A MONOIDAL STRUCTURE ON THE CATEGORY OF
... is always a left B-comodule, in fact we have a forgetful functor B CA → B C. The following natural question arises: is there a monoidal structure on B CA that is compatible with the one on B C, by which we mean that the forgetful functor is strongly monoidal. Monoidal structures on a more general ca ...
... is always a left B-comodule, in fact we have a forgetful functor B CA → B C. The following natural question arises: is there a monoidal structure on B CA that is compatible with the one on B C, by which we mean that the forgetful functor is strongly monoidal. Monoidal structures on a more general ca ...
Hochschild cohomology
... cosimplicial objects and its associated (co-)chain complex. In fact, Hochschild cohomology can be described by the cohomology of a cochain complex associated to a cosimplicial bimodule, that is a cosimplicial object in the category of bimodules. Definition. Let ∆ be the category whose objects are th ...
... cosimplicial objects and its associated (co-)chain complex. In fact, Hochschild cohomology can be described by the cohomology of a cochain complex associated to a cosimplicial bimodule, that is a cosimplicial object in the category of bimodules. Definition. Let ∆ be the category whose objects are th ...
Coxeter functors and Gabriel's theorem
... a quadruple of subspaces. Consider the space W= V ®V and in it the graph of/, that is, the subspace Et of pairs (£,/£), where f e V. The mapping/is described by a quadruple of subspaces in W, namely £·, = V @ 0, Et = 0 φ V, E3 = {(f, t) | 4 e V}(E3 is the diagonal) and E, = {(£,/?) I £ e V}- the gra ...
... a quadruple of subspaces. Consider the space W= V ®V and in it the graph of/, that is, the subspace Et of pairs (£,/£), where f e V. The mapping/is described by a quadruple of subspaces in W, namely £·, = V @ 0, Et = 0 φ V, E3 = {(f, t) | 4 e V}(E3 is the diagonal) and E, = {(£,/?) I £ e V}- the gra ...
Relative and Modi ed Relative Realizability Introduction
... idea is, that instead of doing realizability with one partial combinatory algebra A one uses an inclusion of partial combinatory algebras A] A (such that there are combinators k s 2 A] which also serve as combinators for A) the principal point being that \(A] -) computable" functions may also ac ...
... idea is, that instead of doing realizability with one partial combinatory algebra A one uses an inclusion of partial combinatory algebras A] A (such that there are combinators k s 2 A] which also serve as combinators for A) the principal point being that \(A] -) computable" functions may also ac ...
Slides - The Department of Mathematics and Statistics
... • Fundamental questions of economic theory require a treatment different from that which they have found thus far in the literature • Application of mathematical theory of games of strategy to economic problems provides a new approach to a number of questions as yet unsettled • Intent of this is oft ...
... • Fundamental questions of economic theory require a treatment different from that which they have found thus far in the literature • Application of mathematical theory of games of strategy to economic problems provides a new approach to a number of questions as yet unsettled • Intent of this is oft ...
A Grothendieck site is a small category C equipped with a
... 5) The Nisnevich site Nis|S has the same underlying category as the étale site, namely all étale maps V → S and morphisms between them. A Nisnevich cover is a family of étale maps Vα → V such that every morphism Sp(K) → V lifts to some Vα where K is any field. 6) A flat cover of a scheme T is a ...
... 5) The Nisnevich site Nis|S has the same underlying category as the étale site, namely all étale maps V → S and morphisms between them. A Nisnevich cover is a family of étale maps Vα → V such that every morphism Sp(K) → V lifts to some Vα where K is any field. 6) A flat cover of a scheme T is a ...
Derived Representation Theory and the Algebraic K
... grading coming from the graded structure on π∗ , and the p-index refers tothe homological degree. This spectral sequence is referred to as the Künneth spectral sequence for this situation. Remark: Throughout this paper, all Hom and smash product spectra will be computed using only cofibrant module ...
... grading coming from the graded structure on π∗ , and the p-index refers tothe homological degree. This spectral sequence is referred to as the Künneth spectral sequence for this situation. Remark: Throughout this paper, all Hom and smash product spectra will be computed using only cofibrant module ...
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... (2) If L ∈ / W (L0 ) then Ĝ(L) is disjoint from Ĝ(L0 ). Main Problem: Find a classification of Ĝ(E) = Ĝunip. . The above reduction is reminiscent of the classification of conjugacy classes in algebraic groups. The above theorem can be seen as separating the semi-simple and unipotent parts. Note ...
