DOCSLIB.ORG

III a Functional Approach to General Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
III a Functional Approach to General Topology III A Functional Approach to General Topology Maria Manuel Clementino, Eraldo Giuli and Walter Tholen In this chapter we wish to present a categorical approach to fundamental con- cepts of General Topology, by providing a category with an additional structure which allows us to display more directly the geometricX properties of the objects of regarded as spaces. Hence, we study topological properties for them, such as HausdorffX separation, compactness and local compactness, and we describe im- portant topological constructions, such as the compact-open topology for function spaces and the Stone-Cechˇ compactification. Of course, in a categorical setting, spaces are not investigated \directly" in terms of their points and neighbourhoods, as in the traditional set-theoretic setting; rather, one exploits the fact that the re- lations of points and parts inside a space become categorically special cases of the relation of the space to other objects in its category. It turns out that many stability properties and constructions are established more economically in the categorical rather than the set-theoretic setting, leave alone the much greater level of generality and applicability. The idea of providing a category with some kind of topological structure is cer- tainly not new. So-called Grothendieck topologies (see Chapter VII) and, more gen- erally, Lawvere-Tierney topologies are fundamental for the geometrically-inspired construction of topoi. Specifically, these structures provide a notion of closure and thereby a notion of closed subobject, for every object in the category, such that all morphisms become \continuous". The notion of Dikranjan-Giuli closure oper- ator [17] axiomatizes this idea and can be used to study topological properties categorically (see, for example, [9, 12]). Here we go one step further and follow the approach first outlined in [48]. Hence, we provide the given category with a factorization structure and a special class of morphisms of which we think as of the closed morphisms, satisfying threeF basic axioms; however, there is no a-priori provision of \closure" of subob- jects. Depending on the parameter , we introduce and study basic topological properties as mentioned previously, butF encounter also more advanced topics, such as exponentiability. The following features distinguish our presentation from the treatment of the same topological themes in existing topology books: 1. We emphasize the object-morphism interplay: every object notion corresponds to a morphism notion, and vice versa. For example, compact spaces \are" proper 104 III. A Functional Approach to General Topology maps, and conversely. Consequently, every theorem on compact spaces \is" a the- orem on proper maps, and conversely. locally open separated closed dense ( , 2.1) ( +, 7.1) (2.2) (10.6) F F @ @ initial separated proper final surjective (6.5) ( 0, 4.2) ( ∗, 3.1) (8.6) ( , 1.2) F F E @ @@ @ @ @ @ @ @ @ @ @ @ absolutely Tychonoff exponentiable quotient closed (6.1) ( exp, 10.1) (9.1) (6.1) F locally local perfect homeomorphism (8.1) (7.4) perfect biquotient @ ( ∗ 0, 5.2) (10.3) F \F @ @ @ @ @ @ @@ @@ closed open proper stable open stable embedding embedding surjection surjection + + ( 0 = , 2.1) ( , 7.1) ( ∗ ∗, 10.3) ( ∗, 10.3) F F\M F \M F \E F \E Table 1 2. We reach a wide array of applications not only by considering various cat- egories, such as the categories of topological spaces, of locales (see Chapter II), etc., but also by varying the notion of closed morphism within the same category. 105 For example, in our setting, compact spaces behave just like discrete spaces, and perfect maps just like local homeomorphisms, because these notions arise from a common generalization. 3. Our categorical treatment is entirely constructive. In particular, we avoid the use of the Axiom of Choice (also when we interpret the categorical theory in the category of topological spaces), and we restrict ourselves to finitary properties (with the exception of some general remarks on the Tychonoff Theorem and the existence of the Stone-Cechˇ compactification at the end of the chapter). We expect the Reader to be familiar with basic categorical notions, such as adjoint functor and limit, specifically pullback, and we also assume familiarity with basic topological notions, such as topological space, neighbourhood, closure of subsets, continuous map. The Reader will find throughout the chapter mostly easy exercises; those marked with are deemed to be more demanding. ∗ locally Hausdorff (10.6) -Haus -Comp F(4.3) F (3.3) @ @ @ @ @ -AC -Tych exponentiable F(6.1) F(6.1) (10.4) -LCompHaus F (8.1) -CompHaus F (5.3) Table 2 106 III. A Functional Approach