Results 11  20
of
20
The hunting of the Hopf ring
 Homology, Homotopy Appl
"... We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E ∗ (−). Our description is as a graded and completed version of a TallWraith monoid. The E ∗cohomology of a space X is a module for this TallWrai ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E ∗ (−). Our description is as a graded and completed version of a TallWraith monoid. The E ∗cohomology of a space X is a module for this TallWraith monoid. We also show that the corresponding Hopf ring of unstable cooperations is a module for the TallWraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.
In this not...
"... Abstract. If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C → D we understand a functor which, when composed with the forgetful functor D → Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined ..."
Abstract
 Add to MetaCart
Abstract. If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C → D we understand a functor which, when composed with the forgetful functor D → Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined by Dcoalgebra objects of C. Let
Algebraic MetaTheories and . . .
"... Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [ ..."
Abstract
 Add to MetaCart
Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [3] and its multisorted version [26], and also to synthesise a new version of the Nominal Algebra of Gabbay and Mathijssen [41] and the Nominal Equational Logic of Clouston and Pitts [8] for reasoning about languages with namebinding operators. Based on these case studies and further preliminary investigations, we contend that Sol can make an impact in the problem of engineering logics for modern computational languages. For example, our proposed research on secondorder equational logic will provide foundations for designing a secondorder extension of the Maude system [37], a firstorder semantic and logical framework used in formal software engineering for specification and programming. Our research strategy can be visualised as follows: (I)
The Syntax of Coherence
, 1999
"... This article tackles categorical coherence within a twodimensional generalization of Lawvere’s functorial semantics. 2theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded, many coherence results become simple statements about the qu ..."
Abstract
 Add to MetaCart
This article tackles categorical coherence within a twodimensional generalization of Lawvere’s functorial semantics. 2theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded, many coherence results become simple statements about the quasiYoneda lemma and 2theorymorphisms. Given two 2theories and a 2theorymorphism between them, we explore the induced relationship between the corresponding 2categories of algebras. The strength of the induced quasiadjoints are classified by the strength of the 2theorymorphism. These quasiadjoints reflect the extent to which one structure can be replaced by another. A twodimensional analogue of the Kronecker product is defined and constructed. This operation allows one to generate new coherence laws from old ones. 1
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES I
, 2006
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
Abstract
 Add to MetaCart
Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. In all these considerations we lay stress on the role of the monoidal structure, and the difference between this approach and the approach using (in general nonmonoidal) abelian categories as models for categories of quasicoherent sheaves on noncommutative schemes. Contents
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES I
, 2007
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
Abstract
 Add to MetaCart
Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. In all these considerations we lay stress on the role of the monoidal structure, and the difference between this approach and the approach using (in general nonmonoidal) abelian categories as models for categories of quasicoherent sheaves on noncommutative schemes. Contents
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES
, 2007
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
Abstract
 Add to MetaCart
Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way. Contents
COLIMITS OF REPRESENTABLE ALGEBRAVALUED FUNCTORS
, 711
"... Abstract. If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C → D we understand a functor which, when composed with the forgetful functor D → Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined ..."
Abstract
 Add to MetaCart
Abstract. If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C → D we understand a functor which, when composed with the forgetful functor D → Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined by Dcoalgebra objects of C. Let