1 The Hyland-Schalke functor G Rel. 2 Weakenings
|
|
- Darleen Clarke
- 6 years ago
- Views:
Transcription
1 1 The Hyland-Schalke functor G Rel Let G denote an appropriate category of games and strategies, and let Rel denote the category of sets and relations. It is known (Hyland, Schalke, 1996) that the following define a functor F : G Rel: F() = P ( ) F B = {(s, s B ) : s is a P -play} It can be shown that this functor F is faithful (see Calderon and McCusker, 2010 for a proof). This surprising fact will be very useful for us: if we want to show that a particular diagram commutes in G, we can normally do so more easily by applying the functor F to the diagram and showing that the resulting diagram commutes in Rel. Notation 1.1. We shall simplify our notation by writing P rather than F() and writing rather than F(). 2 Weakenings Let = (M, λ, P ) be a game. Then we have two notions of a weakening of - informally, a game that is easier than, from the player P s point of view. 1. n antecedent weakening of is a game = (M, λ, P ), where P P such that for every O-play s occuring in both P and P, the set of P -responses to s in is precisely the set of P -responses to s in and for every P -play t occurring in both P and P, the set of O-responses to t in is a subset of the set of O-responses to t in. 2. precedent strengthening of is a game = (M, λ, P ) where P P such that for every O-play s occuring in both P and P, the set of player responses to s in is a subset of the set of P -responses to s in, and such that for every P -play t occuring in both P and P the set of O-responses to t in is precisely the set of O-responses to t in. 1
2 In other words, a precedent strengthening of is a game in which the player has more choices at each move, but the opponent has the same choices, while an antecedent weakening of is a game in which the player has the same choices at each move, but the opponent has fewer choices. We can express the definition of an antecedent weakening more compactly as follows: (M, λ, P ) is an antecedent weakening of (M, λ, P ) if P P and the following condition holds: whenever s P is an O-play and a M is a P -move such that sa P, then sa P. This condition says that player P has all the moves available to her in that she had in P - the other conditions mentioned above follow automatically from the fact that P P. The important fact about both types of weakening is that we have copycat morphisms,. From here on, we will use the term weakening to refer to an antecedent weakening. If is a weakening of, we will write : for the copycat morphism. Proposition 2.1. Let be a weakening of. Then : has a right inverse. Proof. The inverse for is the partial strategy str:. This is also constructed as a copycat strategy; the only difference is that if O makes a move in that does not occur in P then the player P has no reply. We now claim that str = id. Formally: = {s P : if s is even, then s = s } str = { s P : if s is even, then s = s } We wish to show that str = id. In the style of section 1, we shall show that str = P in Rel. It is easy to see that = {(s, s) : s P } P P str = {(s, s) : s P } P P It is then easy to see that str = P P, as we wanted. s a sanity check, if we try to take the composition the other way round then we get P P, which is not the identity unless P = P. 2
3 3 The weakening monad of a game 3.1 Weakening pullback of a game Let be a game. Define Suppose B is a game and : B is a morphism. P B = {s B : s } P B is prefix closed, since is, so B = (M B, λ B, P B ) is a well-formed game. We claim that B is a weakening of B. Indeed, suppose s P B is an O-play and sa P B. By the definition of P B, there must be some s such that s B = s; choosing s minimal, we may assume that the last move in s coincides with the last move of s. Since s is an O-play, this last move is an O-move in B and therefore a P -move in B. Therefore, sa since is a strategy and so sa = sa B P B. We call B the weakening pullback of along the morphism. Next, we define a morphism : B. s sets, = (noting that, by definition, P B ). We claim that commutes. Once again, we check this in Rel. If we note that = PB P B P B and that = as sets, then it is clear that = in Rel, and so = in G. The weakening B and the morphism satisfy a universal property - namely, if B is a weakening of B and : B is a morphism such that commutes, then B is a weakening of B and B B (1) (2) 3