to General Topology Our basic hypotheses on the category we are working with are formulated as axioms (F0)-(F2) in 1.2 (axioms for factorization systems), (F3)-(F5) in 2.1 (axioms for closed maps), (F6)-(F8) in 11.1 (axioms giving closures), (F9) in 11.2 (infinite product axiom). The morphism notions discussed in this chapter are summarized in Table 1, which also indicates the relevant subsections and the abbreviations used. Upward- directed lines indicate implications; some of these may need extra (technical) hy- potheses. Table 2 replicates part of Table 1 at the object level. Many of these notions get discussed throughout the chapter in standard exam- ples, as introduced in Section 2. We list some of them here in summary form, for the Reader's convenience: op: topological spaces, = closed maps (2.3), T F opopen: topological spaces, = open maps (2.8), T F opclopen: topological spaces, = maps preserving clopen sets (2.6), T F opZariski: topological spaces, = Zariski-closed sets (2.7), T oc: locales, = closedF maps (2.4), LbGrp: abelian groups,F = homomorphisms whose restriction to the A torsion subgroupsF is surjective (2.9), any topos with a universal closure operator (2.5), any finitely-complete category with a Dikranjan-Giuli closure operator (2.5), any lextensive category with summand-preserving morhisms (2.6), any comma-category of any of the preceding categories (2.10). The authors are grateful for valuable comments received from many colleagues and the anonymous referees which helped us getting this chapter into its current form. We particularly thank Jorge Picado, AleˇsPultr and Peter Johnstone for their interest in resolving the problem of characterizing open maps of locales in terms of closure (see Section 7.3), which led to the comprehensive solution given in [31]. 1. Subobjects, images, preimages 1.1. Motivation. A mapping f : X Y of sets may be decomposed as ! Z (1) e > @ m }} @@ }} @ X / Y f with a surjection e followed by an injection m, simply by taking Z = f[X], the image of f; e maps like f, and m is an inclusion map. Hence, f = m e with an epimorphism e and a monomorphism m of the category et of sets. If· f is a S 1. Subobjects, images, preimages 107 homomorphism of groups X, Y , then Z is a subgroup of Y , and e and m become epi- and monomorphisms in the category rp of groups, respectively. If f is a continuous mapping of topological spaces XG, Y , then Z may be endowed with the subspace topology inherited from Y , and e and m are now epi- and monomorphisms in the category op of topological spaces, respectively. However, the situation in op is rather differentT from that in et and rp: applying the decomposition (1) toT a monomorphism f, in et and Srp we obtainG an isomorphism e, but not so in op, simply because theS et-inverseG of the continuous bijective mapping e may failT to be continuous. S In what follows we therefore consider a (potentially quite special) class of monomorphisms and, symmetrically, a (potentially quite special) class Mof epimorphisms in a category and assume the existence of ( ; )-decompositionsE for all morphisms, as follows.X E M 1.2. Axioms for factorization systems. Throughout this chapter we work in a finitely-complete category with two distinguished classes of morphisms and such that X E M (F0) is a class of monomorphisms and is a class of epimorphisms in , andM both are closed under compositionE with isomorphisms; X (F1) every morphism f decomposes as f = m e with m , e ; (F2) every e is orthogonal to every m · (written2 M as e 2m E), that is: given any2 morphismsE u; v with m u =2 Mv e, then there? is a uniquely determined morphism w making the· following· diagram commutative: u / (2) · Ñ@ · e Ñ m Ñ w Ñ / · v · Any pair ( ; ) satisfying conditions (F0)-(F2) is referred to as a proper (orthog- onal) factorizationE M system of ; \proper" gets dropped if one allows and to be arbitrary morphism classes,X i.e., if one drops the \mono" and \epi"M conditionE in (F0). In that case the unicity requirement for w in (F2) becomes essential; however it is redundant in our situation. Exercises. 1. Show that every category has two trivial but generally non-proper factorization systems: (Iso; All), (All; Iso).X 2. Show that (Epi; Mono) is a proper factorization system in et and rp, but not in op. S G 3. FindT a class such that (Epi; ) is a proper factorization system in op. 4. Show that f M: X Y is an epimorphismM in the full subcategory aus Tof Hausdorff spaces in op if and! only if f is dense (i.e., its image meets everyH non-empty open set in Y ).T Find a class such that (Epi; ) is a proper factorization system in aus. Compare this withM op. M H T 108 III. A Functional Approach to General Topology (Here Iso, Epi, Mono denotes the class of all iso-, epi-, monomorphisms in the respective category, and All the class of all morphisms.) 1.3.
