Commutative monads as a theory of distributions
... Recall the notion of monad T = (T, η, µ) on a category E , cf. e.g. [16] 6.1 and 6.2. Recall also the notion of T -algebra A = (A, α) for such monad; here, α : T (A) → A is the structure map for the given algebra. There is a notion of morphism of algebras (A, α) → (B, β ), cf. loc.cit., so that we h ...
... Recall the notion of monad T = (T, η, µ) on a category E , cf. e.g. [16] 6.1 and 6.2. Recall also the notion of T -algebra A = (A, α) for such monad; here, α : T (A) → A is the structure map for the given algebra. There is a notion of morphism of algebras (A, α) → (B, β ), cf. loc.cit., so that we h ...
categories types and structures
... sense, which provides a unified understanding of various aspects of the theory of programs. This is one of the reasons for the increasing role of category theory in the semantic investigation of programs if compared, say, to the set-theoretic approach. However, the influence of this mathematical dis ...
... sense, which provides a unified understanding of various aspects of the theory of programs. This is one of the reasons for the increasing role of category theory in the semantic investigation of programs if compared, say, to the set-theoretic approach. However, the influence of this mathematical dis ...
TRACES IN SYMMETRIC MONOIDAL CATEGORIES Contents
... This is an example of string diagram notation for monoidal categories, which is “Poincaré dual” to the usual sort of diagrams: instead of drawing objects as vertices and morphisms as arrows connecting these vertices, we draw objects as arrows and morphisms as vertices, often with boxes around them. ...
... This is an example of string diagram notation for monoidal categories, which is “Poincaré dual” to the usual sort of diagrams: instead of drawing objects as vertices and morphisms as arrows connecting these vertices, we draw objects as arrows and morphisms as vertices, often with boxes around them. ...
An introduction to stable homotopy theory “Abelian groups up to
... ` An object X is small in C if ⊕[X, Ai] → [X, Ai] is an isomorphism. An object X is a generator of C (or Ho(C)) if the only localizing subcategory containing X is Ho(C) itself. (A localizing subcategory is a triangulated subcategory which is closed under coproducts.) Example: A is a small generator ...
... ` An object X is small in C if ⊕[X, Ai] → [X, Ai] is an isomorphism. An object X is a generator of C (or Ho(C)) if the only localizing subcategory containing X is Ho(C) itself. (A localizing subcategory is a triangulated subcategory which is closed under coproducts.) Example: A is a small generator ...
Morita equivalence for regular algebras
... following way: Proposition 2.1 The assignments of proposition 1.6 give rise to a biequivalence between the 2-category of regular algebras and regular and colimit-preserving functors, and the bicategory Algreg. This biequivalence is the identity on the objects. Since any biequivalence preserves and r ...
... following way: Proposition 2.1 The assignments of proposition 1.6 give rise to a biequivalence between the 2-category of regular algebras and regular and colimit-preserving functors, and the bicategory Algreg. This biequivalence is the identity on the objects. Since any biequivalence preserves and r ...
HOMOTOPICAL ENHANCEMENTS OF CYCLE CLASS MAPS 1
... Example 5. Suppose U• is a cover given by two open subsets U, V , and F• is a chain complex of injective sheaves. Then the homotopy sheaf condition for DK(F• ) is the fact that F(X) → F(U ) ⊕ F(V ) → F(U ∩ V ) is acyclic. There’s a lot of machinery we would like to have for homotopy sheaves (e.g. s ...
... Example 5. Suppose U• is a cover given by two open subsets U, V , and F• is a chain complex of injective sheaves. Then the homotopy sheaf condition for DK(F• ) is the fact that F(X) → F(U ) ⊕ F(V ) → F(U ∩ V ) is acyclic. There’s a lot of machinery we would like to have for homotopy sheaves (e.g. s ...
Contents 1. Introduction 2 2. The monoidal background 5 2.1
... the previous result to cohomology theories in the sense of Cartan, including the comparison of Sullivan polynomial forms and singular cochains. We should point out that in this paper we do not work out the contravariant symmetric case. It will be dealt with elsewhere as an example of a more systemat ...
... the previous result to cohomology theories in the sense of Cartan, including the comparison of Sullivan polynomial forms and singular cochains. We should point out that in this paper we do not work out the contravariant symmetric case. It will be dealt with elsewhere as an example of a more systemat ...
