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45
SemiAbelian Categories
, 2000
"... The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories ar ..."
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Cited by 37 (3 self)
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The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to "old" exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semiabelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar nonabelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
"... ..."
Model category structures on chain complexes of sheaves
 Trans. Amer. Math. Soc
"... of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently ..."
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Cited by 27 (0 self)
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of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently
A Categorical Quantum Logic
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2005
"... We define a strongly normalising proofnet calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax ca ..."
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Cited by 22 (5 self)
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We define a strongly normalising proofnet calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.
New Model Categories From Old
 J. Pure Appl. Algebra
, 1995
"... . We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categor ..."
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Cited by 13 (5 self)
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. We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas  most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in nonabelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such nonabelian derived functors is the E 2 term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...
Exactly Definable Categories
"... this paper is to show that certain properties ofmodules become more transparent if one views them as exact functors. In particular, one can use the machinery of localization theory for locally coherent Grothendieck categories because Ex(C ..."
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Cited by 12 (7 self)
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this paper is to show that certain properties ofmodules become more transparent if one views them as exact functors. In particular, one can use the machinery of localization theory for locally coherent Grothendieck categories because Ex(C
Green's Theorem on Hall Algebras
 In "Representations of Algebras and Related Topics," CMS Conference Proceedings 19 (1996), American Mathematical Society
"... . Let k be a finite field and a hereditary finitary kalgebra. Let P be the set of isomorphism classes of finite modules. We define a multiplication on the Qspace with basis P by counting the number of submodules U of a given module V with prescribed isomorphism classes both of V=U and U. In thi ..."
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Cited by 11 (0 self)
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. Let k be a finite field and a hereditary finitary kalgebra. Let P be the set of isomorphism classes of finite modules. We define a multiplication on the Qspace with basis P by counting the number of submodules U of a given module V with prescribed isomorphism classes both of V=U and U. In this way we obtain the so called Hall algebra H=H(;Q) with coefficients in Q: Besides H, we are also interested in the subalgebra C generated by the the subset I of all isomorphism classes of simple  modules; this subalgebra is called the corresponding composition algebra, since it encodes the number of composition series of all modules. Recently, J. A. Green has introduced on H (and on C) a comultiplication ffi so that it becomes nearly a bialgebra. Here, "nearly" means that ffi:H!H\Omega\Gamma is in general not an algebra homomorphism for the usual (componentwise) multiplication on H\Omega\Gamma ; instead, there is a slightly twisted multiplication on H\Omega\Gamma which has to be con...
Quantum informationflow, concretely, abstractly
 PROC. QPL 2004
, 2004
"... These ‘lecture notes ’ are based on joint work with Samson Abramsky. I will survey and informally discuss the results of [3, 4, 5, 12, 13] in a pedestrian not too technical way. These include: • ‘The logic of entanglement’, that is, the identification and abstract axiomatization of the ‘quantum info ..."
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Cited by 10 (4 self)
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These ‘lecture notes ’ are based on joint work with Samson Abramsky. I will survey and informally discuss the results of [3, 4, 5, 12, 13] in a pedestrian not too technical way. These include: • ‘The logic of entanglement’, that is, the identification and abstract axiomatization of the ‘quantum informationflow ’ which enables protocols such as quantum teleportation. 1 To this means we defined strongly compact closed categories which abstractly capture the behavioral properties of quantum entanglement. • ‘Postulates for an abstract quantum formalism ’ in which classical informationflow (e.g. token exchange) is part of the formalism. As an example, we provided a purely formal description of quantum teleportation and proved correctness in abstract generality. 2 In this formalism types reflect kinds, contra the essentially typeless von Neumann formalism [25]. Hence even concretely this formalism manifestly improves on the usual one. • ‘A highlevel approach to quantum informatics’. 3 Indeed, the above discussed work can be conceived as aiming to solve: von Neumann quantum formalism � highlevel language lowlevel language. I also provide a brief discussion on how classical and quantum uncertainty can be mixed in the above formalism (cf. density matrices). 4
On The Freyd Categories Of An Additive Category
, 2000
"... To any additive category C, we associate in a functorial way two additive categories A(C), B(C). The category A(C), resp. B(C), is the reflection of C in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration AB(C) i ..."
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Cited by 5 (1 self)
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To any additive category C, we associate in a functorial way two additive categories A(C), B(C). The category A(C), resp. B(C), is the reflection of C in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration AB(C) is the reflection of C in the category of abelian categories and exact functors. We call A(C) and B(C) the Freyd categories of C since the first systematic study of these categories was done by Freyd in the midsixties. The purpose of the paper is to study further the Freyd categories and to indicate their applications to the module theory of an abelian or triangulated category.