Classifying spaces and spectral sequences
... on SSG is obviously locally trivial. One can obtain BG from SSG by collapsing degenerate simplexes, i.e. those joining elements g^ . . .,^ of G with two g^ equal; thus it is related to SSG in precisely the way that reduced suspensions are related to suspensions. But S8G fits into my framework, too. ...
... on SSG is obviously locally trivial. One can obtain BG from SSG by collapsing degenerate simplexes, i.e. those joining elements g^ . . .,^ of G with two g^ equal; thus it is related to SSG in precisely the way that reduced suspensions are related to suspensions. But S8G fits into my framework, too. ...
Lieblich Definition 1 (Category Fibered in Groupoids). A functor F : D
... Continuing the Proof of Schlessinger’s Criterion. Already shown that (H1)-(H3) imply that we have a Hull. Suppose that F has a hull (R, ξ). Then (H3) follows from TR ' TF and R noetherian of finite dimension. Now suppose that we have p0 : A0 → A and p00 : A00 → A in Art(Λ, k) with p00 a surjection. ...
... Continuing the Proof of Schlessinger’s Criterion. Already shown that (H1)-(H3) imply that we have a Hull. Suppose that F has a hull (R, ξ). Then (H3) follows from TR ' TF and R noetherian of finite dimension. Now suppose that we have p0 : A0 → A and p00 : A00 → A in Art(Λ, k) with p00 a surjection. ...
Section 2.1
... You might ask ‘what do we gain from knowing that two functors are adjoint?’ The uniqueness is a crucial part of the answer. Let us return to the example of (c). It would take you only a few minutes to learn what Lie algebras are, what associative algebras are, and what the standard functor G is that ...
... You might ask ‘what do we gain from knowing that two functors are adjoint?’ The uniqueness is a crucial part of the answer. Let us return to the example of (c). It would take you only a few minutes to learn what Lie algebras are, what associative algebras are, and what the standard functor G is that ...
Category Theory: an abstract setting for analogy
... structure of the objects. The result is that, loosely speaking, using the category C, no one can distinguish two groups if they have the same cardinality. Thus the ‘structure’ of an object of C is really just that of a set. This raises the philosophical issue of whether two copies of the same objec ...
... structure of the objects. The result is that, loosely speaking, using the category C, no one can distinguish two groups if they have the same cardinality. Thus the ‘structure’ of an object of C is really just that of a set. This raises the philosophical issue of whether two copies of the same objec ...
s principle
... ’ in the sense of Grothendieck , an alternate diagnosis i s that the use of s implicial sets ( as the combinatorial obj ects whose geometric realization should be exact ) is to o restrictive . ...
... ’ in the sense of Grothendieck , an alternate diagnosis i s that the use of s implicial sets ( as the combinatorial obj ects whose geometric realization should be exact ) is to o restrictive . ...
April 6: Groups Acting on Categories
... b then we call X a twisted strongly if their composition comes from C), H−equivariant object. Throughout the above discussion, we have mostly dealt with a weak/strong action of G on an abelian category C. However, this always induces a weak/strong action on the category of complexes C(C), and from t ...
... b then we call X a twisted strongly if their composition comes from C), H−equivariant object. Throughout the above discussion, we have mostly dealt with a weak/strong action of G on an abelian category C. However, this always induces a weak/strong action on the category of complexes C(C), and from t ...
Categories of Groups and Rings: A Brief Introduction to Category
... for k1 , k2 in the kernel of f , g is a ring homomorphism, and hence an arrow in Rng. Also, as h(k1 + k2 ) = 0A = 0A + 0A = h(k1 ) + h(k2 ), and h(k1 · k2 ) = 0A = 0A · 0A = h(k1 ) · h(k2 ) for k1 , k2 in the kernel of f , h is a ring homomorphism, and hence an arrow in Rng. f ◦ g = f ◦ h is the rin ...
... for k1 , k2 in the kernel of f , g is a ring homomorphism, and hence an arrow in Rng. Also, as h(k1 + k2 ) = 0A = 0A + 0A = h(k1 ) + h(k2 ), and h(k1 · k2 ) = 0A = 0A · 0A = h(k1 ) · h(k2 ) for k1 , k2 in the kernel of f , h is a ring homomorphism, and hence an arrow in Rng. f ◦ g = f ◦ h is the rin ...
850 Oberwolfach Report 15 Equivariant Sheaves on Flag Varieties
... perfect derived category Perf(Ext(IC)) that can be described for a more general class of dg algebras (see [Sch08a]). This yields an algebraic description of the category of B-equivariant perverse sheaves on X. • The algebra Ext(IC) is isomorphic to the endomorphism algebra of the B-equivariant hyper ...
