Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Continuous function wikipedia , lookup
Sheaf (mathematics) wikipedia , lookup
General topology wikipedia , lookup
Homological algebra wikipedia , lookup
Grothendieck topology wikipedia , lookup
Homotopy type theory wikipedia , lookup
Homotopy groups of spheres wikipedia , lookup
SAM III General Topology Lecture 5 Contents Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn Product in a category SAM III General Topology Lecture 5 The definition Preservation of products by the fundamental group SAM — Seminar in Abstract Mathematics [Version 20130306] is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy SAM III General Topology Lecture 5 Contents Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn Product in a category The definition Preservation of products by the fundamental group 1 Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn 2 Product in a category The definition Preservation of products by the fundamental group Homotopy of paths SAM III General Topology Lecture 5 Contents Definition Consider two points x and y in a topological space (X , τ ). Let f : [0, 1] → X and g : [0, 1] → X be two paths from x to y . A homotopy from f to g is a continuous map h : [0, 1]2 → X such that h(x, 0) = f (x) and h(x, 1) = g (x) for all x ∈ [0, 1]. We say that f and g are homotopic, and when there is a homotopy from f to g . Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn Product in a category The definition Preservation of products by the fundamental group Homotopy classes Show that the relation “f is homotopic to g ”, written as f ' g , is an equivalence relation. The equivalence class of a path f under this equivalence relation is called the homotopy class of f . Composition of paths If f is a path from x to y and g is a path from y to z, then we can define a path g · f from x to y as follows: (g · f )(x) = f (x), g (x), 0 6 x 6 21 , 1 6 x 6 1. 2 Show that if f ' f 0 and g ' g 0 then g · f ' g 0 · f 0 . Fundamental groupoid SAM III General Topology Definition A groupoid is a category in which every morphism is an isomorphism. Lecture 5 Contents Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn Product in a category The definition Preservation of products by the fundamental group Fundamental groupoid To a topological space X = (X , τ ) one associates a category Π1 X defined as follows: objects of the category are points in the topological space, while a morphism from x to y is a homotopy class [f ] of a path f from x to y ; composition of morphisms is defined by the formula [g ] ◦ [f ] = [g ◦ f ]; for an object x, the identity morphism is given by the homotopy class [x] where x stands for the path [0, 1] → X , r 7→ x. Verify that Π1 X is indeed a category, and that moreover, it is a groupoid. This groupoid is called the fundamental groupoid of the space. Fundamental group The fundamental group of a topological space X = (X , τ ), at a point x ∈ X , written as π1 (X , x), is the group of automorphisms of x in the fundamental groupoid Π1 X of the space. Show that in a path-connected space, fundamental groups at any two distinct points are isomorphic to each other. Deduce this from a general observation about an abstract groupoid. Convex sets in Rn SAM III General Topology Lecture 5 Contents Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn Product in a category The definition Preservation of products by the fundamental group Definition A subset S of Rn is said to be convex when S is closed under linear combinations c1 v1 + c2 v2 where c1 + c2 = 1 and c1 c2 > 0. Fundmental groupoid of a convex subset of Rn A groupoid is codiscrete when for any two objects x and y there is exactly one morphism x → y . Show that if a subspace of Rn is a convex set then its fundamental groupoid is codiscrete. In general, show that the fundamental groupoid of a topological space is codiscrete if and only if the space is path-connected and the fundamental group at any point is a trivial group. The definition SAM III General Topology Definition Let X and Y be two objects in a category. Their product is defined as a diagram Lecture 5 Contents X o π1 π2 X ×Y /Y Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn Product in a category The definition Preservation of products by the fundamental group such that for any other such diagram X o f C g /Y there is a unique morphism C → X × Y , denoted as (f , g ), such that π1 (f , g ) = f and π2 (f , g ) = g : X π1 π2 /< Y X ×O Y bFo F FF xx x FF (f ,g ) xx f FFF xx g xx C Preservation of products by the fundamental group SAM III General Topology Lecture 5 Contents Homotopy Homotopy of paths Fundamental groupoid Convex subsets of Rn Product in a category The definition Preservation of products by the fundamental group Products in Grpd Define the category Grpd of groupoids and construct products in this category. Products in Top Show that in the category Top of topological spaces, product of two spaces with the canonical projection maps is the same as their categorical product. Functoriality of the fundamental group and preservation of products Define a structure-preserving map between categories, and call it a functor. Show that the process of assigning to a topological space X its fundamental groupoid ΠX defines a functor Top → Grpd. Formulate what does it mean for a functor to preserve products and show that the fundamental groupoid functor is such.