... (2) If L ∈ / W (L0 ) then Ĝ(L) is disjoint from Ĝ(L0 ). Main Problem: Find a classification of Ĝ(E) = Ĝunip. . The above reduction is reminiscent of the classification of conjugacy classes in algebraic groups. The above theorem can be seen as separating the semi-simple and unipotent parts. Note ...
Mathematical Logic
... Let us spell this out in more elementary terms. We have a set X with a realizability relation for equality: for every x, y ∈ X a set [x = y] of realizers of the equality of x and y is given (subject to a few natural conditions). A binary relation R on X is given by, for each x, y ∈ X, a set R(x, y) ...
... Let us spell this out in more elementary terms. We have a set X with a realizability relation for equality: for every x, y ∈ X a set [x = y] of realizers of the equality of x and y is given (subject to a few natural conditions). A binary relation R on X is given by, for each x, y ∈ X, a set R(x, y) ...
1.1.1 Introduction to Axiomatic Systems
... called axioms or postulates, concerning the undefined terms. One obtains a mathematical theory by proving new statements, called theorems, using only the axioms (postulates), logic system, and previous theorems. Definitions are made in the process in order to be more concise. Most early Greeks made ...
... called axioms or postulates, concerning the undefined terms. One obtains a mathematical theory by proving new statements, called theorems, using only the axioms (postulates), logic system, and previous theorems. Definitions are made in the process in order to be more concise. Most early Greeks made ...
Basic Modern Algebraic Geometry
... Example 1.1.3.1 The collection of all groups form a category, the morphisms being the group-homomorphisms. This category is denoted by Grp Example 1.1.3.2 The collection of all Abelian groups form a category, the morphisms being the group-homomorphisms. This category is denoted by Ab. Example 1.1.3. ...
... Example 1.1.3.1 The collection of all groups form a category, the morphisms being the group-homomorphisms. This category is denoted by Grp Example 1.1.3.2 The collection of all Abelian groups form a category, the morphisms being the group-homomorphisms. This category is denoted by Ab. Example 1.1.3. ...
Commutative ring objects in pro-categories and generalized Moore
... We summarize the portions of this paper not previously described. Our work begins in Section 2 by collecting definitions and results on the homotopy theory of operads and spaces of operad structures on objects. A more detailed outline is at the beginning of that section. In Section 3 we flesh out th ...
... We summarize the portions of this paper not previously described. Our work begins in Section 2 by collecting definitions and results on the homotopy theory of operads and spaces of operad structures on objects. A more detailed outline is at the beginning of that section. In Section 3 we flesh out th ...
Commutative ring objects in pro-categories and generalized Moore spectra June 30, 2013
... but not an A(p)-algebra; and so on. (A discussion of the literature on multiplicative properties of Moore spectra can be found in [42, A.6], while multiplicative properties of V (1) can be found in [30]. The higher structure on Moore spectra plays an important role in [38].) These facts and others f ...
... but not an A(p)-algebra; and so on. (A discussion of the literature on multiplicative properties of Moore spectra can be found in [42, A.6], while multiplicative properties of V (1) can be found in [30]. The higher structure on Moore spectra plays an important role in [38].) These facts and others f ...
The braid group action on the set of exceptional sequences
... We consider exceptional sequences of objects in a triangulated category D b , as introduced in the first lecture. In particular, we assume that all left and right mutations are defined in this category. We denote by Hom • (E, F ) the finite dimensional vector space ⊕l Hom (E, F [l]). Moreover, we wr ...
... We consider exceptional sequences of objects in a triangulated category D b , as introduced in the first lecture. In particular, we assume that all left and right mutations are defined in this category. We denote by Hom • (E, F ) the finite dimensional vector space ⊕l Hom (E, F [l]). Moreover, we wr ...
Cohomology as the derived functor of derivations.
... But the right-hand side of this may be transformed, by applying (2.4) to its first term, into p(xx„ ® a(y)a(z)), thus yielding (2.3), and completing the demonstration. More generally, since R is /¿-projective there is a /¿-module R' such that R® R' is /¿-free. Consider R' as an abelian Lie algebra. ...
... But the right-hand side of this may be transformed, by applying (2.4) to its first term, into p(xx„ ® a(y)a(z)), thus yielding (2.3), and completing the demonstration. More generally, since R is /¿-projective there is a /¿-module R' such that R® R' is /¿-free. Consider R' as an abelian Lie algebra. ...
LECTURE 2 1. Finitely Generated Abelian Groups We discuss the
... minimal set of generators with q elements, then A is isomorphic to the free abelian group of rank q. Proof. By induction on the minimal number of generators of A. If A is cyclic (that is, generated by one non-zero element), the conclusion is clear. Suppose that the result holds for all finitely gene ...