4 commutes. Indeed, working in Rel again, diagram (1) tells us that = as sets, which implies that P B P. Noting the definition of P B, it follows that B is the weakening pullback of along, and so diagram (2) commutes. 3.2 The weakening monad We shall want to prove various useful properties of these weakening pullbacks, notably a functoriality condition: if C B are morphisms, then we get a morphism : C B such that ( ) =. This and other useful facts are brought together in the following theorem: Theorem 3.1. Let be a game. Then we have a functor weak : G/ G/ that sends a morphism B to the pullback weakening B and that gives rise to an idempotent monad on G/. Proof. In order to complete the definition of the functor weak, we need to define its action on morphisms. Let C B be morphisms in G (so gives us a morphism from C to B in G/). We define a morphism : C B by = P C B. It is easy to see that is a strategy, so we need to check that = ( ). s usual, we perform this check in Rel. Since ( ) is created according to the pullback recipe given above, we have ( ) = as sets. So, as subsets of P C P, we have: ( ) = = = {(s, t) P C P : u P B. (s, u), (u, t) } Meanwhile, as sets we have = and = (P C P B ). So, as subsets of P C P, we have: = {(s, t) P C P : u P B. (s, u), (u, t) } - in other words, the same thing, but with P B instead of P B. It is clear then that ( ). In the other direction, suppose (s, t) P C P and that there exiss u P B such that (s, u) and (u, t). But now, 4
5 since (u, t), we must have u P B by the definition of P B and therefore (u, t). So far what we have done is show that the morphism does indeed give us a morphism in G/ from C ( ) to B. We still need to prove that what we have defined respects composition and so gives us a functor G/ G/. This means proving the following statement: suppose D υ C B are morphisms in G. This gives us morphisms υ, in G/ as pictured in the diagram below: υ D C B υ We have υ : D C given by υ = υ P D C and : C B given by = P C B, as before. We also have a morphism ( υ) : D B given by ( υ) = ( υ) P D B. We need to show that ( υ) = υ. We have ( υ) = ( υ) P D B, so: ( υ) = ( υ) (P D P B ) = ( υ) (P D P B ) = {(s, t) P D P B : u P C. (s, u) υ, (u, t) } Meanwhile, = P D and υ = υ P B, so υ = {(s, t) P D P B : u P C. (s, u) υ (u, t) } which is the same thing, but with P C rather than P C. Clearly, υ ( υ). In the other direction, suppose that s P D, t P B and that u P C such that (s, u) υ, (u, t). It will suffice to show that u P C. Now note that C is pullback of along and that B is pullback of along. Therefore: P C = {s C : s } P B = {t B : t } Since t P B, there must be some t such that t B = t. Put another way, there must be some v P such that (t, v). We have (u, t), so therefore (u, v), and it follows that u P C. 5
6 Therefore, we have that υ = υ = ( υ). Since the Hyland-Schalke functor is faithful, this means that υ = ( υ). This all means that we can define a functor weak : G/ G/ by C B B B C B We now define the monad structure on G/. The unit transformation η : id G/ weak is given by the weakening morphisms : B B for each object B ; we have already shown that the diagrams commute. To show that this is indeed a natural transformation, we need to show that if we are given some morphism in G/ C B then the following square commutes: To show this, we work in Rel again. On the one hand, we have: = {(s, t) P C P B : u P C. (s, u), (u, t) } = {(s, t) P C P B : (s, t) } = = (P C P B ) 6
7 On the other hand: = {(s, t) P C P B = (P C P B ) : u P B. (s, u), (u, t) } We certainly have. In the other direction, suppose that s P C and t P B such that (s, t). It will suffice to show that s P C. Indeed, by the definition of P B, there is some v P such that (t, v). Therefore, (s, v), and so s P C, as required. Before defining the multiplication µ : weak weak weak in the monad, we look at the identities that this unit transformation η will have to satisfy. If B is an object of G/ then we want the diagrams B B µ (B) and B B B µ (B) B to commute. The arrow : B B in the first diagram is the weakening arrow obtained from the object B of G/ using the natural transformation η, while the arrow : B B in the second diagram is obtained by applying the functor weak to the morphism : B B. In practice, however, there is no need for this distinction: B is the same game as B and the arrows and are the identity. To see this, recall the definitions of B and : P B = {s B : s } = (as sets of plays) Then we should have P B = {s B : s }, but since is the same set as, this is exactly the same as the definition of P B. Since, are identity maps, we observe that we have the strict equality weak weak = weak and we had better define µ to be the identity transformation. This µ is automatically associative and the unit diagrams above hold (since all maps involved are identity maps). We end up with an idempotent monad on G/. 7
