Which Functor Is the Projective Line?

Abstract

school of past century is that a mathematical object is in some sense same as information needed to encapsulate it. For example, we view a complex vector space Vc as 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 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 automorphism defined by J(v) = i v, we obtain pair (Va, J), as desired. Conversely, if we start with 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 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 same vector space, whether that means writing down a complete addition and multiplication table for VC, or first specifying only structure of a real vector space Va, then declaring (via automorphism J) how complex number i will act on Va, and finally letting vector space axioms along with fact i generates C over R do rest of work. Naturally, vector spaces over C are not alone in this respect: almost all mathematical objects have (often extremely useful) feature that they can be described in several different ways. Consider, for a moment, 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 quotient of Z by ideal of even numbers. We could describe it in terms of generators and relations, (ala2 = e); or as unique group of order 2; or as Galois group Gal (C/R). We could, more whimsically, describe it as group whose set of elements is {cabbages, kings} with following multiplication table: