Two papers in categorical topology
... that a singleton has only one structure* This condition is usually satisfied for categories of sets with topological structures of some kind. The categories considered in 3«3 and the preceding paragraph have the following properties in'common. (a) A map f of U" is monomorphic in 7 " if and only if P ...
... that a singleton has only one structure* This condition is usually satisfied for categories of sets with topological structures of some kind. The categories considered in 3«3 and the preceding paragraph have the following properties in'common. (a) A map f of U" is monomorphic in 7 " if and only if P ...
CLASSIFYING SPACES OF MONOIDS – APPLICATIONS IN
... Thus, the bh a defines a natural transformation between the identity functor IC : C → C on C and the constant (in ∗) functor on C. Therefore, C is contractible. ¤ Schori introduced in [S] a sequence Dn of interesting configuration spaces. It was proved in [A-M-S], that D2 is the well-known Dunce Hat ...
... Thus, the bh a defines a natural transformation between the identity functor IC : C → C on C and the constant (in ∗) functor on C. Therefore, C is contractible. ¤ Schori introduced in [S] a sequence Dn of interesting configuration spaces. It was proved in [A-M-S], that D2 is the well-known Dunce Hat ...
Tutorial Sheet 3, Topology 2011
... β is a base for a topology and that in this topology each member of β is both open and closed. (This topology is called the half-open interval topology.) Remark: Apologies if this question was in any way unclear. I should perhaps have said explicitly that the corresponding topology is τ = {U : U = ∅ ...
... β is a base for a topology and that in this topology each member of β is both open and closed. (This topology is called the half-open interval topology.) Remark: Apologies if this question was in any way unclear. I should perhaps have said explicitly that the corresponding topology is τ = {U : U = ∅ ...
Homotopy Theory
... Definition 7.9. The smash product X ∧ Y of two pointed spaces is defined by: X ×Y X ∧Y = X ×∗∪∗×Y Corollary 7.10. If X, Y, Z are pointed spaces and Y is locally compact then a pointed map f : X ∧ Y → Z is continuous if and only if its adjoint fb : X → Map0 (Y, Z) is continuous. This just follows fro ...
... Definition 7.9. The smash product X ∧ Y of two pointed spaces is defined by: X ×Y X ∧Y = X ×∗∪∗×Y Corollary 7.10. If X, Y, Z are pointed spaces and Y is locally compact then a pointed map f : X ∧ Y → Z is continuous if and only if its adjoint fb : X → Map0 (Y, Z) is continuous. This just follows fro ...
NOTE ON COFIBRATION In this overview I assume
... (3) We call a map g : A → B cofibration iff it has LLP with respect to any trivial fibration f : X → Y . (4) Trivial cofibration is a map which is both a weak equivalence and a cofibration. If we now think of category where objects are the maps of topological spaces, what will be the arrows? Intuiti ...
... (3) We call a map g : A → B cofibration iff it has LLP with respect to any trivial fibration f : X → Y . (4) Trivial cofibration is a map which is both a weak equivalence and a cofibration. If we now think of category where objects are the maps of topological spaces, what will be the arrows? Intuiti ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.