8 4 Lax weakening morphisms We now introduce a lax version of our notion of weakening morphism that is stable under composition with isomorphisms: Definition 4.1. Let, B be games. We say that a morphism : B is a lax weakening if it is the composition of a weakening morphism and an isomorphism; in other words, if there exists a weakening B of B and an isomorphism ˆ : B such that the diagram commutes. Proposition 4.2. i) If B is a weakening of B, then : B B is a lax weakening. ii) If : B is an isomorphism, then is a lax weakening. iii) If C B are morphisms such that and are lax weakenings, then : C is a lax weakening. Proof. (i) and (ii) are both obvious. For part (iii), note that we have the diagram ˆ ˆ It will suffice to find a weakening C of C and an isomorphism ˆ : C B making the following diagram commute: C ˆ ˆ ˆ ˆ 8
9 Therefore, it will suffice to prove the lemma which follows. Lemma 4.3 (Lifting of weakenings along isomorphisms). Let B, C be games and let : C B be an isomorphism. Let B be a weakening of B. Then there is a weakening C of C and an isomorphism : C B such that the following diagram commutes: We shall give two proofs of this lemma - one direct proof using the Hyland- Schalke functor and one proof using the idempotent monad structure described in the previous section. First proof. We know that : C B is an isomorphism; therefore, its image P C P B under the Hyland-Schalke functor is an isomorphism in Rel. It is well-known (and easy to show) that the isomorphisms in Rel are precisely those relations that are the graphs of bijective functions. Therefore, we have a bijection : P C P B with the property that i) For all s P C there exists a P -play s such that s C = s and s B = (s) ii) For all P -plays s, s B = (s C ). Let P C = 1 (P B ) P C. We claim that P C is prefix-closed. Let s P C and let t s. By the definition of P C, (s) P B. Since P B is prefix-closed, it will suffice to show that (t) (s). By (i), we know that there exists a P -play s such that s C = s and s B = (s). Since t s, there is some prefix t s such that t C = t. Then, by (ii), we must have (t) = t B s B. Therefore, C = (M C, λ C, P C ) is a well-formed game. We claim that it is a weakening of C. Indeed, suppose s P C is an O-play and that sc is a P -reply in P C. We need to show that sc P C. By the definition of P C, we know that (s) P B. By (i) we know that there exists some P -play s such that s C = s and s B = (s); since s is a P -play, we can deduce that s and (s) have the same sign, and so (s) is an O-play in P B. Examining (i) and (ii) above, we can see that 9
10 (sc) = (s)b for some b P B. Therefore, since B is a weakening of B, we have (sc) P B and so sc P C as desired. Lastly, we define to be given by the set P C B, and we claim that the diagram commutes. This is most easily checked by applying the faithful functor G F Rel Rel op where : Rel Rel op is the identity on objects and flips a relation R B to give a relation R B. If : is a weakening morphism in G then its image under this functor is the graph of the inclusion P P. Meanwhile, the image of under this functor is the graph of the function 1 : P B P C. By the definition of P C, the image of under this functor is the graph of the function 1 PB : P B P C. It is then easy to check that the resulting diagram commutes in Rel: P C P C 1 1 PB P B P B The last thing that we need to do is to check that is an isomorphism. If we apply exactly the same argument to C C and the isomorphism 1 : B C then we get a weakening B of B and an isomorphism ( 1 ) : B C making the following diagram commute: 1 ( 1 ) Now note that P B is defined to be (P C ). Since is a bijection and P C 10
11 was defined to be 1 (P B ), we get that B = B. We now get: B B ( 1 ) C B = B 1 C C B = B 1 C B B = B B Since is an epimorphism, it follows that ( 1 ) = id B. n identical argument shows that ( 1 ) = id C. So is an isomorphism. Second proof. We start off with the diagram C We use the monad on the category G/B. Let C be the pullback of B along the morphism : C B. This gives us the diagram ( ) It remains to show that ( ) is an isomorphism. To show this, let us rearrange the original diagram a bit to make the monadic structure explicit: C B When we apply the functor weak B B to this diagram, we get C B ( ) By definition, = η B : B B, and so = (η B ). Since our monad is idempotent, (η B ) is an isomorphism. is an isomorphism by functoriality and therefore ( ) is an isomorphism, as desired. B 11
where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset
Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More informationSECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES
SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there
More informationPART III.3. IND-COHERENT SHEAVES ON IND-INF-SCHEMES
PART III.3. IND-COHERENT SHEAVES ON IND-INF-SCHEMES Contents Introduction 1 1. Ind-coherent sheaves on ind-schemes 2 1.1. Basic properties 2 1.2. t-structure 3 1.3. Recovering IndCoh from ind-proper maps
More informationMorita Equivalence. Eamon Quinlan
Morita Equivalence Eamon Quinlan Given a (not necessarily commutative) ring, you can form its category of right modules. Take this category and replace the names of all the modules with dots. The resulting
