Aspects of categorical algebra in initialstructure categories
... proj ectives, injectives, or ( coequalizer, mono )-bicategory structures ( ==> homomorphism theorem ) , then the same is valid for any INS-category over L . Finally all important theorems of equationally defined universal algebra ( e.g. existence of free K-algebras, adjointness of algebraic fun ...
... proj ectives, injectives, or ( coequalizer, mono )-bicategory structures ( ==> homomorphism theorem ) , then the same is valid for any INS-category over L . Finally all important theorems of equationally defined universal algebra ( e.g. existence of free K-algebras, adjointness of algebraic fun ...
ON MACKEY TOPOLOGIES IN TOPOLOGICAL ABELIAN
... There has been some effort into finding general conditions on a class of groups that ensure the existence of Mackey coreflections. See, for example, [Chasco et. al., 1999]. On the other hand, there has been some effort into finding a class of topological abelian groups on which the circle group is inject ...
... There has been some effort into finding general conditions on a class of groups that ensure the existence of Mackey coreflections. See, for example, [Chasco et. al., 1999]. On the other hand, there has been some effort into finding a class of topological abelian groups on which the circle group is inject ...
Autonomous categories and linear logic
... use the metatheorem of [Jay, 1989] (see especially Section 5). Both of these allow one to derive the required coherence results by using “elements” and that makes it easy. In all cases, the only question at hand is whether a certain diagram, constructed using only the arrows from the free autonomous ...
... use the metatheorem of [Jay, 1989] (see especially Section 5). Both of these allow one to derive the required coherence results by using “elements” and that makes it easy. In all cases, the only question at hand is whether a certain diagram, constructed using only the arrows from the free autonomous ...
Boyarchenko on associativity.pdf
... Of course, the original associativity axiom is a special case of this one for n = 3. The fact that the special case n = 3 implies all other cases is, of course, almost trivial; nevertheless, it is a prototype of many “coherence theorems” in the theory of monoidal categories and related areas. 2.5. I ...
... Of course, the original associativity axiom is a special case of this one for n = 3. The fact that the special case n = 3 implies all other cases is, of course, almost trivial; nevertheless, it is a prototype of many “coherence theorems” in the theory of monoidal categories and related areas. 2.5. I ...
Algebraic Geometry
... with a and a radical, then the intersection W and W in the sense of schemes is Spec kŒX1 ; : : : ; XnCn0 =.a; a / while their intersection in the sense of varieties is Spec kŒX1 ; : : : ; XnCn0 =rad.a; a0 / (and their intersection in the sense of algebraic spaces is Spm kŒX1 ; : : : ; XnCn0 =.a; ...
... with a and a radical, then the intersection W and W in the sense of schemes is Spec kŒX1 ; : : : ; XnCn0 =.a; a / while their intersection in the sense of varieties is Spec kŒX1 ; : : : ; XnCn0 =rad.a; a0 / (and their intersection in the sense of algebraic spaces is Spm kŒX1 ; : : : ; XnCn0 =.a; ...
ENRICHED MODEL CATEGORIES IN EQUIVARIANT CONTEXTS
... say in the context of simplicial model categories, where we start with a simplicial group G and a simplicial model category M and display equivalences between model categories of G-objects in M and appropriate presheaf categories. The wellinformed reader will immediately notice that other examples i ...
... say in the context of simplicial model categories, where we start with a simplicial group G and a simplicial model category M and display equivalences between model categories of G-objects in M and appropriate presheaf categories. The wellinformed reader will immediately notice that other examples i ...
lecture08
... Referential integrity constraints: if you work for a company, it must exist in the database. Other constraints: peoples’ ages are between 0 and 150. ...
... Referential integrity constraints: if you work for a company, it must exist in the database. Other constraints: peoples’ ages are between 0 and 150. ...
Effective descent morphisms for Banach modules
... category theory, to [3] and [15] for terminology and general results on Banach spaces and to G. Janelidze and W. Tholen [5], [6] and [7] for descent theory. 2. PRELIMINARIES Suppose that V is a fixed symmetric monoidal closed category with tensor product ⊗, unit object I, and internal-hom [−, −]. Re ...
... category theory, to [3] and [15] for terminology and general results on Banach spaces and to G. Janelidze and W. Tholen [5], [6] and [7] for descent theory. 2. PRELIMINARIES Suppose that V is a fixed symmetric monoidal closed category with tensor product ⊗, unit object I, and internal-hom [−, −]. Re ...
