Big Toy Models - Tulane University

... • Models vs. Axioms. Examples: sheaves and toposes, domain-theoretic models of the λ-calculus. • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of quantum states: A toy theory’. ...

... • Models vs. Axioms. Examples: sheaves and toposes, domain-theoretic models of the λ-calculus. • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of quantum states: A toy theory’. ...

COMPUTABLE MODEL THEORY Contents 1. Introduction and

... uses the tools of computability theory to explore algorithmic content (effectiveness) of notions, theorems, and constructions in various areas of ordinary mathematics. In algebra this investigation based on intuitive notion of effectiveness dates back to van der Waerden who in his 1930 book Modern A ...

... uses the tools of computability theory to explore algorithmic content (effectiveness) of notions, theorems, and constructions in various areas of ordinary mathematics. In algebra this investigation based on intuitive notion of effectiveness dates back to van der Waerden who in his 1930 book Modern A ...

Affine group schemes over symmetric monoidal categories

... 2009]. It is therefore natural to ask whether arithmetic geometry can be similarly developed in the general framework of symmetric monoidal categories. In particular, since the theory of finite flat group schemes is closely linked to arithmetic (see [Tate 1997], for instance), they are a natural sta ...

... 2009]. It is therefore natural to ask whether arithmetic geometry can be similarly developed in the general framework of symmetric monoidal categories. In particular, since the theory of finite flat group schemes is closely linked to arithmetic (see [Tate 1997], for instance), they are a natural sta ...

Work It, Wrap It, Fix It, Fold It

... end, a more structured approach (2010) was then developed based upon initial-algebra semantics, a categorical approach to recursion that is widely used in program optimisation (Bird & de Moor, 1997). More specifically, a worker/wrapper theory was developed for programs defined using fold operators, ...

... end, a more structured approach (2010) was then developed based upon initial-algebra semantics, a categorical approach to recursion that is widely used in program optimisation (Bird & de Moor, 1997). More specifically, a worker/wrapper theory was developed for programs defined using fold operators, ...

Morphisms of Algebraic Stacks

... Let X = [U/R] be a presentation of an algebraic stack. Then the properties of the diagonal of X over S, are the properties of the morphism j : R → U ×S U . For example, if X = [S/G] for some smooth group G in algebraic spaces over S then j is the structure morphism G → S. Hence the diagonal is not a ...

... Let X = [U/R] be a presentation of an algebraic stack. Then the properties of the diagonal of X over S, are the properties of the morphism j : R → U ×S U . For example, if X = [S/G] for some smooth group G in algebraic spaces over S then j is the structure morphism G → S. Hence the diagonal is not a ...

On Brauer Groups of Lubin

... Then x is representable by an atomic Azumaya algebra over E if and only if the quadratic form q is nondegenerate and x1 is represented by the Clifford algebra Clq (as an element of the Brauer-Wall group BWpκq. Warning 1.0.12. In the statement of Theorem 1.0.11, the projection map BrpEq Ñ BWpκq and t ...

... Then x is representable by an atomic Azumaya algebra over E if and only if the quadratic form q is nondegenerate and x1 is represented by the Clifford algebra Clq (as an element of the Brauer-Wall group BWpκq. Warning 1.0.12. In the statement of Theorem 1.0.11, the projection map BrpEq Ñ BWpκq and t ...

arXiv:math/0009100v1 [math.DG] 10 Sep 2000

... has the subspace topology from G × G. Again if one of the maps α, β and the inversion are continuous, then the other map is continuous. Let X be a topological space. Then G = X × X is a topological groupoid on X. In which each pair (x, y) is a morphism from x to y and the groupoid composite is defin ...

... has the subspace topology from G × G. Again if one of the maps α, β and the inversion are continuous, then the other map is continuous. Let X be a topological space. Then G = X × X is a topological groupoid on X. In which each pair (x, y) is a morphism from x to y and the groupoid composite is defin ...

Derived Algebraic Geometry XI: Descent

... will henceforth denote simply by ModR ). This was proven in [40] as a consequence of the following more general result: if A is a connective E∞ -ring and C is an A-linear ∞-category which admits an excellent t-structure, then the construction B 7→ LModB (C) satisfies descent for the flat topology (T ...

... will henceforth denote simply by ModR ). This was proven in [40] as a consequence of the following more general result: if A is a connective E∞ -ring and C is an A-linear ∞-category which admits an excellent t-structure, then the construction B 7→ LModB (C) satisfies descent for the flat topology (T ...