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset
More information9. Birational Maps and Blowing Up
72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense
More informationConcentrated Schemes
Concentrated Schemes Daniel Murfet July 9, 2006 The original reference for quasi-compact and quasi-separated morphisms is EGA IV.1. Definition 1. A morphism of schemes f : X Y is quasi-compact if there
More informationPART II.1. IND-COHERENT SHEAVES ON SCHEMES
PART II.1. IND-COHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Ind-coherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. t-structure 3 2. The direct image functor 4 2.1. Direct image
More informationIndCoh Seminar: Ind-coherent sheaves I
IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of
More informationCOMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY
COMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY VIVEK SHENDE A ring is a set R with two binary operations, an addition + and a multiplication. Always there should be an identity 0 for addition, an
More informationLecture 9: Sheaves. February 11, 2018
Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with
More informationBoolean Algebras, Boolean Rings and Stone s Representation Theorem
Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationMODULI TOPOLOGY. 1. Grothendieck Topology
MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided
More informationThe tensor product of commutative monoids
The tensor product of commutative monoids We work throughout in the category Cmd of commutative monoids. In other words, all the monoids we meet are commutative, and consequently we say monoid in place
More information1 Introduction. 2 Categories. Mitchell Faulk June 22, 2014 Equivalence of Categories for Affine Varieties
Mitchell Faulk June 22, 2014 Equivalence of Categories for Affine Varieties 1 Introduction Recall from last time that every affine algebraic variety V A n determines a unique finitely generated, reduced
More informationCategories and Modules
Categories and odules Takahiro Kato arch 2, 205 BSTRCT odules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract
More informationAn Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees
An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees Francesc Rosselló 1, Gabriel Valiente 2 1 Department of Mathematics and Computer Science, Research Institute
More informationPOSTNIKOV EXTENSIONS OF RING SPECTRA
POSTNIKOV EXTENSIONS OF RING SPECTRA DANIEL DUGGER AND BROOKE SHIPLEY Abstract. We give a functorial construction of k-invariants for ring spectra, and use these to classify extensions in the Postnikov
More informationCategory Theory. Travis Dirle. December 12, 2017
Category Theory 2 Category Theory Travis Dirle December 12, 2017 2 Contents 1 Categories 1 2 Construction on Categories 7 3 Universals and Limits 11 4 Adjoints 23 5 Limits 31 6 Generators and Projectives
More informationDerived Algebraic Geometry IX: Closed Immersions
Derived Algebraic Geometry I: Closed Immersions November 5, 2011 Contents 1 Unramified Pregeometries and Closed Immersions 4 2 Resolutions of T-Structures 7 3 The Proof of Proposition 1.0.10 14 4 Closed
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)
More informationDirect Limits. Mathematics 683, Fall 2013
Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry
More informationLECTURE 5: v n -PERIODIC HOMOTOPY GROUPS
LECTURE 5: v n -PERIODIC HOMOTOPY GROUPS Throughout this lecture, we fix a prime number p, an integer n 0, and a finite space A of type (n + 1) which can be written as ΣB, for some other space B. We let
More informationMath 210B. Profinite group cohomology
Math 210B. Profinite group cohomology 1. Motivation Let {Γ i } be an inverse system of finite groups with surjective transition maps, and define Γ = Γ i equipped with its inverse it topology (i.e., the
More informationUNIVERSAL DERIVED EQUIVALENCES OF POSETS
UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for
More informationTHE HEART OF A COMBINATORIAL MODEL CATEGORY
Theory and Applications of Categories, Vol. 31, No. 2, 2016, pp. 31 62. THE HEART OF A COMBINATORIAL MODEL CATEGORY ZHEN LIN LOW Abstract. We show that every small model category that satisfies certain
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationFROM COHERENT TO FINITENESS SPACES
FROM COHERENT TO FINITENESS SPACES PIERRE HYVERNAT Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France. e-mail address: Pierre.Hyvernat@univ-savoie.fr Abstract. This
More information14 Lecture 14: Basic generallities on adic spaces
14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions
More informationDEFINITIONS: OPERADS, ALGEBRAS AND MODULES. Let S be a symmetric monoidal category with product and unit object κ.
