http://homes.dsi.unimi.it/ ghilardi/allegati/dispcesena.pdf
... These notes cover the content of a basic course in propositional algebraic logic given by the author at the italian School of Logic held in Cesena, September 18-23, 2000. They are addressed to people having few background in Symbolic Logic and they are mostly intended to develop algebraic methods fo ...
... These notes cover the content of a basic course in propositional algebraic logic given by the author at the italian School of Logic held in Cesena, September 18-23, 2000. They are addressed to people having few background in Symbolic Logic and they are mostly intended to develop algebraic methods fo ...
Locally ringed spaces and affine schemes
... We have a (contravariant) functor Spec from the category of rings to the category of locally ringed spaces defined as follows. Let ϕ : A −→ B be a morphism of rings. We already know how this induces a morphism f : Spec B −→ Spec A of topological spaces. Let X = Spec B and Y = Spec A. There is a can ...
... We have a (contravariant) functor Spec from the category of rings to the category of locally ringed spaces defined as follows. Let ϕ : A −→ B be a morphism of rings. We already know how this induces a morphism f : Spec B −→ Spec A of topological spaces. Let X = Spec B and Y = Spec A. There is a can ...
ƒkew group —lge˜r—s of pie™ewise heredit—ry
... In this paper, all considered algebras are nite dimensional algebras over an algebraically closed eld k (and, unless otherwise specied, basic and connected). Moreover, all modules are nitely generated left modules. For an algebra A, we denote by mod A the category of nitely generated A-modules ...
... In this paper, all considered algebras are nite dimensional algebras over an algebraically closed eld k (and, unless otherwise specied, basic and connected). Moreover, all modules are nitely generated left modules. For an algebra A, we denote by mod A the category of nitely generated A-modules ...
ON ∗-AUTONOMOUS CATEGORIES OF TOPOLOGICAL
... topologically into a power of R, with R topologized discretely. Among other things, this implies that the topology is generated by (translates of) the open submodules. An ideal I ⊆ R is called large if its intersection with every non-zero ideal is non-zero. An obvious Zorn’s lemma argument shows tha ...
... topologically into a power of R, with R topologized discretely. Among other things, this implies that the topology is generated by (translates of) the open submodules. An ideal I ⊆ R is called large if its intersection with every non-zero ideal is non-zero. An obvious Zorn’s lemma argument shows tha ...
DESCENT OF DELIGNE GROUPOIDS 1. Introduction 1.1. A formal
... Let y ∈ g0 [t], y = ni=1 ai ti satisfy exp(y)(x) = x. Let a be an ideal in g such that [g, a] = 0. Put h = g/a. By induction, we suppose that the claim is correct for the Lie algebra h, therefore ai ∈ a. Then exp(y)(x) = x + [y, x] + dy which easily implies y = 0. ...
... Let y ∈ g0 [t], y = ni=1 ai ti satisfy exp(y)(x) = x. Let a be an ideal in g such that [g, a] = 0. Put h = g/a. By induction, we suppose that the claim is correct for the Lie algebra h, therefore ai ∈ a. Then exp(y)(x) = x + [y, x] + dy which easily implies y = 0. ...
model categories of diagram spectra
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
ON THE REPRESENTABILITY OF ACTIONS IN A SEMI
... Council, a grant of whom made possible Janelidze’s visit to Sydney Received by the editors 2005-02-24 and, in revised form, 2005-06-28. Transmitted by W. Tholen. Published on 2005-08-25. 2000 Mathematics Subject Classification: 18C10, 18D35, 18G15. Key words and phrases: semi-abelian category, variet ...
... Council, a grant of whom made possible Janelidze’s visit to Sydney Received by the editors 2005-02-24 and, in revised form, 2005-06-28. Transmitted by W. Tholen. Published on 2005-08-25. 2000 Mathematics Subject Classification: 18C10, 18D35, 18G15. Key words and phrases: semi-abelian category, variet ...
On the homology and homotopy of commutative shuffle algebras
... and this is k-projective but not projective as a module over the group algebra of the symmetric group Σn , k[Σn ]. Usually one replaces the operad Com by an E∞ -operad to make things homotopy invariant. For instance Mike Mandell showed [M03, 1.8, 1.3] that the normalization functor induces an isomor ...
