0708-1300/Class notes for Tuesday, January 8

Algebraic Topology

Temporarily, at least, we will no longer assume that all structures are manifolds and that all functions are smooth. However, functions will still be assumed to be continuous.

We will also, temporarily, assume that all spaces are pointed spaces. That is, a space X will be assumed to have a distinguished point x, whether mentioned explicitely or not.

General idea of algebraic topology: to find "functors" from topology to algebra.

Informal definition: A category C consists of

1) a collection Obj(C) of objects

2) a class hom(C) of morphisms between these objects, so that for each X₁ and X₂ in Obj(C) we get a set mor{X₁, X₂}

3) Composition laws:

a) a binary function on hom(C): mor(X₁, X₂) x mor (X₂, X₃ --> mor (X₁, X₃)

b) Identities (for every X, there is an I_X in mor(X,X))

4) Compatibility laws:

a) associativity of composition

b) identities are in fact identities.