DEFINITIONS: OPERADS, ALGEBRAS AND MODULES J. P. MAY Let S be a symmetric monoidal category with product and unit object κ. Definition 1. An operad C in S consists of objects C (j), j 0, a unit map η :
More informationLOCALIZATIONS, COLOCALIZATIONS AND NON ADDITIVE -OBJECTS
LOCALIZATIONS, COLOCALIZATIONS AND NON ADDITIVE -OBJECTS GEORGE CIPRIAN MODOI Abstract. Given two arbitrary categories, a pair of adjoint functors between them induces three pairs of full subcategories,
More informationPART I. Abstract algebraic categories
PART I Abstract algebraic categories It should be observed first that the whole concept of category is essentially an auxiliary one; our basic concepts are those of a functor and a natural transformation.
More informationMath 210B. The bar resolution
Math 210B. The bar resolution 1. Motivation Let G be a group. In class we saw that the functorial identification of M G with Hom Z[G] (Z, M) for G-modules M (where Z is viewed as a G-module with trivial
More informationMODEL STRUCTURES ON PRO-CATEGORIES
Homology, Homotopy and Applications, vol. 9(1), 2007, pp.367 398 MODEL STRUCTURES ON PRO-CATEGORIES HALVARD FAUSK and DANIEL C. ISAKSEN (communicated by J. Daniel Christensen) Abstract We introduce a notion
More informationApplications of 2-categorical algebra to the theory of operads. Mark Weber
Applications of 2-categorical algebra to the theory of operads Mark Weber With new, more combinatorially intricate notions of operad arising recently in the algebraic approaches to higher dimensional algebra,
More informationWhat are stacks and why should you care?
What are stacks and why should you care? Milan Lopuhaä October 12, 2017 Todays goal is twofold: I want to tell you why you would want to study stacks in the first place, and I want to define what a stack
More informationSection Higher Direct Images of Sheaves
Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will
More informationDerivations and differentials
Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,
More informationPostnikov extensions of ring spectra
1785 1829 1785 arxiv version: fonts, pagination and layout may vary from AGT published version Postnikov extensions of ring spectra DANIEL DUGGER BROOKE SHIPLEY We give a functorial construction of k invariants
More informationCLASSIFYING TANGENT STRUCTURES USING WEIL ALGEBRAS
Theory and Applications of Categories, Vol. 32, No. 9, 2017, pp. 286 337. CLASSIFYING TANGENT STRUCTURES USING WEIL ALGEBRAS POON LEUNG Abstract. At the heart of differential geometry is the construction
More informationCATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.
CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists
More informationTheories With Duality DRAFT VERSION ONLY
Theories With Duality DRAFT VERSION ONLY John C. Baez Department of athematics, University of California Riverside, CA 9252 USA Paul-André elliès Laboratoire PPS Université Paris 7 - Denis Diderot Case
More informationReview of category theory
Review of category theory Proseminar on stable homotopy theory, University of Pittsburgh Friday 17 th January 2014 Friday 24 th January 2014 Clive Newstead Abstract This talk will be a review of the fundamentals
More informationAzumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras
Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part
More information1. Introduction. Let C be a Waldhausen category (the precise definition
K-THEORY OF WLDHUSEN CTEGORY S SYMMETRIC SPECTRUM MITY BOYRCHENKO bstract. If C is a Waldhausen category (i.e., a category with cofibrations and weak equivalences ), it is known that one can define its
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday
More information1 Categorical Background
1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,
More informationCellularity, composition, and morphisms of algebraic weak factorization systems
Cellularity, composition, and morphisms of algebraic weak factorization systems Emily Riehl University of Chicago http://www.math.uchicago.edu/~eriehl 19 July, 2011 International Category Theory Conference
More informationAlgebraic Geometry
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationRepresentations of quivers
Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice
More informationRecall: a mapping f : A B C (where A, B, C are R-modules) is called R-bilinear if f is R-linear in each coordinate, i.e.,
23 Hom and We will do homological algebra over a fixed commutative ring R. There are several good reasons to take a commutative ring: Left R-modules are the same as right R-modules. [In general a right
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationINTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES
INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES 1. Why correspondences? This part introduces one of the two main innovations in this book the (, 2)-category of correspondences as a way to encode
More informationThe Bianchi Identity in Path Space
The Bianchi Identity in Path Space Matt Noonan January 15, 2007 this is a test. Abstract 1 Two Geometric Problems Throughout this paper, we will be interested in local problems; therefore, we will take
More informationGorenstein algebras and algebras with dominant dimension at least 2.