... perfect derived category Perf(Ext(IC)) that can be described for a more general class of dg algebras (see [Sch08a]). This yields an algebraic description of the category of B-equivariant perverse sheaves on X. • The algebra Ext(IC) is isomorphic to the endomorphism algebra of the B-equivariant hyper ...
A construction of real numbers in the category of categories
... The existence of a natural number object allows to develop the integers Z and the rational numbers Qd through a series of applications of the axioms above, as well as to define the usual operations of sum and product making Qd have the properties of a field. The construction is the same as the one c ...
... The existence of a natural number object allows to develop the integers Z and the rational numbers Qd through a series of applications of the axioms above, as well as to define the usual operations of sum and product making Qd have the properties of a field. The construction is the same as the one c ...
Categories - University of Oregon
... {(a, b) | a ∈ A, b ∈ B} such that for every a ∈ A there exists a unique b ∈ B with (a, b) ∈ f (of course we always write f (a) = b instead of (a, b) ∈ f !) For example, the empty set ∅ is a set, hence an object in the category of sets. Given any other set A, Homsets (∅, A) consists of exactly one mo ...
... {(a, b) | a ∈ A, b ∈ B} such that for every a ∈ A there exists a unique b ∈ B with (a, b) ∈ f (of course we always write f (a) = b instead of (a, b) ∈ f !) For example, the empty set ∅ is a set, hence an object in the category of sets. Given any other set A, Homsets (∅, A) consists of exactly one mo ...
Model structures for operads
... commute with the structure morphisms. Similarly we can define the subcategory of reduced operads Op0 (C) whose objects have P (0) = I and whose morphisms have f0 = id : P (0) → Q(0). Theorem (Berger-Moerdijk, ’03). Let C be a cofibrantly generated monoidal model category with cofibrant unit such tha ...
... commute with the structure morphisms. Similarly we can define the subcategory of reduced operads Op0 (C) whose objects have P (0) = I and whose morphisms have f0 = id : P (0) → Q(0). Theorem (Berger-Moerdijk, ’03). Let C be a cofibrantly generated monoidal model category with cofibrant unit such tha ...
Natural associativity and commutativity
... while associativity is an isomorphism a natural in its arguments A,B, and C. The general associative law again shows that any two iterated products F and F' of the n arguments A,, ...,A, are naturally isomorphic, under a natural isomorphism F z F' given by "iteration" of a. We then ask: what conditi ...
... while associativity is an isomorphism a natural in its arguments A,B, and C. The general associative law again shows that any two iterated products F and F' of the n arguments A,, ...,A, are naturally isomorphic, under a natural isomorphism F z F' given by "iteration" of a. We then ask: what conditi ...
1. Basics 1.1. Definitions. Let C be a symmetric monoidal (∞,2
... The unit 1 ∈ A is dualized to the trace map A → C. In other words, A is a Frobenius algebra. Theorem. Homotopy fixed points of the SO2 action on fully dualizable algebras are identified with Frobenius algebras. From Wedderburn’s theorem one can see that any fully dualizable algebra has a Frobenius s ...
... The unit 1 ∈ A is dualized to the trace map A → C. In other words, A is a Frobenius algebra. Theorem. Homotopy fixed points of the SO2 action on fully dualizable algebras are identified with Frobenius algebras. From Wedderburn’s theorem one can see that any fully dualizable algebra has a Frobenius s ...
MATH 432B/537: Elementary Number Theory Section
... this material might already be familiar to you (congruences come up in courses on abstract algebra, for instance). Once we have this foundation, there are many different directions we could follow. Some topics I would like to cover include: finding roots of polynomial congruences; the Quadratic Reci ...
... this material might already be familiar to you (congruences come up in courses on abstract algebra, for instance). Once we have this foundation, there are many different directions we could follow. Some topics I would like to cover include: finding roots of polynomial congruences; the Quadratic Reci ...
9 Direct products, direct sums, and free abelian groups
... Let {Gi }i∈I be a family of groups, and let S = i∈I Gi be the disjoint union of sets of elements of these groups. A word in S is a sequence a1 a2 . . . ak where k ≥ 0 and a1 , a2 , . . . , ak ∈ S. Consider the equivalence relation of words generated by the following conditions: 1) if eGi is the triv ...
... Let {Gi }i∈I be a family of groups, and let S = i∈I Gi be the disjoint union of sets of elements of these groups. A word in S is a sequence a1 a2 . . . ak where k ≥ 0 and a1 , a2 , . . . , ak ∈ S. Consider the equivalence relation of words generated by the following conditions: 1) if eGi is the triv ...