... and this is k-projective but not projective as a module over the group algebra of the symmetric group Σn , k[Σn ]. Usually one replaces the operad Com by an E∞ -operad to make things homotopy invariant. For instance Mike Mandell showed [M03, 1.8, 1.3] that the normalization functor induces an isomor ...
256B Algebraic Geometry
... varieties. The organizing framework for this class will be a 2-dimensional topological field theory called the B-model. Topics will include 1. Vector bundles and coherent sheaves 2. Cohomology, derived categories, and derived functors (in the differential graded setting) 3. Grothendieck-Serre dualit ...
... varieties. The organizing framework for this class will be a 2-dimensional topological field theory called the B-model. Topics will include 1. Vector bundles and coherent sheaves 2. Cohomology, derived categories, and derived functors (in the differential graded setting) 3. Grothendieck-Serre dualit ...
Cohomology of Categorical Self-Distributivity
... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...
... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...
arXiv:1510.01797v3 [math.CT] 21 Apr 2016 - Mathematik, Uni
... a cofinite ideal. This extension is no longer an inverse equivalence to the (restricted) dual algebra functor, but only a left adjoint. In addition, A• underlies a Hopf algebra if A does. Attempts have been made to generalize this to more general commutative rings than just fields. In [2] and [6] th ...
... a cofinite ideal. This extension is no longer an inverse equivalence to the (restricted) dual algebra functor, but only a left adjoint. In addition, A• underlies a Hopf algebra if A does. Attempts have been made to generalize this to more general commutative rings than just fields. In [2] and [6] th ...
On *-autonomous categories of topological modules.
... surjective. But it is also injective since the original pairing on (A, X) was non-singular. The result is that hom(G, T ) ∼ = X so that we recover (A, X). Much more can be said; the details can be found in [Barr & Kleisli (2001)] as well in the note [Barr, (unpublished)]. / Hom(G, T ) surjective. A ...
... surjective. But it is also injective since the original pairing on (A, X) was non-singular. The result is that hom(G, T ) ∼ = X so that we recover (A, X). Much more can be said; the details can be found in [Barr & Kleisli (2001)] as well in the note [Barr, (unpublished)]. / Hom(G, T ) surjective. A ...
A Meaningful Justification for the Representational Theory of
... description of scientific topic, is then used to justify the representational theory of measurement, currently the dominant theory of measurement. The meaningfulness concept is based on the following four intuitive principles about scientific topics. Principle 1. Scientific topics have both qualitat ...
... description of scientific topic, is then used to justify the representational theory of measurement, currently the dominant theory of measurement. The meaningfulness concept is based on the following four intuitive principles about scientific topics. Principle 1. Scientific topics have both qualitat ...
Introduction The following thesis plays a central role in deformation
... The situation is dramatically simpler if we wish to study not arbitrary n-categories, but n-groupoids. An n-category C is called an n-groupoid if every k-morphism in C is invertible. If X is any topological space, then the n-category π≤n X is an example of an n-groupoid: for example, the 1-morphisms ...
... The situation is dramatically simpler if we wish to study not arbitrary n-categories, but n-groupoids. An n-category C is called an n-groupoid if every k-morphism in C is invertible. If X is any topological space, then the n-category π≤n X is an example of an n-groupoid: for example, the 1-morphisms ...
Cyclic Homology Theory, Part II
... where g ∈ Sn , g(1) = 1 is chosen in such way that g(12 . . . n)g−1 is teh cycle which we want to sendo to A⊗n . The map in the opposite direction is α 7→ (12 . . . n) ⊗ α. and the one composition is identity on A• and the second one is homotopic to the identity. Corollary 1.22. HL• (gl(A)) ...
... where g ∈ Sn , g(1) = 1 is chosen in such way that g(12 . . . n)g−1 is teh cycle which we want to sendo to A⊗n . The map in the opposite direction is α 7→ (12 . . . n) ⊗ α. and the one composition is identity on A• and the second one is homotopic to the identity. Corollary 1.22. HL• (gl(A)) ...
Recognisable Languages over Monads
... A, and the restriction of the multiplication operation from A to the subfunctor: mulA |T0 A : T0 A → A It is easy to show, using associativity, that if T0 spans A, then A is uniquely determined by its T0 -reduct. In particular, if T0 is complete, then every operation T0 A → A extends to at most one ...
