FELL BUNDLES ASSOCIATED TO GROUPOID MORPHISMS §1
... In his Memoir [F1], J. M. G. Fell generalizes Mackey’s theory of unitary representations of group extensions to a natural enrichment of the concept of Banach *-algebra, called Banach *-algebraic bundle. Given a normal subgroup K of G, he constructs a bundle B over H = G/K with the fiber over the neu ...
... In his Memoir [F1], J. M. G. Fell generalizes Mackey’s theory of unitary representations of group extensions to a natural enrichment of the concept of Banach *-algebra, called Banach *-algebraic bundle. Given a normal subgroup K of G, he constructs a bundle B over H = G/K with the fiber over the neu ...
Galois actions on homotopy groups of algebraic varieties
... Definition 1.3 A map f W X ! Y in Top is said to be a weak equivalence if it gives an isomorphism 0 X ! 0 Y on path components, and for all x 2 X , the maps n .f /W n .X; x/ ! n .Y; f x/ are all isomorphisms. We give S the model structure of Goerss and Jardine [11, Theorem V.7.6]; in particular ...
... Definition 1.3 A map f W X ! Y in Top is said to be a weak equivalence if it gives an isomorphism 0 X ! 0 Y on path components, and for all x 2 X , the maps n .f /W n .X; x/ ! n .Y; f x/ are all isomorphisms. We give S the model structure of Goerss and Jardine [11, Theorem V.7.6]; in particular ...
Examples - Stacks Project
... coordinate axes) on the affine plane. Then let X1 be the (reduced) full preimage of X0 on the blow-up of the plane (X1 has three rational components forming a chain). Then blow up the resulting surface at the two singularities of X1 , and let X2 be the reduced preimage of X1 (which has five rational ...
... coordinate axes) on the affine plane. Then let X1 be the (reduced) full preimage of X0 on the blow-up of the plane (X1 has three rational components forming a chain). Then blow up the resulting surface at the two singularities of X1 , and let X2 be the reduced preimage of X1 (which has five rational ...
Presenting distributive laws
... Presenting distributive laws In the current chapter, we study distributive laws of monads over functors. These capture interaction between algebraic structure and observable behaviour in a systematic way. There are several benefits of this approach, recalled in more detail in Section 3.5: a distribu ...
... Presenting distributive laws In the current chapter, we study distributive laws of monads over functors. These capture interaction between algebraic structure and observable behaviour in a systematic way. There are several benefits of this approach, recalled in more detail in Section 3.5: a distribu ...
UNIVERSAL COVERS OF TOPOLOGICAL MODULES AND A
... element 0 of additive group structure of M . We now apply the construction M (W ) of section 2 to the additive group of M and W . Proposition 3.5. Let R be a simply connected topological ring with identity 1R . Let M be a topological left R-module and W an open, path connected neighbourhood of the i ...
... element 0 of additive group structure of M . We now apply the construction M (W ) of section 2 to the additive group of M and W . Proposition 3.5. Let R be a simply connected topological ring with identity 1R . Let M be a topological left R-module and W an open, path connected neighbourhood of the i ...
Light leaves and Lusztig`s conjecture 1 Introduction
... We remark that (as we said before) we believe that Fiebig’s conjecture is equivalent to Lusztig’s conjecture and this would imply that in Theorem 3, when W is an affine Weyl group we could replace “if” by “if and only if”. The missing part (the only if) could be relevant only if we could find counte ...
... We remark that (as we said before) we believe that Fiebig’s conjecture is equivalent to Lusztig’s conjecture and this would imply that in Theorem 3, when W is an affine Weyl group we could replace “if” by “if and only if”. The missing part (the only if) could be relevant only if we could find counte ...
COMPLEXES OF INJECTIVE kG-MODULES 1. Introduction Let k be
... stmod kG ←− Db (mod kG) ←− Db (proj kG). This means that K(Inj kG) can be regarded as the appropriate “big” category for Db (mod kG), whereas D(Mod kG) has too few compact objects for this purpose. In this sense, K(Inj kG) is a nicer category to work in than D(Mod kG). From the point of view of alge ...
... stmod kG ←− Db (mod kG) ←− Db (proj kG). This means that K(Inj kG) can be regarded as the appropriate “big” category for Db (mod kG), whereas D(Mod kG) has too few compact objects for this purpose. In this sense, K(Inj kG) is a nicer category to work in than D(Mod kG). From the point of view of alge ...
DIALGEBRAS Jean-Louis LODAY There is a notion of
... [L4]. The next step would consist in computing the dialgebra homology of the augmentation ideal of K[GL(A)], for an associative algebra A. Here is the content of this article. In the first section we introduce the notion of associative dimonoid, or dimonoid for short, and develop the calculus in a ...
... [L4]. The next step would consist in computing the dialgebra homology of the augmentation ideal of K[GL(A)], for an associative algebra A. Here is the content of this article. In the first section we introduce the notion of associative dimonoid, or dimonoid for short, and develop the calculus in a ...