Spectra of Small Categories and Infinite Loop Space Machines
... For the rest of this paper, the realization of a simplicial category A will mean the special realization A → |τ A| and we will denote it simply by |A|. Note 3.6 There is another way to remedy the problem of the cofibration condition. As it is shown in [8], one could work with the standard realizatio ...
... For the rest of this paper, the realization of a simplicial category A will mean the special realization A → |τ A| and we will denote it simply by |A|. Note 3.6 There is another way to remedy the problem of the cofibration condition. As it is shown in [8], one could work with the standard realizatio ...
OPERADS, FACTORIZATION ALGEBRAS, AND (TOPOLOGICAL
... physics, more precisely, the mathematical study of (topological) quantum field theories. We will first recall Atiyah’s axiomatic approach to topological field theories using categorical language. We will encounter operads as a framework useful to abstracting algebraic structures. We will in particul ...
... physics, more precisely, the mathematical study of (topological) quantum field theories. We will first recall Atiyah’s axiomatic approach to topological field theories using categorical language. We will encounter operads as a framework useful to abstracting algebraic structures. We will in particul ...
2-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES
... Our goal is to demonstrate that every cobordism is diffeomorphic to the composition and disjoint union of this set of generating morphisms. We will do this by defining a normal form for a connected cobordism determined only by the numbers of in- and out- boundaries and the number of holes, and expla ...
... Our goal is to demonstrate that every cobordism is diffeomorphic to the composition and disjoint union of this set of generating morphisms. We will do this by defining a normal form for a connected cobordism determined only by the numbers of in- and out- boundaries and the number of holes, and expla ...
Abstract and Variable Sets in Category Theory
... elements. Concrete sets are typically obtained as extensions of attributes. Thus to be a member of a concrete set C is precisely to possess a certain attribute A, in short, to be an A. (It is for this reason that Peano used ∈, the first letter of Greek εστι, “is”, to denote membership.) The identity ...
... elements. Concrete sets are typically obtained as extensions of attributes. Thus to be a member of a concrete set C is precisely to possess a certain attribute A, in short, to be an A. (It is for this reason that Peano used ∈, the first letter of Greek εστι, “is”, to denote membership.) The identity ...
categories - Andrew.cmu.edu
... some deep and surprising connections. 1970s Applications were already appearing in computer science, linguistics, cognitive science, philosophy, and many other areas. One very striking thing about the field is that it has such wide-ranging applications. In fact, it turns out to be a kind of universa ...
... some deep and surprising connections. 1970s Applications were already appearing in computer science, linguistics, cognitive science, philosophy, and many other areas. One very striking thing about the field is that it has such wide-ranging applications. In fact, it turns out to be a kind of universa ...
W-TYPES IN HOMOTOPY TYPE THEORY 1. Introduction This paper
... Recently, Voevodsky has shown that the category of simplicial sets provides a model of type theory ; more precisely, of the Calculus of Constructions. In this model, types are interpreted as Kan complexes and type dependencies are interpreted as Kan fibrations. One of the main new features of this m ...
... Recently, Voevodsky has shown that the category of simplicial sets provides a model of type theory ; more precisely, of the Calculus of Constructions. In this model, types are interpreted as Kan complexes and type dependencies are interpreted as Kan fibrations. One of the main new features of this m ...
(pdf)
... We shall see how this can be done in Section 6. Monads can also be thought of as theories. As we will see, there is a notion of an algebra of a monad. These are the structured objects produced by the monad-asprocess. In the ultrafilter example above, an algebra would be a set X together with a map f ...
... We shall see how this can be done in Section 6. Monads can also be thought of as theories. As we will see, there is a notion of an algebra of a monad. These are the structured objects produced by the monad-asprocess. In the ultrafilter example above, an algebra would be a set X together with a map f ...
THE ZEN OF ∞-CATEGORIES Contents 1. Derived categories
... which are Quillen equivalent to (CatsSet )Bergner and thus likewise present the homotopy theory of homotopy theories (by virtue of Theorem 2.2). Purely as a matter of terminology, objects of any of these model categories – or more precisely, their weak equivalence classes – have come to be referred ...
... which are Quillen equivalent to (CatsSet )Bergner and thus likewise present the homotopy theory of homotopy theories (by virtue of Theorem 2.2). Purely as a matter of terminology, objects of any of these model categories – or more precisely, their weak equivalence classes – have come to be referred ...
Finite MTL
... A totally ordered MTL-algebra (MTL-chain) is archimedean if for every x ≤ y < 1, there exists n ∈ N such that y n ≤ x. A forest is a poset X such that for every a ∈ X the set ↓ a = {x ∈ X | x ≤ a} is a totally ordered subset of X. A p-morphism is a morphism of posets f : X → Y satisfying the followi ...