Load more

Recommended publications
  • Or Dense in Itself)Ifx Has No Isolated Points
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 11, Pages 3607{3616 S 0002-9939(03)06660-7 Article electronically published on February 24, 2003 TYCHONOFF EXPANSIONS BY INDEPENDENT FAMILIES WANJUN HU (Communicated by Alan Dow) Abstract. A method for Tychonoff expansions using independent families is introduced. Using this method we prove that every countable Tychonoff space which admits a partition into infinitely many open-hereditarily irre- solvable dense subspaces has a Tychonoff expansion that is !-resolvable but not strongly extraresolvable. We also show that, under Luzin's Hypothesis ! ! (2 1 =2 ), there exists an !-resolvable Tychonoff space of size !1 which is not maximally resolvable. Introduction A topological space X is called crowded (or dense in itself)ifX has no isolated points. E. Hewitt defined a resolvable space as a space which has two disjoint dense subsets (see [17]). A space is called κ-resolvable [3] if it admits κ-many pairwise disjoint dense sets. The cardinal ∆(X):=minfjUj : U is a non-empty open subset of Xg is called the dispersion character of the space X.ThespaceX is called maximally resolvable [2] if it is ∆(X)-resolvable; if every crowded subspace of X is not resolvable, then X is called hereditarily irresolvable (HI). Analogously, X is open-hereditarily irresolvable (or OHI) if every non-empty open subset of X is irresolvable. V.I. Malykhin introduced a similar concept of extraresolvable (see [20] + and [6]): a space is extraresolvable if there exists a family fDα : α<∆(X) g of dense susbets of X such that Dα \ Dβ is nowhere dense in X for distinct α and β.
    [Show full text]
  • Chapter 7 Separation Properties
    Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and .
    [Show full text]
  • On Α-Normal and Β-Normal Spaces
    Comment.Math.Univ.Carolin. 42,3 (2001)507–519 507 On α-normal and β-normal spaces A.V. Arhangel’skii, L. Ludwig Abstract. We define two natural normality type properties, α-normality and β-normality, and compare these notions to normality. A natural weakening of Jones Lemma immedi- ately leads to generalizations of some important results on normal spaces. We observe that every β-normal, pseudocompact space is countably compact, and show that if X is a dense subspace of a product of metrizable spaces, then X is normal if and only if X is β-normal. All hereditarily separable spaces are α-normal. A space is normal if and only if it is κ-normal and β-normal. Central results of the paper are contained in Sections 3 and 4. Several examples are given, including an example (identified by R.Z. Buzyakova) of an α-normal, κ-normal, and not β-normal space, which is, in fact, a pseudocompact topological group. We observe that under CH there exists a locally compact Hausdorff hereditarily α-normal non-normal space (Theorem 3.3). This example is related to the main result of Section 4, which is a version of the famous Katˇetov’s theorem on metrizability of a compactum the third power of which is hereditarily normal (Corollary 4.3). We also present a Tychonoff space X such that no dense subspace of X is α-normal (Section 3). Keywords: normal, α-normal, β-normal, κ-normal, weakly normal, extremally discon- nected, Cp(X), Lindel¨of, compact, pseudocompact, countably compact, hereditarily sep- arable, hereditarily α-normal, property wD, weakly perfect, first countable Classification: 54D15, 54D65, 54G20 §0.
    [Show full text]
  • On the Covering Dimension of Subspaces
    proceedings of the american mathematical society Volume 72, Number 3, December 1978 ON THE COVERINGDIMENSION OF SUBSPACES OF PRODUCT OF SORGENFREY LINES1 AU A. FORA Abstract. Let 5 denote the Sorgenfrey line. Then the following results are proved in this paper: (i) If X is a nonempty subspace of 5*°, then dim X = 0. (ii) For any nonempty separable space X c 5"°, dimA""1 = 0 for any cardinal m. 1. Introduction. The question of whether dimiA' X Y) < dim X + dim Y for topological spaces X and Y has long been considered (see e.g., [G, pp. 263 and 277]). By dim X, or the covering dimension of X, we mean the least integer, tj, such that each finite cozero cover of X has a finite cozero refinement of order n. (A cover is of order n if and only if each point of the space is contained in at most n + 1 elements of the cover. All spaces considered are completely regular.) Researchers have long worked on the above problem, and only recently Wage [W] and Przymusinski [P] constructed a Lindelöf space X such that dim X = 0 and X2 is normal but has dim X2 > 0. The aim of this paper is to prove that no product of subspaces of Sorgenfrey lines can serve as a counterexample to the product conjecture. Another aim is to give a full answer to one of the questions raised by Mrowka [Mr2J in the conference of 1972 which says: "We still do not know if subspaces of S" (n = 2, 3, .
