M275 Notes 1.2 Category Theory
نویسنده
چکیده
Lecture notes from M275 at SJSU, a graduate-level course at SJSU in Algebraic Topology taught by Professor Richard Kulbelka. These notes were taken using Vim and/or GVim equipped with L A T E X-Suite, which sped up typesetting significantly; latexmk with the flag-pvc was useful for compiling these notes in real-time. 1.1 The Question Question. Given two topological spaces, can we determine if they are homeomorphic? In short, this is very hard. Exercise. Show that R ≈ [0, 1] with the usual topology. Solution. Let I = [0, 1]. If we remove 0 from I, then I is still connected. On the other hand, no point in R has this property, so we lose, since homeomorphism preserves connectedness. Definition. A category C will consist of a " class " of objects ob(C), as well as a " class " of morphisms mor(C), such that for every pair of objects there is associated a (possibly empty) set of morphisms mor(X, Y). Example. The category G of groups, with ob(G) being " the set of all groups " , except this leads to a contradiction, so we call it a class instead. In fact, if the objects of a category are a set, it is called a " small category " or " kitty-gory ". Example. A is the category of abelian groups, category V of vector spaces, category R of rings.
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