... A, and the restriction of the multiplication operation from A to the subfunctor: mulA |T0 A : T0 A → A It is easy to show, using associativity, that if T0 spans A, then A is uniquely determined by its T0 -reduct. In particular, if T0 is complete, then every operation T0 A → A extends to at most one ...
ON QUANTIC CONUCLEI IN ORTHOMODULAR LATTICES
... hypothesis, and this of course is equal to the RHS of the last inequality. Therefore φ = uz where z = φ(0) = j(1)⊥ ; i.e., if a is an arbitrary element of L we have φ(a) = a ∨ j(1)⊥ but φ(a) = j(a⊥ )⊥ hence j(a⊥ )⊥ = a ∨ j(1)⊥ and j(a) = a ∧ j(1) = mj(1) . The converse is of course trivial. We want ...
... hypothesis, and this of course is equal to the RHS of the last inequality. Therefore φ = uz where z = φ(0) = j(1)⊥ ; i.e., if a is an arbitrary element of L we have φ(a) = a ∨ j(1)⊥ but φ(a) = j(a⊥ )⊥ hence j(a⊥ )⊥ = a ∨ j(1)⊥ and j(a) = a ∧ j(1) = mj(1) . The converse is of course trivial. We want ...
ON QUANTIC CONUCLEI IN ORTHOMODULAR LATTICES
... for orthomodular lattices or in a more general context, namely, in any quantic lattice. We believe that the last idea is the best choice since we can in fact introduce the concept of quantale. So, let us begin with the denition of a quantic lattice. 1.2. Definition. Let Q be a complete lattice (Q ...
... for orthomodular lattices or in a more general context, namely, in any quantic lattice. We believe that the last idea is the best choice since we can in fact introduce the concept of quantale. So, let us begin with the denition of a quantic lattice. 1.2. Definition. Let Q be a complete lattice (Q ...
Notes
... respectively. Hence the forgetful functor RS → Top preserves all limits and colimits. To reflect geometry better, we want a notion of functions vanishing at a point. The functions that vanish at a point should form a unique maximal ideal, in other words: the stalks should be local rings. This is equ ...
... respectively. Hence the forgetful functor RS → Top preserves all limits and colimits. To reflect geometry better, we want a notion of functions vanishing at a point. The functions that vanish at a point should form a unique maximal ideal, in other words: the stalks should be local rings. This is equ ...
Higher regulators and values of L
... Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists a universal cohomology theory H~(X, Z(i)), satisfying Poincare duality and related to Qui ...
... Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists a universal cohomology theory H~(X, Z(i)), satisfying Poincare duality and related to Qui ...
Divided power structures and chain complexes
... 2.0.1. Divided power structures in the simplicial context. On the homotopy groups of simplicial commutative rings there are divided power operations and it is this instance of divided power structures that we will investigate in this paper. In the context of the action of the Steenrod algebra on coh ...
... 2.0.1. Divided power structures in the simplicial context. On the homotopy groups of simplicial commutative rings there are divided power operations and it is this instance of divided power structures that we will investigate in this paper. In the context of the action of the Steenrod algebra on coh ...
Hopf algebras
... 1. The category Set whose objects are sets, and where the set of morphisms between two sets is given by all mappings between those sets. 2. Let k be a commutative ring, then Mk denotes the category with as objects all (right) kmodules, and with as morphisms between two k-modules all k-linear mapping ...
... 1. The category Set whose objects are sets, and where the set of morphisms between two sets is given by all mappings between those sets. 2. Let k be a commutative ring, then Mk denotes the category with as objects all (right) kmodules, and with as morphisms between two k-modules all k-linear mapping ...
On fusion categories - Annals of Mathematics
... semisimple rigid tensor (=monoidal) category with finitely many simple objects and finite dimensional spaces of morphisms, such that the endomorphism algebra of the neutral object is k (see [BaKi]). Fusion categories arise in several areas of mathematics and physics – conformal field theory, operator a ...
... semisimple rigid tensor (=monoidal) category with finitely many simple objects and finite dimensional spaces of morphisms, such that the endomorphism algebra of the neutral object is k (see [BaKi]). Fusion categories arise in several areas of mathematics and physics – conformal field theory, operator a ...
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.