    [Show full text]
  • Applications of the Stone-Cech Compactification to Free Topological Groups
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 55, Number 1, February 1976 APPLICATIONS OF THE STONE-CECH COMPACTIFICATION TO FREE TOPOLOGICAL GROUPS J. P. L. HARDY, SIDNEY A. MORRIS1 AND H. B. THOMPSON Abstract. In this note the Stone-Cech compactification is used to produce short proofs of two theorems on the structure of free topological groups. The first is: The free topological group on any Tychonoff space X contains, as a closed subspace, a homeomorphic copy of the product space X". This is a generalization of a result of B. V. S. Thomas. The second theorem proved is C. Joiner's, Fundamental Lemma. 1. Introduction. Definition. Let A be any topological space. Then the compact Hausdorff space BX is said to be the Stone-Cech compactification of X if there exists a continuous map B: X —»BX such that for any continuous map of A into any compact Hausdorff space ATthere exists a unique continuous map $: BX -» K such that <&B= <j>. While BX exists and is unique for any topological space A, it is of particular interest when A is a Tychonoff (= completely regular Hausdorff) space, for then B is an embedding of A in BX and we can consider A to be a subspace of BX. (For details, see Kelley [5].) Definition. Let A be any Tychonoff space. Then the Hausdorff topological group F(A) is said to be the free topological group on A if A is a subspace of F(A") and for any continuous map <bof A into any topological group G there exists a unique continuous homomorphism 4>: F(A) —*G such that $|A = <f>.
    [Show full text]
  • On Λ-Perfect Maps
    Turkish Journal of Mathematics Turk J Math (2017) 41: 1087 { 1091 http://journals.tubitak.gov.tr/math/ ⃝c TUB¨ ITAK_ Research Article doi:10.3906/mat-1512-86 On λ-perfect maps Mehrdad NAMDARI∗, Mohammad Ali SIAVOSHI Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran Received: 22.12.2015 • Accepted/Published Online: 20.10.2016 • Final Version: 25.07.2017 Abstract: λ-Perfect maps, a generalization of perfect maps (i.e. continuous closed maps with compact fibers) are presented. Using Pλ -spaces and the concept of λ-compactness some classical results regarding λ-perfect maps will be extended. In particular, we show that if the composition fg is a λ-perfect map where f; g are continuous maps with fg well-defined, then f; g are α-perfect and β -perfect, respectively, on appropriate spaces, where α, β ≤ λ. Key words: λ-Compact, λ-perfect, Pλ -space, Lindel¨ofnumber 1. Introduction A perfect map is a kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps. Let X; Y be two topological spaces such that X is Hausdorff. A continuous map f : X ! Y is said to be a perfect map provided that f is closed and surjective and each fiber f −1(y) is compact in X . In [3], the authors study a generalization of compactness, i.e. λ-compactness. Motivated by this concept we are led to generalize the concept of perfect mapping.
    [Show full text]
  • De Rham Cohomology of Smooth Manifolds
    VU University, Amsterdam Bachelorthesis De Rham Cohomology of smooth manifolds Supervisor: Author: Prof. Dr. R.C.A.M. Patrick Hafkenscheid Vandervorst Contents 1 Introduction 3 2 Smooth manifolds 4 2.1 Formal definition of a smooth manifold . 4 2.2 Smooth maps between manifolds . 6 3 Tangent spaces 7 3.1 Paths and tangent spaces . 7 3.2 Working towards a categorical approach . 8 3.3 Tangent bundles . 9 4 Cotangent bundle and differential forms 12 4.1 Cotangent spaces . 12 4.2 Cotangent bundle . 13 4.3 Smooth vector fields and smooth sections . 13 5 Tensor products and differential k-forms 15 5.1 Tensors . 15 5.2 Symmetric and alternating tensors . 16 5.3 Some algebra on Λr(V ) ....................... 16 5.4 Tensor bundles . 17 6 Differential forms 19 6.1 Contractions and exterior derivatives . 19 6.2 Integrating over topforms . 21 7 Cochains and cohomologies 23 7.1 Chains and cochains . 23 7.2 Cochains . 24 7.3 A few useful lemmas . 25 8 The de Rham cohomology 29 8.1 The definition . 29 8.2 Homotopy Invariance . 30 8.3 The Mayer-Vietoris sequence . 32 9 Some computations of de Rham cohomology 35 10 The de Rham Theorem 40 10.1 Singular Homology . 40 10.2 Singular cohomology . 41 10.3 Smooth simplices . 41 10.4 De Rham homomorphism . 42 10.5 de Rham theorem . 44 1 11 Compactly supported cohomology and Poincar´eduality 47 11.1 Compactly supported de Rham cohomology . 47 11.2 Mayer-Vietoris sequence for compactly supported de Rham co- homology . 49 11.3 Poincar´eduality .