LOCALIZATION OF ALGEBRAS OVER COLOURED OPERADS
... C-coloured collections in E with their morphisms, for a fixed set of colours C. A C-coloured operad P in E is a C-coloured collection equipped with a unit map I −→ P (c; c) for each c in C and, for every (n + 1)-tuple of colours (c1 , . . . , cn ; c) and n given tuples (a1,1 , . . . , a1,k1 ; c1 ), ...
... C-coloured collections in E with their morphisms, for a fixed set of colours C. A C-coloured operad P in E is a C-coloured collection equipped with a unit map I −→ P (c; c) for each c in C and, for every (n + 1)-tuple of colours (c1 , . . . , cn ; c) and n given tuples (a1,1 , . . . , a1,k1 ; c1 ), ...
(pdf)
... In October, 1999, I organized a conference here in honor of MacLane’s 90th birthday. I’ll repeat how I started my talk then. “A great deal of modern mathematics would quite literally be unthinkable without the language of categories, functors, and natural transformations introduced by Eilenberg and ...
... In October, 1999, I organized a conference here in honor of MacLane’s 90th birthday. I’ll repeat how I started my talk then. “A great deal of modern mathematics would quite literally be unthinkable without the language of categories, functors, and natural transformations introduced by Eilenberg and ...
PDF
... is an affine group scheme of multiplicative units. And going in the opposite direction, the algebra of natural transformations from an affine group scheme to its affine 1-space is a commutative Hopf algebra, with coalgebra structure given by dualising the group structure of the affine group scheme. ...
... is an affine group scheme of multiplicative units. And going in the opposite direction, the algebra of natural transformations from an affine group scheme to its affine 1-space is a commutative Hopf algebra, with coalgebra structure given by dualising the group structure of the affine group scheme. ...
THE COTANGENT STACK 1. Introduction 1.1. Let us fix our
... classes of lifts and whose H −1 is infinitesimal automorphisms of our S −→ X . Furthermore, after giving an appropriate descent theory for derived categories, one can talk about the pull-back of the tangent complex of X along S −→ X (possibly not a smooth morphism), and one finds that it is represen ...
... classes of lifts and whose H −1 is infinitesimal automorphisms of our S −→ X . Furthermore, after giving an appropriate descent theory for derived categories, one can talk about the pull-back of the tangent complex of X along S −→ X (possibly not a smooth morphism), and one finds that it is represen ...
Lecture 4 Supergroups
... In ordinary geometry we can associate to any group scheme G a Lie algebra, commonly denoted by Lie(G), which is identified with the tangent space to the group scheme G at the identity. This is an extremely important construction, since it allows us to linearize problems, by transfering our question ...
... In ordinary geometry we can associate to any group scheme G a Lie algebra, commonly denoted by Lie(G), which is identified with the tangent space to the group scheme G at the identity. This is an extremely important construction, since it allows us to linearize problems, by transfering our question ...
HHG-published (pdf, 416 KiB) - Infoscience
... The key problem that we must solve before defining homotopic Hopf–Galois extensions is to determine how to compute the homotopy coinvariants of a coaction, in particular when taking multiplicative structure into account. Our discussion of this problem forms the heart of this paper. We begin in Secti ...
... The key problem that we must solve before defining homotopic Hopf–Galois extensions is to determine how to compute the homotopy coinvariants of a coaction, in particular when taking multiplicative structure into account. Our discussion of this problem forms the heart of this paper. We begin in Secti ...
Fredrik Dahlqvist and Alexander Kurz. Positive coalgebraic logic
... where S0 is the functor sending a distributive lattice to the poset of its prime filters, and P0 is the functor sending a poset to the distributive lattice of its upsets. The following observation will be of fundamental importance in what follows: the adjunction S0 a P0 : Posop → DL which is the bac ...
... where S0 is the functor sending a distributive lattice to the poset of its prime filters, and P0 is the functor sending a poset to the distributive lattice of its upsets. The following observation will be of fundamental importance in what follows: the adjunction S0 a P0 : Posop → DL which is the bac ...
TYPES, SETS AND CATEGORIES
... 4 If we express the assertion q ≺ p by saying that q is below p, then (2) may be expressed as: for any proposition p ∈ M, p is identical with the proposition every proposition below p is true. 5 Russell [1908]. ...
... 4 If we express the assertion q ≺ p by saying that q is below p, then (2) may be expressed as: for any proposition p ∈ M, p is identical with the proposition every proposition below p is true. 5 Russell [1908]. ...
4. Morphisms
... Proposition 4.7 (Morphisms between affine varieties). Let U be an open subset of an affine variety X, and let Y ⊂ An be another affine variety. Then the morphisms f : U → Y are exactly the maps of the form ...