Gorenstein algebras and algebras with dominant dimension at least 2. M. Auslander Ø. Solberg Department of Mathematics Brandeis University Waltham, Mass. 02254 9110 USA Institutt for matematikk og statistikk
More informationLecture 9 - Faithfully Flat Descent
Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,
More informationTCC Homological Algebra: Assignment #3 (Solutions)
TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationEquivalence of the Combinatorial Definition (Lecture 11)
Equivalence of the Combinatorial Definition (Lecture 11) September 26, 2014 Our goal in this lecture is to complete the proof of our first main theorem by proving the following: Theorem 1. The map of simplicial
More informationp,q H (X), H (Y ) ), where the index p has the same meaning as the
There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore
More informationTHE H-PRINCIPLE, LECTURE 14: HAEFLIGER S THEOREM CLASSIFYING FOLIATIONS ON OPEN MANIFOLDS
THE H-PRINCIPLE, LECTURE 14: HAELIGER S THEOREM CLASSIYING OLIATIONS ON OPEN MANIOLDS J. RANCIS, NOTES BY M. HOYOIS In this lecture we prove the following theorem: Theorem 0.1 (Haefliger). If M is an open
More informationin path component sheaves, and the diagrams
Cocycle categories Cocycles J.F. Jardine I will be using the injective model structure on the category s Pre(C) of simplicial presheaves on a small Grothendieck site C. You can think in terms of simplicial
More informationDuality in Probabilistic Automata
Duality in Probabilistic Automata Chris Hundt Prakash Panangaden Joelle Pineau Doina Precup Gavin Seal McGill University MFPS May 2006 Genoa p.1/40 Overview We have discovered an - apparently - new kind
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationLECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES
LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain
More informationINTRO TO TENSOR PRODUCTS MATH 250B
INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two A-modules. For a third A-module Z, consider the
More informationOMEGA-CATEGORIES AND CHAIN COMPLEXES. 1. Introduction. Homology, Homotopy and Applications, vol.6(1), 2004, pp RICHARD STEINER
Homology, Homotopy and Applications, vol.6(1), 2004, pp.175 200 OMEGA-CATEGORIES AND CHAIN COMPLEXES RICHARD STEINER (communicated by Ronald Brown) Abstract There are several ways to construct omega-categories
More informationIn the special case where Y = BP is the classifying space of a finite p-group, we say that f is a p-subgroup inclusion.
Definition 1 A map f : Y X between connected spaces is called a homotopy monomorphism at p if its homotopy fibre F is BZ/p-local for every choice of basepoint. In the special case where Y = BP is the classifying
More informationNotes on the definitions of group cohomology and homology.
Notes on the definitions of group cohomology and homology. Kevin Buzzard February 9, 2012 VERY sloppy notes on homology and cohomology. Needs work in several places. Last updated 3/12/07. 1 Derived functors.
More informationAn introduction to Yoneda structures
An introduction to Yoneda structures Paul-André Melliès CNRS, Université Paris Denis Diderot Groupe de travail Catégories supérieures, polygraphes et homotopie Paris 21 May 2010 1 Bibliography Ross Street
More informationTHE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS
THE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS DANIEL LITT Let us fix the following notation: 1. Notation and Introduction K is a number field; L is a CM field with totally real subfield L + ; (A,
More informationCategory Theory. Categories. Definition.