    [Show full text]
  • Midterm Exam
    Prof. Alexandru Suciu MATH 4565 TOPOLOGY Spring 2018 MIDTERM EXAM 1. Consider the following two topologies on R2: (i) The Zariski topology, T1, where a base of open sets is formed by the comple- ments of zero sets of polynomials in two variables. (ii) The topology T2, the weakest topology where all straight lines are closed sets. 2 2 Show that (R ; T1) and (R ; T2) are not homeomorphic. 2. Define a topology T on the closed interval X = [−1; 1] by declaring a set to be open if it either does not contain the point 0, or it does contain (−1; 1). (a) Verify that T is indeed a topology. (b) What are the limit points of X? (c) Is X a T0 space? (I.e., given a pair of distinct points in X, does at least one of them have a neighborhood not containing the other?) (d) Is X a T1 space? (I.e., given a pair of distinct points in X, does each one of them have a neighborhood not containing the other?) (e) Show that X is compact. (f) Show that X locally path-connected. 3. Prove or disprove the following: (a) If X and Y are path-connected, then X × Y is path-connected. (b) If A ⊂ X is path-connected, then A is path-connected. (c) If X is locally path-connected, and A ⊂ X, then A is locally path-connected. (d) If X is path-connected, and f : X ! Y is continuous, then f(X) is path- connected. (e) If X is locally path-connected, and f : X ! Y is continuous, then f(X) is locally path-connected.
    [Show full text]
  • Category Theory Course
    Category Theory Course John Baez September 3, 2019 1 Contents 1 Category Theory: 4 1.1 Definition of a Category....................... 5 1.1.1 Categories of mathematical objects............. 5 1.1.2 Categories as mathematical objects............ 6 1.2 Doing Mathematics inside a Category............... 10 1.3 Limits and Colimits.......................... 11 1.3.1 Products............................ 11 1.3.2 Coproducts.......................... 14 1.4 General Limits and Colimits..................... 15 2 Equalizers, Coequalizers, Pullbacks, and Pushouts (Week 3) 16 2.1 Equalizers............................... 16 2.2 Coequalizers.............................. 18 2.3 Pullbacks................................ 19 2.4 Pullbacks and Pushouts....................... 20 2.5 Limits for all finite diagrams.................... 21 3 Week 4 22 3.1 Mathematics Between Categories.................. 22 3.2 Natural Transformations....................... 25 4 Maps Between Categories 28 4.1 Natural Transformations....................... 28 4.1.1 Examples of natural transformations........... 28 4.2 Equivalence of Categories...................... 28 4.3 Adjunctions.............................. 29 4.3.1 What are adjunctions?.................... 29 4.3.2 Examples of Adjunctions.................. 30 4.3.3 Diagonal Functor....................... 31 5 Diagrams in a Category as Functors 33 5.1 Units and Counits of Adjunctions................. 39 6 Cartesian Closed Categories 40 6.1 Evaluation and Coevaluation in Cartesian Closed Categories. 41 6.1.1 Internalizing Composition................. 42 6.2 Elements................................ 43 7 Week 9 43 7.1 Subobjects............................... 46 8 Symmetric Monoidal Categories 50 8.1 Guest lecture by Christina Osborne................ 50 8.1.1 What is a Monoidal Category?............... 50 8.1.2 Going back to the definition of a symmetric monoidal category.............................. 53 2 9 Week 10 54 9.1 The subobject classifier in Graph.................
