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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 80 (9 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:...
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 67 (9 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.
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
"... ..."
EXACT CATEGORIES
, 2008
"... We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We th ..."
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Cited by 23 (0 self)
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We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then
© Hindawi Publishing Corp. REFLEXIVE AND DIHEDRAL (CO)HOMOLOGY OF A PREADDITIVE CATEGORY
, 1998
"... Abstract. The group dihedral homology of an algebra over a field with characteristic zero was introduced by Tsygan (1983). The dihedral homology and cohomology of an algebra with involution over commutative ring with identity, associated with the small category, were studied by Krasauskas et al. (19 ..."
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Abstract. The group dihedral homology of an algebra over a field with characteristic zero was introduced by Tsygan (1983). The dihedral homology and cohomology of an algebra with involution over commutative ring with identity, associated with the small category, were studied by Krasauskas et al. (1988), Loday (1987), and Lodder (1993). The aim of this work is concerned with dihedral and reflexive (co)homology of small preadditive category. We also define the free product of involutive algebras associated with this category and study its dihedral homology group. Finally, following Perelygin (1990), we show that a small preadditive category is Morita equivalence.
HUMAN CONSCIOUSNESS AND ARTIFICIAL INTELLIGENCE
"... In this monograph we present a novel approach to the problems raised by higher complexity in both nature and the human society, by considering the most complex levels of objective existence as ontological metalevels, such as those present in the creative human minds and civilised, modern societies. ..."
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In this monograph we present a novel approach to the problems raised by higher complexity in both nature and the human society, by considering the most complex levels of objective existence as ontological metalevels, such as those present in the creative human minds and civilised, modern societies. Thus, a ‘theory ’ about theories is called a ‘metatheory’. In the same sense that a statement about propositions is a higherlevel 〈proposition 〉 rather than a simple proposition, a global process of subprocesses is a metaprocess, and the emergence of higher levels of reality via such metaprocesses results in the objective existence of ontological metalevels. The new concepts suggested for understanding the emergence and evolution of life, as well as human consciousness, are in terms of globalisation of multiple, underlying processes into the metalevels of their existence. Such concepts are also useful in computer aided ontology and computer science [1],[194],[197]. The selected approach for our broad– but also indepth – study of the fundamental, relational structures and functions present in living, higher organisms and of the extremely complex processes and metaprocesses of the human mind combines new concepts from three recently developed, related mathematical fields: Algebraic Topology (AT), Category Theory (CT) and Higher Dimensional Algebra