functors of artin ringso
... In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, ■)->■F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism ...
... In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, ■)->■F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism ...
Symmeric self-adjoint Hopf categories and a categorical Heisenberg double June 17, 2014
... adjoints which satify certain relations making it a Heisenberg double action. Using the SSH structure on a category we can attempt to directly categorify the proof of Proposition 4.4 by replacing these relations with canonically constructed 2-isomorphisms (5.1) in §5. The nice feature of these 2-iso ...
... adjoints which satify certain relations making it a Heisenberg double action. Using the SSH structure on a category we can attempt to directly categorify the proof of Proposition 4.4 by replacing these relations with canonically constructed 2-isomorphisms (5.1) in §5. The nice feature of these 2-iso ...
UNIQUENESS METHODS IN CONSTRUCTIVE POTENTIAL
... arrow is right-smoothly contravariant and freely null. T. R. Robinson’s construction of contraparabolic, non-discretely linear subrings was a milestone in pure Lie theory. This leaves open the question of associativity. Hence it has long been known that L00 ≥ 0 [7]. We wish to extend the results of ...
... arrow is right-smoothly contravariant and freely null. T. R. Robinson’s construction of contraparabolic, non-discretely linear subrings was a milestone in pure Lie theory. This leaves open the question of associativity. Hence it has long been known that L00 ≥ 0 [7]. We wish to extend the results of ...
1. Let G be a sheaf of abelian groups on a topological space. In this
... for every U ⊂ V . Show that H 1 (X, G) = 0 for every flasque sheaf G. (2) Recall that a sheaf G on a paracompact Hausdorff space X is called fine if for every locally finite covering {Ui } of X, there exist ϕi : G → G such that P a family of endomorphisms {x ∈ X | (ϕi )x 6= 0} ⊂ Ui , and that ϕi = 1 ...
... for every U ⊂ V . Show that H 1 (X, G) = 0 for every flasque sheaf G. (2) Recall that a sheaf G on a paracompact Hausdorff space X is called fine if for every locally finite covering {Ui } of X, there exist ϕi : G → G such that P a family of endomorphisms {x ∈ X | (ϕi )x 6= 0} ⊂ Ui , and that ϕi = 1 ...
Simplicial Objects and Singular Homology
... For each dimension n we can take a standard n-simplex 4n in the space n , labelling the vertices (0, 1, ..., n). This standard n-simplex is the convex hull of the standard basis of n along with the origin (labelled 0). The standard n-simplex can be expressed by barycentric coordinates relative to th ...
... For each dimension n we can take a standard n-simplex 4n in the space n , labelling the vertices (0, 1, ..., n). This standard n-simplex is the convex hull of the standard basis of n along with the origin (labelled 0). The standard n-simplex can be expressed by barycentric coordinates relative to th ...
Collated Notes on TQFT.pdf
... Define idx to be the constant path at x, idx (t) = x. Define p−1 (t) = p(1 − t). Composition is not associative or unital, but it becomes so after passage to equivalence classes. Verifications that we leave to the reader (or the first chapter of “A concise course in algebraic topology”1) show that Π ...
... Define idx to be the constant path at x, idx (t) = x. Define p−1 (t) = p(1 − t). Composition is not associative or unital, but it becomes so after passage to equivalence classes. Verifications that we leave to the reader (or the first chapter of “A concise course in algebraic topology”1) show that Π ...
Chern Character, Loop Spaces and Derived Algebraic Geometry
... The conclusion of the observation above is that there should exist an interesting notion of categorical sheaves, which are sheaves of categories rather than sheaves of vector spaces, useful for a geometric description of objects underlying elliptic cohomology. In this work we have been interested in ...
... The conclusion of the observation above is that there should exist an interesting notion of categorical sheaves, which are sheaves of categories rather than sheaves of vector spaces, useful for a geometric description of objects underlying elliptic cohomology. In this work we have been interested in ...