... Proposition 4.7 (Morphisms between affine varieties). Let U be an open subset of an affine variety X, and let Y ⊂ An be another affine variety. Then the morphisms f : U → Y are exactly the maps of the form ...
Generalized Cohomology
... In the 1950s, several examples of generalized (co)homology theories were discovered. Each of them has its own geometric origin but it turns out that they can be expressed as homotopy sets by using the notion of spectrum. Before we list the axioms for generalized homology and cohomology, let us take ...
... In the 1950s, several examples of generalized (co)homology theories were discovered. Each of them has its own geometric origin but it turns out that they can be expressed as homotopy sets by using the notion of spectrum. Before we list the axioms for generalized homology and cohomology, let us take ...
Constructing quantales and their modules from monoidal
... P(M) arose in Lambek’s [10] formal language theory and Girard’s [6] linear logic. The left P(M)-modules P(X) can be found in the labeled transition systems of Ambra.msky and Vickers [1], but there the a.ction was induced by a ...
... P(M) arose in Lambek’s [10] formal language theory and Girard’s [6] linear logic. The left P(M)-modules P(X) can be found in the labeled transition systems of Ambra.msky and Vickers [1], but there the a.ction was induced by a ...
monoidal category that is also a model category i
... Given any monoidal category, one has categories of monoids and of modules over a given monoid. If we are working in a monoidal model category, we would like these associated categories also to be model categories, so that we can have a homotopy theory of rings and modules. The first results on this ...
... Given any monoidal category, one has categories of monoids and of modules over a given monoid. If we are working in a monoidal model category, we would like these associated categories also to be model categories, so that we can have a homotopy theory of rings and modules. The first results on this ...
Topological balls. - Mathematics and Statistics
... it is a closed subspace of DB . It is well known that these spaces are reflexive in the sense / B 00 are topological and algebraic isomorphisms. that the obvious evaluation maps B See, for example, [Kleisli, Künzi], (2.12) and (4.2). We want to demonstrate that given a ball B, there are both a weak ...
... it is a closed subspace of DB . It is well known that these spaces are reflexive in the sense / B 00 are topological and algebraic isomorphisms. that the obvious evaluation maps B See, for example, [Kleisli, Künzi], (2.12) and (4.2). We want to demonstrate that given a ball B, there are both a weak ...
twisted free tensor products - American Mathematical Society
... correspondence from px.b.s to tf.p.s. The total space of a p.c.b. may have more than one representation as a t.f.p. 3. The construction of a twisted free tensor product. In this section we associate with every t.f.p. A * , FX, a differential graded algebra, which we call a twisted free tensor produc ...
... correspondence from px.b.s to tf.p.s. The total space of a p.c.b. may have more than one representation as a t.f.p. 3. The construction of a twisted free tensor product. In this section we associate with every t.f.p. A * , FX, a differential graded algebra, which we call a twisted free tensor produc ...
Set theory is the branch of mathematical logic that studies sets
... property. The order in which the elements of a set or multiset are listed is irrelevant (unlike for a sequence or tuple). Combining these two ideas into an example ...
... property. The order in which the elements of a set or multiset are listed is irrelevant (unlike for a sequence or tuple). Combining these two ideas into an example ...
THE COBORDISM HYPOTHESIS - UT Mathematics
... defined abelian groups based on the notion of a bordism, though it was Thom [T] who made the first systematic computations of bordism groups using homotopy theory. There are many variations according to the type of manifold: oriented, spin, framed, etc. Theory and computation of bordism groups were ...
... defined abelian groups based on the notion of a bordism, though it was Thom [T] who made the first systematic computations of bordism groups using homotopy theory. There are many variations according to the type of manifold: oriented, spin, framed, etc. Theory and computation of bordism groups were ...
Chapter 7 Duality
... in SmS , admits a duality involution. In particular, if S = Spec(k), and if one has resolution of singularities for k-varieties, then the category DM(k) has a duality involution, making DM(k) a rigid triangulated tensor category. We then give some applications of the duality involution: in (7.4.4)-( ...
... in SmS , admits a duality involution. In particular, if S = Spec(k), and if one has resolution of singularities for k-varieties, then the category DM(k) has a duality involution, making DM(k) a rigid triangulated tensor category. We then give some applications of the duality involution: in (7.4.4)-( ...
strongly complete logics for coalgebras
... Abstract. Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly comp ...
... Abstract. Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly comp ...
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.