Category Theory Category theory is a general mathematical theory of structures, systems of structures and relationships between systems of structures. It provides a unifying and economic mathematical modeling
More informationLecture 2: Syntax. January 24, 2018
Lecture 2: Syntax January 24, 2018 We now review the basic definitions of first-order logic in more detail. Recall that a language consists of a collection of symbols {P i }, each of which has some specified
More informationRepresentable presheaves
Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom
More informationLOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT
LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated
More informationHomology and Cohomology of Stacks (Lecture 7)
Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic
More informationSome remarks on Frobenius and Lefschetz in étale cohomology
Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)
More informationDerived Algebraic Geometry III: Commutative Algebra
Derived Algebraic Geometry III: Commutative Algebra May 1, 2009 Contents 1 -Operads 4 1.1 Basic Definitions........................................... 5 1.2 Fibrations of -Operads.......................................
More informationPEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION. The Peano axioms
PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION The Peano axioms The following are the axioms for the natural numbers N. You might think of N as the set of integers {0, 1, 2,...}, but it turns
More information0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
More informationBEN KNUDSEN. Conf k (f) Conf k (Y )
CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X Y is an injective
More informationSubpullbacks and Po-flatness Properties of S-posets
Journal of Sciences, Islamic Republic of Iran 25(4): 369-377 (2014) University of Tehran, ISSN 1016-1104 http://jsciences.ut.ac.ir Subpullbacks and Po-flatness Properties of S-posets A. Golchin * and L.
More informationLECTURE 6: THE BOUSFIELD-KUHN FUNCTOR
LECTURE 6: THE BOUSFIELD-KUHN FUNCTOR Let V be a finite space of type n, equipped with a v n -self map v Σ t V V. In the previous lecture, we defined the spectrum Φ v (X), where X is a pointed space. By
More informationTopological Groupoids and Exponentiability
Topological Groupoids and Exponentiability Susan Niefield (joint with Dorette Pronk) July 2017 Overview Goal: Study exponentiability in categories of topological groupoid. Starting Point: Consider exponentiability
More informationCategory theory and set theory: algebraic set theory as an example of their interaction
Category theory and set theory: algebraic set theory as an example of their interaction Brice Halimi May 30, 2014 My talk will be devoted to an example of positive interaction between (ZFC-style) set theory
More informationh M (T ). The natural isomorphism η : M h M determines an element U = η 1
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli
More informationHomework 3: Relative homology and excision
Homework 3: Relative homology and excision 0. Pre-requisites. The main theorem you ll have to assume is the excision theorem, but only for Problem 6. Recall what this says: Let A V X, where the interior
More informationCongruence Boolean Lifting Property
Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;
More informationMath Homotopy Theory Hurewicz theorem
Math 527 - Homotopy Theory Hurewicz theorem Martin Frankland March 25, 2013 1 Background material Proposition 1.1. For all n 1, we have π n (S n ) = Z, generated by the class of the identity map id: S
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More informationMonads Need Not Be Endofunctors
Monads Need Not Be Endofunctors Thorsten Altenkirch, University of Nottingham James Chapman, Institute of Cybernetics, Tallinn Tarmo Uustalu, Institute of Cybernetics, Tallinn ScotCats, Edinburgh, 21 May
More informationThus we get. ρj. Nρj i = δ D(i),j.
1.51. The distinguished invertible object. Let C be a finite tensor category with classes of simple objects labeled by a set I. Since duals to projective objects are projective, we can define a map D :
More informationSimulations in Coalgebra
Simulations in Coalgebra Jesse Hughes Dept. Philosophy, Technical Univ. Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. J.Hughes@tm.tue.nl Bart Jacobs Dept. Computer Science, Univ. Nijmegen,
More informationA Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries
A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries Johannes Marti and Riccardo Pinosio Draft from April 5, 2018 Abstract In this paper we present a duality between nonmonotonic
More informationAnother way to proceed is to prove that the function field is purely transcendental. Now the coordinate ring is
3. Rational Varieties Definition 3.1. A rational function is a rational map to A 1. The set of all rational functions, denoted K(X), is called the function field. Lemma 3.2. Let X be an irreducible variety.
More informationAn introduction to locally finitely presentable categories
An introduction to locally finitely presentable categories MARU SARAZOLA A document born out of my attempt to understand the notion of locally finitely presentable category, and my annoyance at constantly
More informationALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.
ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add
More information