Frobenius monads and pseudomonoids
... many score years. A lot of information about a group G is contained in its characters. Characters are group morphisms from G into the multiplicative monoid of an appropriate field k. In other words, we find a category (in this case the category of monoids) where G and k both live as objects so that ...
... many score years. A lot of information about a group G is contained in its characters. Characters are group morphisms from G into the multiplicative monoid of an appropriate field k. In other words, we find a category (in this case the category of monoids) where G and k both live as objects so that ...
18. Fibre products of schemes The main result of this section is
... The main result of this section is: Theorem 18.1. The category of schemes admits fibre products. A key part of the proof is to pass from the local case (in which case all three schemes are affine) to the global case. To do this, we need to be able to construct morphisms, by constructing them locally ...
... The main result of this section is: Theorem 18.1. The category of schemes admits fibre products. A key part of the proof is to pass from the local case (in which case all three schemes are affine) to the global case. To do this, we need to be able to construct morphisms, by constructing them locally ...
Introduction to universal algebra
... - an arity function ar : Σ → N . Defn: A Σ -algebra (or a model for Σ ) A is: - a set |A| (the carrier), A ar(f ) → |A| . - for each f ∈ Σ , a function f : |A| Defn: A Σ -algebra homomorphism h : A → B is a function h : |A| → |B| such that ...
... - an arity function ar : Σ → N . Defn: A Σ -algebra (or a model for Σ ) A is: - a set |A| (the carrier), A ar(f ) → |A| . - for each f ∈ Σ , a function f : |A| Defn: A Σ -algebra homomorphism h : A → B is a function h : |A| → |B| such that ...
Covering Groupoids of Categorical Rings - PMF-a
... A group-groupoid is a group object in the category of groupoids [7]; equivalently, it is an internal category and hence an internal groupoid in the category of groups ([18], [1]) . An alternative name, quite generally used, is “2-group”, see for example [4]. Recently for example in [1], [12] and [13 ...
... A group-groupoid is a group object in the category of groupoids [7]; equivalently, it is an internal category and hence an internal groupoid in the category of groups ([18], [1]) . An alternative name, quite generally used, is “2-group”, see for example [4]. Recently for example in [1], [12] and [13 ...
Topological Field Theories
... ] n (M, N ) has image contained in one connected component, say [W ], so that it can be considered as a map f : X → BDiff(W ). This classifies a bundle f ∗ EDiff(W ) → X with fiber Diff(W ). Considering the associated bundle E := EDiff(W ) ×Diff(W ) W and its pullback f ∗ E, we can say that maps f : ...
... ] n (M, N ) has image contained in one connected component, say [W ], so that it can be considered as a map f : X → BDiff(W ). This classifies a bundle f ∗ EDiff(W ) → X with fiber Diff(W ). Considering the associated bundle E := EDiff(W ) ×Diff(W ) W and its pullback f ∗ E, we can say that maps f : ...
Abelian Categories
... • Note that in the category of monoids, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes. Therefore, the notion of kernel studied in monoid theory is slightly different. Conversely, in the category of rings, there are no ...
... • Note that in the category of monoids, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes. Therefore, the notion of kernel studied in monoid theory is slightly different. Conversely, in the category of rings, there are no ...
Cluster categories for topologists - University of Virginia Information
... The goal of this paper is to introduce topological triangulated orbit categories, and in particular, the motivating example of topological cluster categories. In doing so, we hope to explain the fundamental ideas of triangulated orbit categories to readers from a more homotopy-theoretic, rather than ...
... The goal of this paper is to introduce topological triangulated orbit categories, and in particular, the motivating example of topological cluster categories. In doing so, we hope to explain the fundamental ideas of triangulated orbit categories to readers from a more homotopy-theoretic, rather than ...
A Brief Overview of Topological Quantum Field Theory
... Alternately, starting with a Lie group, there is a algebraic-geometric way to construct a symplectic manifold (from a line bundle) which preserves much of the group structure. From this, we will find that the Borel-Weil theorem gives a method of quantization — a physical interpretation of a purely ...
