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28
A rigidity theorem for Ltheory
, 1984
"... This paper was written in October, 1983, but never published. I am making it available on the internet in this form as a response to the requests for copies of the preprint received in the intervening years. As long as these servers are running, a dvi file of this paper will be available by gopher a ..."
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This paper was written in October, 1983, but never published. I am making it available on the internet in this form as a response to the requests for copies of the preprint received in the intervening years. As long as these servers are running, a dvi file of this paper will be available by gopher at gopher.math.uwo.ca and on the world wide web at
Presheaves of Chain Complexes
, 2003
"... This paper was written to express a personal attitude about derived categories of presheaves and sheaves of chain complexes, much of which has existed for some time but has not previously appeared in the literature ..."
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This paper was written to express a personal attitude about derived categories of presheaves and sheaves of chain complexes, much of which has existed for some time but has not previously appeared in the literature
Localizations of Transfors
, 1998
"... Let C , D and E be ndimensional teisi, i.e., higherdimensional Graycategorical structures. The following questions can be asked. Does a left qtransfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right qtransfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, doe ..."
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Let C , D and E be ndimensional teisi, i.e., higherdimensional Graycategorical structures. The following questions can be asked. Does a left qtransfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right qtransfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, does a functor C\Omega D ! E induce a functor D\Omega C ! E? For c; c 0 elements of C whose (k \Gamma 1)sources and (k \Gamma 1)targets agree, does a qtransfor C ! D induce a qtransfor C (c;c 0 ) ! D (d;d 0 ), for appropriate d;d 0 2 D ? For c; c 0 2 C and d;d 0 2 D whose (k \Gamma 1)sources and (k \Gamma 1)targets agree, does a qtransfor C\Omega D ! E induce a (q+k+1)transfor C (c;c 0 )\Omega D (d;d 0 ) ! E(e;e 0 ), for appropriate e; e 0 2 E? I give answers to these questions in the cases where ndimensional teisi and their tensor product have been defined, i.e., for n 3, and in some cases for n up to 5 which do not need all data and axioms...
Butterflies II: Torsors for 2group stacks
"... We study torsors over 2groups and their morphisms. In particular, we study the first nonabelian cohomology group with values in a 2group, which we reinterpret in terms of gerbes bound by a crossed module, a notion originally due to Debremaeker. Butterfly diagrams encode morphisms of 2groups and ..."
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We study torsors over 2groups and their morphisms. In particular, we study the first nonabelian cohomology group with values in a 2group, which we reinterpret in terms of gerbes bound by a crossed module, a notion originally due to Debremaeker. Butterfly diagrams encode morphisms of 2groups and we employ them to examine the functorial behavior of nonabelian cohomology under change of coefficients. Our main result is to provide a geometric version of this map by lifting a gerbe along the “fraction ” (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at
The Verdier hypercovering theorem
, 2010
"... The Verdier hypercovering theorem is a traditional and widely used method of approximating the morphisms [X, Y] between two objects in homotopy categories of simplicial sheaves and presheaves by simplicial homotopy classes of maps. ..."
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The Verdier hypercovering theorem is a traditional and widely used method of approximating the morphisms [X, Y] between two objects in homotopy categories of simplicial sheaves and presheaves by simplicial homotopy classes of maps.
Simplicial Torsors
, 2001
"... The interpretation by Duskin and Glenn of abelian sheaf cohomology as connected components of a category of torsors is extended to homotopy classes. This is simultaneously an extension of Verdier's version of Cech cohomology to homotopy. 1. ..."
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The interpretation by Duskin and Glenn of abelian sheaf cohomology as connected components of a category of torsors is extended to homotopy classes. This is simultaneously an extension of Verdier's version of Cech cohomology to homotopy. 1.
Integral Homology of
"... E. If l 2 k and F (l; y) = 0 has no rational solutions, denote by k(!) the quadratic extension of k inside the algebraic closure k for which F (l; !) = 0. Our main result is the following. Theorem. For all i 1, H i (PGL 2 (A); Z) = M p2E 2p=0 H i (PGL 2 (k); Z) \Phi M p2E;2p6=0 p\Gammap H ..."
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E. If l 2 k and F (l; y) = 0 has no rational solutions, denote by k(!) the quadratic extension of k inside the algebraic closure k for which F (l; !) = 0. Our main result is the following. Theorem. For all i 1, H i (PGL 2 (A); Z) = M p2E 2p=0 H i (PGL 2 (k); Z) \Phi M p2E;2p6=0 p\Gammap H i (k \Theta ; Z) \Phi M l2k;F (l;y)=0 has no rat. sol.