... A totally ordered MTL-algebra (MTL-chain) is archimedean if for every x ≤ y < 1, there exists n ∈ N such that y n ≤ x. A forest is a poset X such that for every a ∈ X the set ↓ a = {x ∈ X | x ≤ a} is a totally ordered subset of X. A p-morphism is a morphism of posets f : X → Y satisfying the followi ...
Lecture 14: Bordism categories The definition Fix a nonnegative1
... all manifolds Y, X are required to carry X(n)-structures. Examples include stable tangential structures, such as orientation and spin, as well as unstable tangential structures, such as n-framings. We follow the notational convention of Exercise 9.71. ...
... all manifolds Y, X are required to carry X(n)-structures. Examples include stable tangential structures, such as orientation and spin, as well as unstable tangential structures, such as n-framings. We follow the notational convention of Exercise 9.71. ...
K-theory of Waldhausen categories
... Chb (C) of bounded chain complexes with objects in C is a Waldhausen category, where the cofibrations are the degreewise monomorphisms and weak equivalences are quasi-isomorphisms in A. Proposition 3.1 If C is closed under kernel of surjections then we have K0 (Chb (C)) ' K0 (C). Proof. If C. is a c ...
... Chb (C) of bounded chain complexes with objects in C is a Waldhausen category, where the cofibrations are the degreewise monomorphisms and weak equivalences are quasi-isomorphisms in A. Proposition 3.1 If C is closed under kernel of surjections then we have K0 (Chb (C)) ' K0 (C). Proof. If C. is a c ...
The Functor Category in Relation to the Model Theory of Modules
... pp-formulas = finitely generated subfunctors of representables: If φ is a pp-formula, Fφ is a finitely generated subfunctor of a representable functor in fp(mod(R), Ab). If F is a finitely generated subfunctor of a representable functor, then there exists pp-formula φ such that F ∼ = Fφ . ...
... pp-formulas = finitely generated subfunctors of representables: If φ is a pp-formula, Fφ is a finitely generated subfunctor of a representable functor in fp(mod(R), Ab). If F is a finitely generated subfunctor of a representable functor, then there exists pp-formula φ such that F ∼ = Fφ . ...
On the logic of generalised metric spaces
... Distributive laws and equivalence of DU with [[−, Ω], Ω] ...
... Distributive laws and equivalence of DU with [[−, Ω], Ω] ...
FREE GROUPS - Stanford University
... are obvious. The problem with associativity is that when three words are juxtaposed, you must perform simplifications in two different orders, so it isn’t immediate that you always end up with the same reduced word. This is a technical difficulty, which must be handled one way or another. I’ll follo ...
... are obvious. The problem with associativity is that when three words are juxtaposed, you must perform simplifications in two different orders, so it isn’t immediate that you always end up with the same reduced word. This is a technical difficulty, which must be handled one way or another. I’ll follo ...
EXAMPLE SHEET 3 1. Let A be a k-linear category, for a
... 7. Prove that the forgetful functor Bialg Ñ Coalg has a left adjoint given by C ÞÑ T pCq. 8. Let V be a braided monoidal category. Show that the monoidal category MonpVq is braided and its forgetful functor into V is braided iff the braiding of V is a symmetry. Deduce a similar result for ComonpVq. ...
... 7. Prove that the forgetful functor Bialg Ñ Coalg has a left adjoint given by C ÞÑ T pCq. 8. Let V be a braided monoidal category. Show that the monoidal category MonpVq is braided and its forgetful functor into V is braided iff the braiding of V is a symmetry. Deduce a similar result for ComonpVq. ...
Frobenius algebras and monoidal categories
... Frobenius monoids in a monoidal category Theorem Suppose A is a monoid in V a n d e:Aæ æÆ I is a morphism. The following six conditions are equivalent and define Frobenius monoid: (a) there exists r : I æ æÆ A ƒ A such that (A ƒ m) o (r ƒ A) = (m ƒ A) o (A ƒ r) and ( A ƒ e ) o r = h = (e ƒ A) o r ; ...
... Frobenius monoids in a monoidal category Theorem Suppose A is a monoid in V a n d e:Aæ æÆ I is a morphism. The following six conditions are equivalent and define Frobenius monoid: (a) there exists r : I æ æÆ A ƒ A such that (A ƒ m) o (r ƒ A) = (m ƒ A) o (A ƒ r) and ( A ƒ e ) o r = h = (e ƒ A) o r ; ...
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.