... minimal set of generators with q elements, then A is isomorphic to the free abelian group of rank q. Proof. By induction on the minimal number of generators of A. If A is cyclic (that is, generated by one non-zero element), the conclusion is clear. Suppose that the result holds for all finitely gene ...
The Etingof-Kazhdan construction of Lie bialgebra deformations.
... • Outline an explicit construction for quantisation of a finite dimensional Lie bialgebra. • Demonstrate that this construction is the quantum double of a QUE algebra. • Brief remarks on applications to quantisation of r-matrices and the infinite dimensional case. Next talk (Michael Wong): Generaliz ...
... • Outline an explicit construction for quantisation of a finite dimensional Lie bialgebra. • Demonstrate that this construction is the quantum double of a QUE algebra. • Brief remarks on applications to quantisation of r-matrices and the infinite dimensional case. Next talk (Michael Wong): Generaliz ...
Symmetric Spectra Talk
... functor [X, Y ] → [ΣX, ΣY ] is an isomorphism when dimX ≤ 2connY . This theorem says that πn+r (S n ) → πn+r+1 (S n+1 ) is an isomorphism for n > r + 1. There is also a result which states that [X, Y ] has a natural group structure for dimX ≤ 2conn(Y ). So in a range of dimensions and connectivity, ...
... functor [X, Y ] → [ΣX, ΣY ] is an isomorphism when dimX ≤ 2connY . This theorem says that πn+r (S n ) → πn+r+1 (S n+1 ) is an isomorphism for n > r + 1. There is also a result which states that [X, Y ] has a natural group structure for dimX ≤ 2conn(Y ). So in a range of dimensions and connectivity, ...
Belief Propagation in Monoidal Categories
... compute a (possibly partial) contraction, (#P-hard) solve the word problem (are two diagrams equivalent, i.e. do they have the same interpretation) or compute a normal form for a diagram, (undecidable) solve the implementability problem (construct a word equivalent to a target using a library of all ...
... compute a (possibly partial) contraction, (#P-hard) solve the word problem (are two diagrams equivalent, i.e. do they have the same interpretation) or compute a normal form for a diagram, (undecidable) solve the implementability problem (construct a word equivalent to a target using a library of all ...
What is a Logic? - UCSD CSE - University of California San Diego
... logics in use there, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic [17, 31, 19]. The soundness aspect of sound reasoning is addressed by axiomatizing the notion of satisfaction, and the reasoning aspect is addressed by call ...
... logics in use there, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic [17, 31, 19]. The soundness aspect of sound reasoning is addressed by axiomatizing the notion of satisfaction, and the reasoning aspect is addressed by call ...
Introduction to derived algebraic geometry
... Example 9. Derived Artin stacks have derived fiber products; thus, we can obtain derived Artin stacks by starting with ordinary Artin stacks and taking their derived intersection. Let us return to our original motivating example of “finite-dimensional B-modules” for B an associative kalgebra. Let us ...
... Example 9. Derived Artin stacks have derived fiber products; thus, we can obtain derived Artin stacks by starting with ordinary Artin stacks and taking their derived intersection. Let us return to our original motivating example of “finite-dimensional B-modules” for B an associative kalgebra. Let us ...
ALGEBRAS AND MODULES IN MONOIDAL MODEL CATEGORIES
... One model for structured ring spectra is given by the S-algebras of [11]. This example has the special feature that every object is ®brant, which makes it easier to form model structures of modules and algebras. There are other new theories such as `symmetric ring spectra' [13], `functors with smash ...
... One model for structured ring spectra is given by the S-algebras of [11]. This example has the special feature that every object is ®brant, which makes it easier to form model structures of modules and algebras. There are other new theories such as `symmetric ring spectra' [13], `functors with smash ...
12 Super Lie Groups and Super Lie Algebras
... of objects and morphisms, examples include the category of sets (being the objects) and functions (being the morphisms), the category of smooth manifolds (with the smooth maps as morphisms), and the category of supermanifolds. Finite products exist in all three examples, and terminal objects are, re ...
... of objects and morphisms, examples include the category of sets (being the objects) and functions (being the morphisms), the category of smooth manifolds (with the smooth maps as morphisms), and the category of supermanifolds. Finite products exist in all three examples, and terminal objects are, re ...
Moduli Problems for Ring Spectra - International Mathematical Union
... The situation is dramatically simpler if we wish to study not arbitrary ncategories, 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 ncategories, 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 ...
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.