Math 434-501 CRN 15098/ Math 542-501 CRN 15099 Fall 2015

August 24th.

the cross cap

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August 24, 2015. board1






Right about here, I sketched the definition of a category. While we hardly will use this idea, it is mathematically important, so I will review it here. I forgot to photograph it. A category consists of a collection of objects such that for any two objects A and B, there is a set of morphisms, Hom(A,B). And the morphisms satisfy the following properties. The object A is the source of the morphism f (denoted s(f)) which is in Hom(A,B). Meanwhile, the object B is called the target of f (denoted t(f)). In the set Hom(A,A), there is an identity 1(A). Morphisms can be composed. That is that there is an operation from Hom(A,B) X Hom(B, C) to Hom( A, C), called composition. So if t(f)=s(g), then gf is a called the composite morphism. Composition of morphisms is associative: h(gf)=(hg)f. Here f is in Hom(A,B), while g is in Hom(B, C), and h is in Hom(C, D). The identities act as follows: f.1(A)=f and 1(B).f =f. Here I am using "." to denote composition. Mathematics is replete with categories.







Math 434-501 CRN 15098, Math 542-501 CRN 15099 August 24, 2015 ends

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