... Alternately, starting with a Lie group, there is a algebraic-geometric way to construct a symplectic manifold (from a line bundle) which preserves much of the group structure. From this, we will find that the Borel-Weil theorem gives a method of quantization — a physical interpretation of a purely ...
a new look at means on topological spaces fc
... X to be a polyhedron and only required conditions (i), (ii) above up to homotopy One of his principal conclusions was that if X is compact and admits a (homotopy) n-mean for a,ll n, then X is contractible In 1962, Eckmann, together with Tudor Ganea and the author, returned to the study of n-means in ...
... X to be a polyhedron and only required conditions (i), (ii) above up to homotopy One of his principal conclusions was that if X is compact and admits a (homotopy) n-mean for a,ll n, then X is contractible In 1962, Eckmann, together with Tudor Ganea and the author, returned to the study of n-means in ...
Weights for Objects of Monoids
... product in K. This is due to the fact that a composition of strict cones with the lax monoidal functors gives rise to the so-called strict-lax cones (that commutes strictly with the co-algebraic morphisms and in a lax way only with the algebraic ones) that form a category having the subcategory of s ...
... product in K. This is due to the fact that a composition of strict cones with the lax monoidal functors gives rise to the so-called strict-lax cones (that commutes strictly with the co-algebraic morphisms and in a lax way only with the algebraic ones) that form a category having the subcategory of s ...
Weights for Objects of Monoids
... sends the generalized concept from an arbitrary 2-category back to the original one in Cat. For example, a 0-cell T together with two 2-cells η : 1 ⇒ T and µ : T 2 ⇒ T is a monad in a 2-category K iff it is sent by any representable functor to a monad in Cat or, equivalently, iff its image under the ...
... sends the generalized concept from an arbitrary 2-category back to the original one in Cat. For example, a 0-cell T together with two 2-cells η : 1 ⇒ T and µ : T 2 ⇒ T is a monad in a 2-category K iff it is sent by any representable functor to a monad in Cat or, equivalently, iff its image under the ...
Reporting Category: Number Sense and Operations
... 12. What is the missing number that makes the number sentence below true? ...
... 12. What is the missing number that makes the number sentence below true? ...
Here - Mathematisches Institut der Universität Bonn
... vortex counting, etc. They lead to new relations — similar to the relation between Seiberg-Witten and Donaldson-Witten invariants that also represent two faces of the same physical system — and bridges between different areas of mathematics. In each of my lectures, I will try to focus on one particu ...
... vortex counting, etc. They lead to new relations — similar to the relation between Seiberg-Witten and Donaldson-Witten invariants that also represent two faces of the same physical system — and bridges between different areas of mathematics. In each of my lectures, I will try to focus on one particu ...
Hilbert`s First and Second Problems and the foundations of
... A strengthening of countable compactness, not shared by all compact spaces, is that of sequential compactness: every sequence has a convergent subsequence. These three concepts agree for all metrizable spaces (those spaces whose topology is given by a distance function to the non-negative reals that ...
... A strengthening of countable compactness, not shared by all compact spaces, is that of sequential compactness: every sequence has a convergent subsequence. These three concepts agree for all metrizable spaces (those spaces whose topology is given by a distance function to the non-negative reals that ...
Fractional Exponent Functors and Categories of Differential Equations
... Lawvere suggested that, utilizing these fractional exponent functors, it should be possible to prove that the category of 2ODEs in E is a topos (see [25]); one aim of the present article is to carry out this suggestion of Lawvere’s, and prove that the 2ODEs over E do indeed form a topos. We study so ...
... Lawvere suggested that, utilizing these fractional exponent functors, it should be possible to prove that the category of 2ODEs in E is a topos (see [25]); one aim of the present article is to carry out this suggestion of Lawvere’s, and prove that the 2ODEs over E do indeed form a topos. We study so ...
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.