Enough, already, ofcategory theory. What does category theory tell us about the process ofalgebrization that underlies the paired-algebras framework?
Let CAlg denote the category whose objects are commutative algebras and whose arrows are algebra homomorphisms. Given any commutative
136 APPENDIX A. SOME CATEGORY THEORY Set Aff Lin CAlg U T M G L A
Figure A.1: A ladder ofadjunctions
algebra, if we forget how to multiply, we are left with a linear space; thus, there is an obvious forgetful functor from CAlg to Lin. Let’s denote that functor M, on the grounds the it forgets how to multiply. The process of algebrization is precisely a functor A that is left adjoint to the forgetful functor M. That is, writing A(X) for the symmetric algebra Sym(X), we have one-to-one correspondences
Lin(X,M(G))←→X,GCAlg(A(X), G).
Figure A.1 shows a ladder ofadjunctions ofwhich we have explained the top two steps. On the right, we have forgetful functors going down,
M: CAlg → Lin and T : Lin → Aff. On the left, their left adjoints go back up, linearizationL: Aff →Lin and algebrization A: Lin→CAlg.
It is helpful to add on one more step at the bottom of the ladder. LetU denote the forgetful functor fromAff toSet, the functor that takes an affine space and forgets everything about it except for the underlying set of points — U for underlying. The left adjoint of U is the functor G that, given any set S, produces the “affinization” or “geometrization” ofS, an affine space for which the points inS form a barycentric frame. Note that we again have one-to-one correspondences
Set(S,U(A))←→S,A Aff(G(S), A).
So we have a ladder with four rungs, each adjacent pair of rungs connected by a forgetful functor, going down, and its left adjoint, going back up. Going down is always easy. Stepping up from Set to Aff is also conceptually easy, although the resulting affine spaces can be quite large. The other two upward steps, linearization and algebrization, are more subtle. But note that jumping up fromSettoLin, taking two steps at once, is easy; given a setS, the linear spaceL(G(S)) is simply a linear space that hasS as a basis. And
A.5. A LADDER OF ADJUNCTIONS 137 jumping all the way up the ladder from Set to CAlg is easy as well; given a set S, the commutative algebra A(L(G(S))) is simply the algebraR[S] of all polynomials whose variables lie in S. Ifwe jump up from a set S to any rung, we simply get the free thing of that type that is generated by S: the f ree affine space withS as a barycentric reference frame, the free linear space with S as a basis, or the free commutative algebra with S as its generating variables. The subtlety comes only when we have already stepped up and we want to step farther up.
Indeed, recall that the easiest concrete construction that we found for linearizing an affine space A involved choosing a barycentric reference frame for A and then forming L(A) as the linear space with those frame points as a basis. In terms ofthe ladder in Figure A.1, choosing a barycentric frame for A means finding some set S with G(S) = A. Note that this is not at all the same as computing U(A), since the functors G and U are adjoints, not inverses. Having found some S with G(S) = A, we can then jump up from Set toLin in one easy step, by forming the linear space that has S as a basis. Thus, jumping up from the ground is so easy that we use it as a subroutine when stepping up one rung.
Given a linear space X, jumping up from Set is also the easiest way to construct the symmetric algebra A(X) = Sym(X). We choose some basis for X; that is, we find some set S with X = L(G(S)). We then construct
A(X), in a big jump back up, as the polynomial algebraR[S].
This ability to step up by backing down to ground level and then jumping one higher is a special property ofthe particular ladder ofadjunctions in Figure A.1. For example, we couldn’t employ the same strategy to step up above CAlg, ifwe added a new rung at the top ofthe ladder, since not every commutative algebra is the polynomial algebra generated by some set ofvariables. One ofthe things that make Aff and Lin simple is that every object has a frame or basis; but that doesn’t hold for CAlg.
Exercise A.5-1 In preparation for the day when locations over A might join sites over A as basic objects in CAGD, so that we can divide by points as well as multiply by them, extend the adjunction ladder ofFigure A.1 with one more rung at the top.
Hint: First replace the former top rung CAlg by the smaller category EAlg of entire algebras, that is, commutative algebras that are free of zero divisors and that hence, when viewed as rings, are integral domains (a.k.a. entire rings). And verify that the functorsMandAform an adjunction also between Lin and EAlg. We can then add, as a new top rung, the category RFld consisting ofthose fields that include the real numbers as a subfield. The forgetful functor D: RFld→EAlg forgets how to divide, while its left adjoint Q: EAlg →RFldforms quotients.
138 APPENDIX A. SOME CATEGORY THEORY