´Etale cohomology of schemes and analytic spaces
... for all (i, j), there exists a unique s ∈ F (X) such that s|Xi = si for every i. Comment. If i = j, the notation si|Xi ×X Xi is confusing, because there are two arrows, namely, the two projections, from Xi ×X Xi to X. Therefore when i = j the condition si|Xi ×X Xj = sj|Xi ×X Xj should be understood ...
... for all (i, j), there exists a unique s ∈ F (X) such that s|Xi = si for every i. Comment. If i = j, the notation si|Xi ×X Xi is confusing, because there are two arrows, namely, the two projections, from Xi ×X Xi to X. Therefore when i = j the condition si|Xi ×X Xj = sj|Xi ×X Xj should be understood ...
The bounded derived category of an algebra with radical squared zero
... finite dimensional if x∈Q0 dimk M (x) is finite. A morphism f : M → N of k-representations of Q consists of a family of k-linear maps f (x) : M (x) → N (x) with x ∈ Q0 such that M (α)f (y) = f (x)N (α) for all arrows α : x → y in Q. The k-representations of Q form a hereditary abelian k-category, de ...
... finite dimensional if x∈Q0 dimk M (x) is finite. A morphism f : M → N of k-representations of Q consists of a family of k-linear maps f (x) : M (x) → N (x) with x ∈ Q0 such that M (α)f (y) = f (x)N (α) for all arrows α : x → y in Q. The k-representations of Q form a hereditary abelian k-category, de ...
www.math.uwo.ca
... reflects fibrations. For example, if A and B are spaces of holomorphic maps and A → B is a map such that the induced map sA → sB of mapping spaces is a Kan fibration, as might follow from some homotopy-theoretic arguments, then A → B itself is a Serre fibration (and conversely). Also, sA → sB is a ...
... reflects fibrations. For example, if A and B are spaces of holomorphic maps and A → B is a map such that the induced map sA → sB of mapping spaces is a Kan fibration, as might follow from some homotopy-theoretic arguments, then A → B itself is a Serre fibration (and conversely). Also, sA → sB is a ...
full text (.pdf)
... ( nite) +, , , 0, and 1, any closed semiringPis a -continuous Kleene algebra. In fact, in the treatment of 1, 10], the sole purpose of seems to be to de ne . A more descriptive name for closed semirings might be !-complete idempotent semirings. These algebras are strongly related to several classe ...
... ( nite) +, , , 0, and 1, any closed semiringPis a -continuous Kleene algebra. In fact, in the treatment of 1, 10], the sole purpose of seems to be to de ne . A more descriptive name for closed semirings might be !-complete idempotent semirings. These algebras are strongly related to several classe ...
127 A GENERALIZATION OF BAIRE CATEGORY IN A
... Now for all U1 , U2 ∈ U (2) if U1 6= U2 then U1 ∩ U2 = ∅. Further, if N is nowhere dense in C, and A is any subset of C, then both N × A and A × N are nowhere dense in C 2 . Therefore if B is a set of first ωα -category in C, and A is any subset of C then both B × A and A × B are sets of first ωα -c ...
... Now for all U1 , U2 ∈ U (2) if U1 6= U2 then U1 ∩ U2 = ∅. Further, if N is nowhere dense in C, and A is any subset of C, then both N × A and A × N are nowhere dense in C 2 . Therefore if B is a set of first ωα -category in C, and A is any subset of C then both B × A and A × B are sets of first ωα -c ...
Chapter III. Basic theory of group schemes. As we have seen in the
... For commutative group schemes [n] is usually called “multiplication by n”. (3.2) The definitions given in (3.1) are sometimes not so practicable. For instance, to define a group scheme one would have to give a scheme G, then one needs to define the morphisms m, i and e, and finally one would have to ...
... For commutative group schemes [n] is usually called “multiplication by n”. (3.2) The definitions given in (3.1) are sometimes not so practicable. For instance, to define a group scheme one would have to give a scheme G, then one needs to define the morphisms m, i and e, and finally one would have to ...
HOMOLOGY OF LIE ALGEBRAS WITH Λ/qΛ COEFFICIENTS AND
... As an application, Hopf formulas for the second and the third homologies of a Lie algebra with Λ/qΛ coefficients are proved. The condition for the existence of the universal q-central relative extension of a Lie epimorphism [Kh] and the description of the kernel of such extension in terms of relative ...
... As an application, Hopf formulas for the second and the third homologies of a Lie algebra with Λ/qΛ coefficients are proved. The condition for the existence of the universal q-central relative extension of a Lie epimorphism [Kh] and the description of the kernel of such extension in terms of relative ...
2.1. Functions on affine varieties. After having defined affine
... removable singularity theorem. 2.2. Sheaves. We have seen in lemma 2.1.8 that regular functions on affine varieties are defined in terms of local properties: they are set-theoretic functions that can locally be written as quotients of polynomials. Local constructions of function-like objects occur i ...
... removable singularity theorem. 2.2. Sheaves. We have seen in lemma 2.1.8 that regular functions on affine varieties are defined in terms of local properties: they are set-theoretic functions that can locally be written as quotients of polynomials. Local constructions of function-like objects occur i ...