Notes

... The second aim is to define spaces of local shtukas in this setup. Let’s go back to the analogy between function fields and number fields. In the function field context, we have moduli spaces of shtukas. These are associated to data (G, {µ1 , . . . , µm }) where G is a reductive group and the µi are ...

... The second aim is to define spaces of local shtukas in this setup. Let’s go back to the analogy between function fields and number fields. In the function field context, we have moduli spaces of shtukas. These are associated to data (G, {µ1 , . . . , µm }) where G is a reductive group and the µi are ...

SCHEMES 01H8 Contents 1. Introduction 1 2. Locally ringed spaces

... Lemma 5.4. Let R be a ring. Let M be an R-module. Let M OSpec(R) -modules associated to M . (1) We have Γ(Spec(R), OSpec(R) ) = R. f) = M as an R-module. (2) We have Γ(Spec(R), M (3) For every f ∈ R we have Γ(D(f ), OSpec(R) ) = Rf . f) = Mf as an Rf -module. (4) For every f ∈ R we have Γ(D(f ), M f ...

... Lemma 5.4. Let R be a ring. Let M be an R-module. Let M OSpec(R) -modules associated to M . (1) We have Γ(Spec(R), OSpec(R) ) = R. f) = M as an R-module. (2) We have Γ(Spec(R), M (3) For every f ∈ R we have Γ(D(f ), OSpec(R) ) = Rf . f) = Mf as an Rf -module. (4) For every f ∈ R we have Γ(D(f ), M f ...

Algebraic D-groups and differential Galois theory

... logarithmic derivatives on G. So our general diﬀerential Galois theory is in a sense subsumed by the very concept of an algebraic D-group. Details of the above will be given in Sections 2 and 3, including a “Tannakian” approach and an examination of diﬀerent manifestations of the differential Galois ...

... logarithmic derivatives on G. So our general diﬀerential Galois theory is in a sense subsumed by the very concept of an algebraic D-group. Details of the above will be given in Sections 2 and 3, including a “Tannakian” approach and an examination of diﬀerent manifestations of the differential Galois ...

Non-archimedean analytic spaces

... If A is a (strictly) k -affinoid algebra then M(A) is called a (strictly) k -affinoid space. A morphism M(A) → M(B) is a map of the form M(φ) for a bounded k -homomorphism φ : B → A. The category of k -affinoid spaces is denoted k -Aff . We will later provide affinoid spaces X with a richer structur ...

... If A is a (strictly) k -affinoid algebra then M(A) is called a (strictly) k -affinoid space. A morphism M(A) → M(B) is a map of the form M(φ) for a bounded k -homomorphism φ : B → A. The category of k -affinoid spaces is denoted k -Aff . We will later provide affinoid spaces X with a richer structur ...

Andr´e-Quillen (co)Homology, Abelianization and Stabilization

... If C and D are model categories and F : C −→ D takes fibrations to fibration and preserves weak equivalences between fibrant objects, define its total right derived functor by RF : HoC −→ HoD RF(X ) = F(f (X )) Maria Basterra ...

... If C and D are model categories and F : C −→ D takes fibrations to fibration and preserves weak equivalences between fibrant objects, define its total right derived functor by RF : HoC −→ HoD RF(X ) = F(f (X )) Maria Basterra ...

ETALE COHOMOLOGY AND THE WEIL CONJECTURES Sommaire 1.

... Remark 2.3.1. Recall that codim X Y is not necessarily equal to dim X −dim Y , even for X irreducible: take X = Spec k[[t]][u], Y = V (tu − 1). Then dim Y is the Krull dimension of the field k[[t]][u]/(tu−1) = k[[t]][ 1t ], hence zero. On the other hand dim X = dim k[[t]]+1 = 2 and codim X Y is the ...

... Remark 2.3.1. Recall that codim X Y is not necessarily equal to dim X −dim Y , even for X irreducible: take X = Spec k[[t]][u], Y = V (tu − 1). Then dim Y is the Krull dimension of the field k[[t]][u]/(tu−1) = k[[t]][ 1t ], hence zero. On the other hand dim X = dim k[[t]]+1 = 2 and codim X Y is the ...

Structured Stable Homotopy Theory and the Descent Problem for

... E2 -term of the spectral sequence in terms of group cohomology, it does not give us understanding of the equivariant behavior of the K-theory spectra. Further, there is no interpretation of the entire E2 -term as an appropriate Ext-group. It is not really a descent spectral sequence. One could then ...