CORE VARIETIES, EXTENSIVITY, AND RIG GEOMETRY 1
... aims to assist the development of those subjects by revealing characteristic ways in which their categories differ from others. (Such considerations will be important in order to carry out Grothendieck’s 1973 program [2] for simplifying the foundations of algebraic geometry.) An axiomatic theory oft ...
... aims to assist the development of those subjects by revealing characteristic ways in which their categories differ from others. (Such considerations will be important in order to carry out Grothendieck’s 1973 program [2] for simplifying the foundations of algebraic geometry.) An axiomatic theory oft ...
Notes on categories - Math User Home Pages
... • Dually, there is a notion of coslice categories (C ↑ c), or categories under a fixed object. Just take the previous example and turn all arrows around. Pointed categories arise in this way; Set∗ is (Set ↑ •), &c. Also Algk = (Ring ↑ k); the terminal object is any algebraic closure of k. The analog ...
... • Dually, there is a notion of coslice categories (C ↑ c), or categories under a fixed object. Just take the previous example and turn all arrows around. Pointed categories arise in this way; Set∗ is (Set ↑ •), &c. Also Algk = (Ring ↑ k); the terminal object is any algebraic closure of k. The analog ...
Version 1.0.20
... One forms a category of presheaves, Pre(X ), by defining the morphisms to be the natural transformations. By convention, the elements of F (U ) are called sections of F on U . This is also denoted Γ(F ,U ). One may define the notion of a presheaf of groups, rings and so on similarly. The key idea he ...
... One forms a category of presheaves, Pre(X ), by defining the morphisms to be the natural transformations. By convention, the elements of F (U ) are called sections of F on U . This is also denoted Γ(F ,U ). One may define the notion of a presheaf of groups, rings and so on similarly. The key idea he ...
Some definitions that may be useful
... With Alex Chirvasitu, we asked the opposite question: what if you fix K and vary G? The construction is obviously functorial in G, and the functor groupoids → groupoids is not obviously representable. In fact, it isn’t representable at all by an honest groupoid, usually, but it is ...
... With Alex Chirvasitu, we asked the opposite question: what if you fix K and vary G? The construction is obviously functorial in G, and the functor groupoids → groupoids is not obviously representable. In fact, it isn’t representable at all by an honest groupoid, usually, but it is ...
FINITE CATEGORIES WITH TWO OBJECTS A paper should have
... associativity is automatic. The identity morphisms are the unique arrows from each object to itself. The first question is: What can you say if there are two morphisms A → A? You can ask rhetorical questions. Then you might need new definitions to make it easier to talk about. Also this gives you a ...
... associativity is automatic. The identity morphisms are the unique arrows from each object to itself. The first question is: What can you say if there are two morphisms A → A? You can ask rhetorical questions. Then you might need new definitions to make it easier to talk about. Also this gives you a ...
EXAMPLE SHEET 1 1. If k is a commutative ring, prove that b k
... 17. Let A be an algebra over a field k. A right A-module M is rational if for each m P M the orbit m ¨ A “ tm ¨ x : x P Au is finite dimensional. Given a coalgebra C, prove that the functor ComodpCq Ñ Mod-C ˚ is an isomorphism into the full subcategory of rational right C ˚ -modules. If A is finite ...
... 17. Let A be an algebra over a field k. A right A-module M is rational if for each m P M the orbit m ¨ A “ tm ¨ x : x P Au is finite dimensional. Given a coalgebra C, prove that the functor ComodpCq Ñ Mod-C ˚ is an isomorphism into the full subcategory of rational right C ˚ -modules. If A is finite ...
Mid Term Game - Harrison High School
... You go to the fair and buy a one pound bag cotton candy for $3 and a four pound bag of cotton candy for $6. If 6 bags are purchased for a total cost of $30, how many four pound bags were ...
... You go to the fair and buy a one pound bag cotton candy for $3 and a four pound bag of cotton candy for $6. If 6 bags are purchased for a total cost of $30, how many four pound bags were ...
LINEABILITY WITHIN PROBABILITY THEORY SETTINGS 1
... special or (as sometimes it is called) “pathological” property (for exam40 ...
... special or (as sometimes it is called) “pathological” property (for exam40 ...
15. The functor of points and the Hilbert scheme Clearly a scheme
... transformations between hX and F are in natural correspondence with the elements of F (X). (2) h is an equivalence of categories with a full subcategory of Hom(C ◦ , D). Proof. Given a natural transformation α : hX −→ F, we assign the element α(iX ), where iX : X −→ X is the identity map. The invers ...
... transformations between hX and F are in natural correspondence with the elements of F (X). (2) h is an equivalence of categories with a full subcategory of Hom(C ◦ , D). Proof. Given a natural transformation α : hX −→ F, we assign the element α(iX ), where iX : X −→ X is the identity map. The invers ...
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.