. Received by the editors 20050301. Transmitted by F. W. Lawvere, W. Tholen and R.J. Wood. Reprint published on 20050615. ..."
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The first author gratefully acknowledges the support he has received from the NSERC of Canada for the last thirty seven years. Received by the editors 20050301. Transmitted by F. W. Lawvere, W. Tholen and R.J. Wood. Reprint published on 20050615.
RESEARCH ARTICLE EXTENSION THEORIES FOR MONOIDS
"... Leech extensions by a centralizing functor which satisfy the other requirements given. 2. The functor called F in Theorem 8 is called F in the proof. n Y ..."
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Leech extensions by a centralizing functor which satisfy the other requirements given. 2. The functor called F in Theorem 8 is called F in the proof. n Y
Journal of Homotopy and Related Structures, vol. 2(1), 2007, pp.109–126 A COMPARISON THEOREM FOR SIMPLICIAL RESOLUTIONS
"... It is well known that Barr and Beck’s definition of comonadic homology makes sense also with a functor of coefficients taking values in a semiabelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Her ..."
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It is well known that Barr and Beck’s definition of comonadic homology makes sense also with a functor of coefficients taking values in a semiabelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induced class of projectives only.