Which Functor Is the Projective Line?
| Autor: | Daniel K. Biss |
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| Rok vydání: | 2003 |
| Předmět: | |
| Zdroj: | The American Mathematical Monthly. 110:574-592 |
| ISSN: | 1930-0972 0002-9890 |
| Popis: | the school of the past century is that a mathematical object is in some sense the same as the information needed to encapsulate it. For example, we view a complex vector space Vc as the same as a real vector space VR equipped with an automorphism J satisfying J2 = -Id. By "the same," we mean that it is possible to pass from one description to the other and back without any loss of information. Indeed, given a complex vector space Vc, we can view it as a real vector space VR, and by letting J : VR -> V. be the automorphism defined by J(v) = i v, we obtain the pair (Va, J), as desired. Conversely, if we start with the information (Va, J), we can define a complex structure on Va by declaring (a + bi) v = a v + b J(v). Thus, these two notions of complex vector space convey precisely the same information. More pedantically, a complex vector space is a collection of elements and a collection of rules-rules that dictate how to add vectors, and how to multiply a vector by a complex scalar. Any method of writing down a particular set of rules gives the same vector space, whether that means writing down a complete addition and multiplication table for VC, or first specifying only the structure of a real vector space Va, then declaring (via the automorphism J) how the complex number i will act on Va, and finally letting the vector space axioms along with the fact i generates C over R do the rest of the work. Naturally, vector spaces over C are not alone in this respect: almost all mathematical objects have the (often extremely useful) feature that they can be described in several different ways. Consider, for a moment, the two-element group Z/2. There is a seemingly endless list of ways to specify-and, accordingly, study-this group. We might describe it, as our notation suggests, as the quotient of Z by the ideal of even numbers. We could describe it in terms of generators and relations, (ala2 = e); or as the unique group of order 2; or as the Galois group Gal (C/R). We could, more whimsically, describe it as the group whose set of elements is {cabbages, kings} with the following multiplication table |
| Databáze: | OpenAIRE |
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