Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

‘there is no ontology here’ 399schemes as functors, and call these functor schemes. There is a functorh : Scheme O → Scheme Vwith two nice properties (Mumford, 1988, §.II.6).²⁸1. The scheme maps from any X, O X to any Y, O Y in Scheme O correspondexactly to the scheme maps h(X) → h(Y) in Scheme V .2. Every functor scheme is isomorphic to the image h(X) of at least oneringed space scheme X, O X .We say that a functor scheme V corresponds to the ringed space scheme Xif V is isomorphic to h(X). ButthenV also corresponds to every ringedspace scheme isomorphic to X. The key to working this way is that allthe ringed space schemes corresponding to X are already isomorphic (asringed space schemes). For many purposes you need not decide whichcategory you are in. You can go back and forth by the functor h. Asone germane application this passage preserves cohomology. Cohomologyof schemes can be defined purely in terms of categorical relations betweenschemes. Exactly the same relations exist in the category of ringed spaceschemes and the category of functor schemes. Many calculations of cohomology,and applications of cohomology, work identically verbatim in the twocategories.14.5.4 Isomorphism versus equivalenceLike the two definitions of the category of schemes, so the two definitionsin Section 14.2.4 of the category of sheaves on a space M agree only up toequivalence. Equivalence is isomorphism of categories up to isomorphism inthe categories.Isomorphism is defined the same way for categories as for any kind ofstructure: A functor F : C → D is an isomorphism if it has an inverse, namelya functor G : D → C composing with F to give the identity functors GF = 1 Cand FG = 1 D .Inotherwords,forallobjectsC of C, andD of D:GF(C) = CFG(D) = Dand the same for morphisms. Categories are isomorphic if there is an isomorphismbetween them.A functor F : C → D is an equivalence if it has a quasi-inverse, a functor Gwith composites isomorphic to identity functors: GF ∼ = 1 C and FG ∼ = 1 D .In²⁸ These imply equivalence as defined in the next section, given a suitable axiom of choice.