A note on actions of a monoidal category
... addition that, at least in the simplest case where V is symmetric monoidal closed, to give an action of V on A for which both − ∗ A and X ∗ − have right adjoints is equivalently to give a V-category A that is both tensored and cotensored. In fact, under these conditions, there is a right adjoint not ...
... addition that, at least in the simplest case where V is symmetric monoidal closed, to give an action of V on A for which both − ∗ A and X ∗ − have right adjoints is equivalently to give a V-category A that is both tensored and cotensored. In fact, under these conditions, there is a right adjoint not ...
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS -modules. 20. KZ functor, II: image
... 20.1. Some general facts about DX -modules. Let X be a smooth algebraic variety and let DX denote the sheaf of differential operators on X. Consider the subcategory Loc(X) ⊂ DX -mod consisting of all DX -modules that are coherent sheaves on X. This is a Serre subcategory in DX -mod. As we have seen i ...
... 20.1. Some general facts about DX -modules. Let X be a smooth algebraic variety and let DX denote the sheaf of differential operators on X. Consider the subcategory Loc(X) ⊂ DX -mod consisting of all DX -modules that are coherent sheaves on X. This is a Serre subcategory in DX -mod. As we have seen i ...
foundations of algebraic geometry class 38
... We have many motivations for doing this. In no particular order: (1) It “globalizes” what we did before. (2) If 0 → F → G → H → 0 is a short exact sequence of quasicoherent sheaves on X, then we know that 0 → π∗ F → π∗ G → π∗ H is exact, and higher pushforwards will extend this to a long exact sequ ...
... We have many motivations for doing this. In no particular order: (1) It “globalizes” what we did before. (2) If 0 → F → G → H → 0 is a short exact sequence of quasicoherent sheaves on X, then we know that 0 → π∗ F → π∗ G → π∗ H is exact, and higher pushforwards will extend this to a long exact sequ ...
Fascicule 1
... and codomain, Banaschewski and Herrlich [5] characterized full subcategories of “suitable” categories which can be specified by such injectivity: they are precisely the subcategories closed under products, subobjects, and filtered colimits. Recently the same result was proved for all locally finitel ...
... and codomain, Banaschewski and Herrlich [5] characterized full subcategories of “suitable” categories which can be specified by such injectivity: they are precisely the subcategories closed under products, subobjects, and filtered colimits. Recently the same result was proved for all locally finitel ...
Whole and Part in Mathematics
... distribution of masses placed at the vertices of P. This fact is generalized to Choquet's theorem, an important result of functional analysis (see, e.g. Edwards 1965). This theorem asserts that any point of a compact convex subset C in a normed space is the barycentre (in a suitably extended sense) ...
... distribution of masses placed at the vertices of P. This fact is generalized to Choquet's theorem, an important result of functional analysis (see, e.g. Edwards 1965). This theorem asserts that any point of a compact convex subset C in a normed space is the barycentre (in a suitably extended sense) ...
4 Choice axioms and Baire category theorem
... f (x) = 0 for any given x. Here is the know-how. We exercise the “nonseparable topological random number generator”, getting (xn )n , xn ∈ R, and check the equalities f (x1 ) = 0, f (x2 ) = 0, . . . until a solution is found. If the equation has a solution (at least one) then it should occur in the ...
... f (x) = 0 for any given x. Here is the know-how. We exercise the “nonseparable topological random number generator”, getting (xn )n , xn ∈ R, and check the equalities f (x1 ) = 0, f (x2 ) = 0, . . . until a solution is found. If the equation has a solution (at least one) then it should occur in the ...
ADEQUATE SUBCATEGORIES
... This paper introduces and studies the notion of a left adequate subcategory of an arbitrary category (and the dual notion). Definition follows. Let a be a full subcategory of e. For any object X of (, let Map ((, X) denote the contravariant functor on a into the category of all sets and all function ...
... This paper introduces and studies the notion of a left adequate subcategory of an arbitrary category (and the dual notion). Definition follows. Let a be a full subcategory of e. For any object X of (, let Map ((, X) denote the contravariant functor on a into the category of all sets and all function ...
SPLIT STRUCTURES To our friend Aurelio Carboni for his 60th
... them. For example, rel has many nice properties and structures (studied by Carboni and Walters [C&W87]) that are inherited by splitting of idempotents and they transfer to ccdsup via the equivalence with kar(rel). Since the real category of interest for completely distributive lattices, ccd, has for ...
... them. For example, rel has many nice properties and structures (studied by Carboni and Walters [C&W87]) that are inherited by splitting of idempotents and they transfer to ccdsup via the equivalence with kar(rel). Since the real category of interest for completely distributive lattices, ccd, has for ...
Advanced Course on Topological Quantum Field Theories
... In this case the corresponding topological invariants are knot and link invariants as the Jones polynomial and its generalizations. TQFTs provide a new point of view to study topological invariants of 3- and 4-dimensional manifolds. Topological Quantum Field Theories are at the moment in the heart o ...
... In this case the corresponding topological invariants are knot and link invariants as the Jones polynomial and its generalizations. TQFTs provide a new point of view to study topological invariants of 3- and 4-dimensional manifolds. Topological Quantum Field Theories are at the moment in the heart o ...
HOW TO PROVE THAT A NON-REPRESENTABLE FUNCTOR IS
... material. That aside, if you find anything in these notes that’s definitely false, please let me know. The purpose of this short set of notes is to discuss some old qual problems related to representability of functors C → Sets. In particular, we will show how to prove that a non-representable funct ...
... material. That aside, if you find anything in these notes that’s definitely false, please let me know. The purpose of this short set of notes is to discuss some old qual problems related to representability of functors C → Sets. In particular, we will show how to prove that a non-representable funct ...
TAG Lecture 2: Schemes
... we can define the symmetric A-algebra SymA (M) and that it has the appropriate universal property. (What would that be?) Let A = S be the sphere spectrum and let M = ∨n S (∨ = coproduct or wedge). What is Spec(SymS (M))? That is, what functor does it represent? 2. Suppose n = 1 and x ∈ SymS (M) is r ...
... we can define the symmetric A-algebra SymA (M) and that it has the appropriate universal property. (What would that be?) Let A = S be the sphere spectrum and let M = ∨n S (∨ = coproduct or wedge). What is Spec(SymS (M))? That is, what functor does it represent? 2. Suppose n = 1 and x ∈ SymS (M) is r ...
Descent and Galois theory for Hopf categories
... category is called an H-Galois category extension of its coinvariants if a collection of canonical maps is invertible; some equivalent conditions are given in Theorem 3.5 and in this case descent data and relative Hopf modules are isomorphic categories, leading to the desired descent theory if some ...
... category is called an H-Galois category extension of its coinvariants if a collection of canonical maps is invertible; some equivalent conditions are given in Theorem 3.5 and in this case descent data and relative Hopf modules are isomorphic categories, leading to the desired descent theory if some ...
Equivariant homotopy theory, model categories
... composite (G/K ⊗ −)H to take generating cofibrations to cofibrations and generating acyclic cofibrations to acyclic cofibrations. The form stated in the proposition is crucial for comparing the F -model structure with the orbit diagrams in C. A toy example, where condition iii) does not hold for H = ...
... composite (G/K ⊗ −)H to take generating cofibrations to cofibrations and generating acyclic cofibrations to acyclic cofibrations. The form stated in the proposition is crucial for comparing the F -model structure with the orbit diagrams in C. A toy example, where condition iii) does not hold for H = ...
How to quantize infinitesimally-braided symmetric monoidal categories
... VectK be a faithful exact functor, where “VectK ” means “finitely-generated projective K-modules”. Then there is a K-linear coalgebra A and (C, F ) is equivalent as a category to (A-corep, Forget). Moreover, structure on (C, F ) determines structure on A. The usual reconstruction is to get an algebr ...
... VectK be a faithful exact functor, where “VectK ” means “finitely-generated projective K-modules”. Then there is a K-linear coalgebra A and (C, F ) is equivalent as a category to (A-corep, Forget). Moreover, structure on (C, F ) determines structure on A. The usual reconstruction is to get an algebr ...
PARTIALIZATION OF CATEGORIES AND INVERSE BRAID
... braid monoid, defined and studied in [EL]. The idea is the following: the braid group can be realized as the mapping class group (of homeomorphisms with compact support) of a punctured plane. In Section 4 we define the category, whose objects are punctured planes with different punctures and whose m ...