Math 8211 Homework 2 PJW
... As practice, but not part of the homework, make sure you can do questions in Rotman apart from the ones listed below, such as 2.19a. Assignment questions: Rotman pages 64-69: 2.13, 2.18, 2.20 (assume without proof Proposition 2.42). Rotman pages 94-97: 2.28, 2.34, 2.36(i) Questions 1, 2, 3 below. 0. ...
... As practice, but not part of the homework, make sure you can do questions in Rotman apart from the ones listed below, such as 2.19a. Assignment questions: Rotman pages 64-69: 2.13, 2.18, 2.20 (assume without proof Proposition 2.42). Rotman pages 94-97: 2.28, 2.34, 2.36(i) Questions 1, 2, 3 below. 0. ...
Categorical and Kripke Semantics for Constructive S4 Modal Logic
... Tarski-style interpretation. The ‘semantic value’ is given by the set of worlds at which a formula is valid. This form of semantics has been very successful for intuitionistic and modal logics alike. More recent and less traditional is the categorical approach. Here, we model not only the ‘semantic ...
... Tarski-style interpretation. The ‘semantic value’ is given by the set of worlds at which a formula is valid. This form of semantics has been very successful for intuitionistic and modal logics alike. More recent and less traditional is the categorical approach. Here, we model not only the ‘semantic ...
Riemann surfaces with boundaries and the theory of vertex operator
... symmetry and Seiberg-Witten theory are among the most famous examples. The results predicted by these physical ideas and intuition also suggest that many seemingly-unrelated mathematical branches are in fact different aspects of a certain yet-to-be-constructed unified theory. The success of physica ...
... symmetry and Seiberg-Witten theory are among the most famous examples. The results predicted by these physical ideas and intuition also suggest that many seemingly-unrelated mathematical branches are in fact different aspects of a certain yet-to-be-constructed unified theory. The success of physica ...
Cyclic A structures and Deligne`s conjecture
... will be the following. First the cyclic symmetry of the form of the Fukaya category must be concretely established. Then applying Theorems D and C it will be an immediate consequence that HH .F.N /; F.N // is a BV algebra, and more specifically that CH .F.N /; F.N // is a BV 1 algebra, or more s ...
... will be the following. First the cyclic symmetry of the form of the Fukaya category must be concretely established. Then applying Theorems D and C it will be an immediate consequence that HH .F.N /; F.N // is a BV algebra, and more specifically that CH .F.N /; F.N // is a BV 1 algebra, or more s ...
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... (r ≥ s > 0) in general, and so we cannot get a good description of the projective limit. However, Theorem 1 allows us to understand the ordinary part of W very well. It will be convenient to abstract the situation slightly. Thus suppose that Mr is a projective system of Λ-modules indexed by positive ...
... (r ≥ s > 0) in general, and so we cannot get a good description of the projective limit. However, Theorem 1 allows us to understand the ordinary part of W very well. It will be convenient to abstract the situation slightly. Thus suppose that Mr is a projective system of Λ-modules indexed by positive ...
PLETHYSTIC ALGEBRA Introduction Consider an example from
... Theorem. There is an O-plethory P 0 that is universal among those that are equipped with a map from P making them P -deformations of Q-rings. Furthermore, P 0 has the property that P -deformations of Q-rings are the same as P 0 -rings that are mtorsion-free. We say P 0 is the amplification of P alon ...
... Theorem. There is an O-plethory P 0 that is universal among those that are equipped with a map from P making them P -deformations of Q-rings. Furthermore, P 0 has the property that P -deformations of Q-rings are the same as P 0 -rings that are mtorsion-free. We say P 0 is the amplification of P alon ...
Some applications of the ultrafilter topology on spaces of valuation
... As in [14], we define the constructible topology on X the topology on X whose basis of open sets is K(X ). We denote by X cons the set X , equipped with the constructible topology. Note that, for Noetherian topological spaces, this definition of constructible topology coincides with the classical on ...
... As in [14], we define the constructible topology on X the topology on X whose basis of open sets is K(X ). We denote by X cons the set X , equipped with the constructible topology. Note that, for Noetherian topological spaces, this definition of constructible topology coincides with the classical on ...
Constellations and their relationship with categories
... A result that we will make frequent use of is the following, which is Lemma 2.3 in [1]. Lemma 2.2 For s, t elements of the constellation P , s·t exists if and only if s·D(t) exists, and D(s · t) = D(s). An important example of a constellation is CX (outlined in (3) above), consisting of partial func ...
... A result that we will make frequent use of is the following, which is Lemma 2.3 in [1]. Lemma 2.2 For s, t elements of the constellation P , s·t exists if and only if s·D(t) exists, and D(s · t) = D(s). An important example of a constellation is CX (outlined in (3) above), consisting of partial func ...
Category theory
Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Category theory can be used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Several terms used in category theory, including the term ""morphism"", are used differently from their uses in the rest of mathematics. In category theory, a ""morphism"" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made.