    [Show full text]
  • Notes on Differential Forms Lorenzo Sadun
    Notes on Differential Forms Lorenzo Sadun Department of Mathematics, The University of Texas at Austin, Austin, TX 78712 arXiv:1604.07862v1 [math.AT] 26 Apr 2016 CHAPTER 1 Forms on Rn This is a series of lecture notes, with embedded problems, aimed at students studying differential topology. Many revered texts, such as Spivak’s Calculus on Manifolds and Guillemin and Pollack’s Differential Topology introduce forms by first working through properties of alternating ten- sors. Unfortunately, many students get bogged down with the whole notion of tensors and never get to the punch lines: Stokes’ Theorem, de Rham cohomology, Poincare duality, and the realization of various topological invariants (e.g. the degree of a map) via forms, none of which actually require tensors to make sense! In these notes, we’ll follow a different approach, following the philosophy of Amy’s Ice Cream: Life is uncertain. Eat dessert first. We’re first going to define forms on Rn via unmotivated formulas, develop some profi- ciency with calculation, show that forms behave nicely under changes of coordinates, and prove Stokes’ Theorem. This approach has the disadvantage that it doesn’t develop deep intuition, but the strong advantage that the key properties of forms emerge quickly and cleanly. Only then, in Chapter 3, will we go back and show that tensors with certain (anti)symmetry properties have the exact same properties as the formal objects that we studied in Chapters 1 and 2. This allows us to re-interpret all of our old results from a more modern perspective, and move onwards to using forms to do topology.
    [Show full text]
  • Arxiv:1807.06986V1 [Math.CT] 18 Jul 2018 1.4
    A NAIVE DIAGRAM-CHASING APPROACH TO FORMALISATION OF TAME TOPOLOGY by notes by misha gavrilovich and konstantin pimenov in memoriam: evgenii shurygin ——————————————————————————————————— ... due to internal constraints on possible architectures of unknown to us functional "mental structures". Misha Gromov. Structures, Learning and Ergosystems: Chapters 1-4, 6. Abstract.— We rewrite classical topological definitions using the category- theoretic notation of arrows and are thereby led to their concise reformulations in terms of simplicial categories and orthogonality of morphisms, which we hope might be of use in the formalisation of topology and in developing the tame topology of Grothendieck. Namely, we observe that topological and uniform spaces are simplicial objects in the same category, a category of filters, and that a number of elementary properties can be obtained by repeatedly passing to the left or right orthogonal (in the sense of Quillen model categories) starting from a simple class of morphisms, often a single typical (counter)example appearing implicitly in the definition. Examples include the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms, and, outside of topology, finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective and surjective (homo)morphisms. Contents 1. Introduction. 3 1.1. Main ideas. 3 1.2. Contents. 4 1.3. Speculations. 6 arXiv:1807.06986v1 [math.CT] 18 Jul 2018 1.4. Surjection: an example . 6 1.5. Intuition/Yoga of orthogonality. 7 1.6. Intuition/Yoga of transcription. 8 A draft of a research proposal. Comments welcome at miishapp@sddf:org.
    [Show full text]
  • Chapter 1 I. Fibre Bundles
    Chapter 1 I. Fibre Bundles 1.1 Definitions Definition 1.1.1 Let X be a topological space and let U be an open cover of X.A { j}j∈J partition of unity relative to the cover Uj j∈J consists of a set of functions fj : X [0, 1] such that: { } → 1) f −1 (0, 1] U for all j J; j ⊂ j ∈ 2) f −1 (0, 1] is locally finite; { j }j∈J 3) f (x)=1 for all x X. j∈J j ∈ A Pnumerable cover of a topological space X is one which possesses a partition of unity. Theorem 1.1.2 Let X be Hausdorff. Then X is paracompact iff for every open cover of X there exists a partition of unity relative to . U U See MAT1300 notes for a proof. Definition 1.1.3 Let B be a topological space with chosen basepoint . A(locally trivial) fibre bundle over B consists of a map p : E B such that for all b B∗there exists an open neighbourhood U of b for which there is a homeomorphism→ φ : p−1(U)∈ p−1( ) U satisfying ′′ ′′ → ∗ × π φ = p U , where π denotes projection onto the second factor. If there is a numerable open cover◦ of B|by open sets with homeomorphisms as above then the bundle is said to be numerable. 1 If ξ is the bundle p : E B, then E and B are called respectively the total space, sometimes written E(ξ), and base space→ , sometimes written B(ξ), of ξ and F := p−1( ) is called the fibre of ξ.
    [Show full text]