... braid monoid, defined and studied in [EL]. The idea is the following: the braid group can be realized as the mapping class group (of homeomorphisms with compact support) of a punctured plane. In Section 4 we define the category, whose objects are punctured planes with different punctures and whose m ...
Algebras of Deductions in Category Theory∗ 1 Logical models from
... is foreign to the Boolean spirit. Since this asymmetry is a consequence of the adjunction involving A ∧ and A ⇒, we should not expect this adjunction for classical logic. If ¬A is de ned as A ⇒ ⊥, it may seem natural to assume in classical logic that A is isomorphic to ¬¬A, or that there is at least ...
... is foreign to the Boolean spirit. Since this asymmetry is a consequence of the adjunction involving A ∧ and A ⇒, we should not expect this adjunction for classical logic. If ¬A is de ned as A ⇒ ⊥, it may seem natural to assume in classical logic that A is isomorphic to ¬¬A, or that there is at least ...
Notes on regular, exact and additive categories
... in Set where p is a surjective map (=regular epimorphism), and let us show that π2 is also a surjective map. Let a be an element in A; there exists then an e ∈ E such that p(e) = f (a). This shows that there is an (e, a) ∈ E ×B A such that π2 (e, a) = a. The same argument still works in the category ...
... in Set where p is a surjective map (=regular epimorphism), and let us show that π2 is also a surjective map. Let a be an element in A; there exists then an e ∈ E such that p(e) = f (a). This shows that there is an (e, a) ∈ E ×B A such that π2 (e, a) = a. The same argument still works in the category ...
On the structure of triangulated categories with finitely many
... the general case. The category CQ is triangulated [19] and, if Q is representationfinite, satisfies a) and b′ ). In a recent article [32], J. Xiao and B. Zhu determined the structure of the Auslander-Reiten quiver of a locally finite triangulated category. In this paper, we obtain the same result with ...
... the general case. The category CQ is triangulated [19] and, if Q is representationfinite, satisfies a) and b′ ). In a recent article [32], J. Xiao and B. Zhu determined the structure of the Auslander-Reiten quiver of a locally finite triangulated category. In this paper, we obtain the same result with ...
Noncommutative Lp-spaces of W*-categories and their applications
... with the corresponding notions of algebras, provided that we either ignore 2-morphisms on the categorical side or add intertwining elements as 2-morphisms on the algebraic side. A *-category is a category enriched over the symmetric monoidal category of complex vector spaces with the algebraic tenso ...
... with the corresponding notions of algebras, provided that we either ignore 2-morphisms on the categorical side or add intertwining elements as 2-morphisms on the algebraic side. A *-category is a category enriched over the symmetric monoidal category of complex vector spaces with the algebraic tenso ...
A very brief introduction to étale homotopy
... over to étale coverings. One would like to apply this construction to a Grothendieck topology on a scheme X, for example, to the small étale site Xét of X. The obvious problem is that open étale “subsets” are rarely contractible. However, it is clear that to recover the space we started with, th ...
... over to étale coverings. One would like to apply this construction to a Grothendieck topology on a scheme X, for example, to the small étale site Xét of X. The obvious problem is that open étale “subsets” are rarely contractible. However, it is clear that to recover the space we started with, th ...
(pdf)
... • Given any formula F, the negation ¬F is a formula; • For any formulae F and G, each of F ∨ G, F ∧ G and F → G is a formula. Definition 2.3. For a set of propositional variables P and a Boolean algebra B, a valuation is a function V : P → B such that V (>) = 1, V (⊥) = 0, V (P ∧ Q) = V (P ) ∩ V (Q) ...
... • Given any formula F, the negation ¬F is a formula; • For any formulae F and G, each of F ∨ G, F ∧ G and F → G is a formula. Definition 2.3. For a set of propositional variables P and a Boolean algebra B, a valuation is a function V : P → B such that V (>) = 1, V (⊥) = 0, V (P ∧ Q) = V (P ) ∩ V (Q) ...
here - Rutgers Physics
... Result: When we take into account the BPS states there is an extremely rich mathematical structure. ...
... Result: When we take into account the BPS states there is an extremely rich mathematical structure. ...
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.