... E2 -term of the spectral sequence in terms of group cohomology, it does not give us understanding of the equivariant behavior of the K-theory spectra. Further, there is no interpretation of the entire E2 -term as an appropriate Ext-group. It is not really a descent spectral sequence. One could then ...

MODEL-CATEGORIES OF COALGEBRAS OVER OPERADS JUSTIN R. SMITH

... • MV C is essentially the largest sub-coalgebra of LV C that is concentrated in nonnegative dimensions. • FV C is the largest pointed irreducible sub-coalgebra of PV C that is concentrated in nonnegative dimensions. ...

... • MV C is essentially the largest sub-coalgebra of LV C that is concentrated in nonnegative dimensions. • FV C is the largest pointed irreducible sub-coalgebra of PV C that is concentrated in nonnegative dimensions. ...

Pobierz - DML-PL

... The degree of a single-valued continuous map allows several descriptions. Some of them, having an intrinsically geometric nature or being purely analytic (see [79]), have a very clear geometric meaning. In the set-valued case, one is forced to apply different techniques. In the first instance, somet ...

... The degree of a single-valued continuous map allows several descriptions. Some of them, having an intrinsically geometric nature or being purely analytic (see [79]), have a very clear geometric meaning. In the set-valued case, one is forced to apply different techniques. In the first instance, somet ...

Étale Cohomology

... Grothendieck was the first to suggest étale cohomology (1960) as an attempt to solve the Weil Conjectures. By that time, it was already known by Serre that if one had a suitable cohomology theory for abstract varieties defined over the complex numbers, then one could deduce the conjectures using th ...

... Grothendieck was the first to suggest étale cohomology (1960) as an attempt to solve the Weil Conjectures. By that time, it was already known by Serre that if one had a suitable cohomology theory for abstract varieties defined over the complex numbers, then one could deduce the conjectures using th ...

[math.QA] 23 Feb 2004 Quantum groupoids and

... of those bialgebroids and their descenders. In particular, we show that their certain quotients have a quasitriangular structure. The infinitesimal analogs of bialgebroids are Lie bialgebroids. We consider Lie bialgebroids which are quasi-classical limits of the bialgebroids related to the dynamical ...

... of those bialgebroids and their descenders. In particular, we show that their certain quotients have a quasitriangular structure. The infinitesimal analogs of bialgebroids are Lie bialgebroids. We consider Lie bialgebroids which are quasi-classical limits of the bialgebroids related to the dynamical ...

D´ ECALAGE AND KAN’S SIMPLICIAL LOOP GROUP FUNCTOR DANNY STEVENSON

... set. Then the comparison map dX → T X is a weak homotopy equivalence. The first published proof of this result was given in [Cegarra-Remedios, 2005], with the authors noting that this fact is stated without proof in [Cordier, 1987] where it is attributed to Zisman (unpublished). When X is a bisimpli ...

... set. Then the comparison map dX → T X is a weak homotopy equivalence. The first published proof of this result was given in [Cegarra-Remedios, 2005], with the authors noting that this fact is stated without proof in [Cordier, 1987] where it is attributed to Zisman (unpublished). When X is a bisimpli ...

Stone duality above dimension zero

... The first two sections of Chapter 2 provide an introduction to the basic theory of latticeordered groups and MV-algebras. These two classes of algebraic structures are tightly related via the equivalence Γ. This connection is exploited in the third section of the chapter. The content of Chapter 2, a ...

... The first two sections of Chapter 2 provide an introduction to the basic theory of latticeordered groups and MV-algebras. These two classes of algebraic structures are tightly related via the equivalence Γ. This connection is exploited in the third section of the chapter. The content of Chapter 2, a ...

Realizability - TU Darmstadt/Mathematik

... Realizability was invented in 1945 by S. C. Kleene as an attempt to make explicit the algorithmic content of constructive proofs. From proofs of existence statements ∃yR(~x, y) one would like to read off a socalled Skolem function, i.e. a function f such that R(~x, f (~x)) holds for all ~x. Assuming ...

... Realizability was invented in 1945 by S. C. Kleene as an attempt to make explicit the algorithmic content of constructive proofs. From proofs of existence statements ∃yR(~x, y) one would like to read off a socalled Skolem function, i.e. a function f such that R(~x, f (~x)) holds for all ~x. Assuming ...

AUTOMORPHISM GROUPS AND PICARD GROUPS OF ADDITIVE

... agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as developed in [2, Chapter 2, §5]. In this paper, we can apply this to any additive full ...

... agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as developed in [2, Chapter 2, §5]. In this paper, we can apply this to any additive full ...

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.