I e-mailed Bill Lawvere a link to this question, and in particular, Tom's answer, and he asked me to post this for him:
I will try to clarify the thread associated with this name for the past 120 years. The clarification involves generalizations of the type that I will need for my research, although I have not yet proved any genuinely new result.
The classical existence theorem of Hilbert has many precise analogues, strengthenings, and generalizations.
Since ‘Stellen’ means ‘places’ in German, one sees immediately that the content is the geometrical one of the existence of places in a space which satisfy given conditions. Because the conditions considered are equations between functions defined on the space, the geometry is intimately related to the algebra of such functions. However, to speak of zeroes (Nullstellen)
is an unnecessary restriction, useful however in limited contexts where there are theorems available concerning factorization et cetera. The question of existence and partial answers also make sense for rigs
(‘rings’ without necessarily an everywhere-defined subtraction, for example in the natural numbers or in real algebraic geometry where one seeks ‘Positivensaetze’.) The usefulness of the restriction to algebras with negatives led to the development of the technique of ideal theory, in particular to the study of the generation of the unit ideal, et cetera. However, from a more conceptual perspective the purpose of the ideals is to give rise to quotient algebras. In that light one sees that the more natural algebraic interpretation of closed subset is as a surjective algebra homomorphism from the algebra of functions on a space to another algebra; in the same spirit the role of points (i.e. the desired Stellen) is to act as general ‘evaluation’ homomorphisms from the same algebra to special algebras. (The concern about maximal ideals and prime ideals comes really from the question of which algebras are special.) The general idea is that the special algebras can be qualitatively smaller than the typical algebras, but such homomorphisms can be proved to exist nonetheless.
Garrett Birkhoff’s 1935 theorem on the ubiquity of subdirectly irreducible algebras implies a qualitative improvement in that there are even enough such generalized points to yield a monomorphic embedding of algebras, as I will explain below. People studying Universal Algebra should consult Birkhoff’s paper in which he states very clearly that his theorem was motivated by work of Hilbert and Noether in algebraic geometry, even though it applies to much more diverse kinds of algebra. There are probably much more recent results in Universal Algebra which apply still to the commutative algebra case.
Hilbert’s original theorem concerned algebraic spaces of arbitrary finite dimension defined over a ground field; he proved that if the existence of points were true for a one-dimensional space, then it would be true for nontrivial spaces of all finite dimensions. (This is suggestive of the more recent theory of O-minimal spaces.) In his ‘algebraically closed’ case, there is only one special algebra, namely the ground field itself which geometrically is the function algebra of a single bare point. However, the theorem extends easily to the case when there is no such hypothesis on the ground field, by allowing arbitrary field extensions that are finite-dimensional as vector spaces to play the role of the special spaces or punctual figure shapes. These results have been further extended (permitting parametric families of spaces) to very general ground rings that are not even fields. (These more general ground rings include all those that are finitely generated as algebras over a smaller ground ring for which the theorem is true.)
By enlarging still further the category of special spaces, namely to general commutative algebras that are finite-dimensional over the ground field, and thus including geometrically not only fat points but also infinitesimal motions as expressed by nilpotent elements, there are in fact enough homomorphisms from any finitely-presented algebra (typically infinite-dimensional) to these special algebras, in the sense that given any two functions $f$ and $g$ in the algebra that are distinct, there exists such an infinitesimally variable point $x$ so that
$f(x)$ is not equal to $g(x)$. Results of this general type I will refer to as a ‘strong Nullstellensatz’. It means algebraically that the given algebra is mapped monomorphically into an infinite product of special algebras. Continuing to a second stage of this resolution, the typical algebra is embedded into an inverse limit involving formal power series at each point. Birkhoff’s theorem is somewhat more precise, insisting always on subdirectly irreducible pieces, whereas the construction just sketched (a ‘coadequacy monad’)is content with subalgebras of subdirectly irreducible algebras, the homomorphisms being typically not surjective and hence more functorial.
In order to express this kind of results in a more fully geometrical way, I recall the method of analysis elaborated before 1960 by Grothendieck for fully revealing the inside of a space (contrary to spurious rumors that category theory treats objects as ‘opaque’). For simplicity, I think of the category $C$ of spaces under consideration as being embedded in a topos, but sufficient would be certain existence and exactness properties that that would imply. We assume given a small subcategory $A$ to serve as ‘figure shapes’; in the case of smooth, analytic, algebraic, real algebraic, et cetera, contexts these figure shapes would typically be taken as those for which the associated function algebras are finitely presented in their appropriate category. Then the inside of any space $X$ is the discretely fibered category $A/X \to A$, this being the functor that assigns to every figure its shape; the maps in the comma category $A/X$, namely the commutative triangles over $X$,
suffice to account for all incidence relations between the figures and hence for the structure of the inside of $X$. Of course, discrete fibrations are equivalent to contravariant set-valued functors, or presheaves, but the discrete fibration formulation seems to be closer to the original geometry; in any case, these discrete fibrations over a given category $A$ constitute a topos in which we assume that $C$ is fully embedded. Grothendieck referred to this analysis of the inside of
$X$ as the ‘functor of points’, which I find misleading because of the 2000-year old tradition according to which points are very special figures. Thus we assume also a subcategory $P$ of $A$. Again, there is an obvious attempt to represent any $X$ as the discrete fibration
$P/X \to P$, but it is intuitively obvious that this representation will probably not be full because the cohesion has been thrown away. (I say cohesion because the classical term ‘continuity’ has been given a particular determination during the past century, which I am not considering; this classical idea is essentially the preservation of incidence relations without tearing them.) Typical examples of my ‘Axiomatic Cohesion’ are obtained by comparing the toposes generated by such a pair $(A,P)$, resulting in a quartet of functors anyone of which determines the other three by adjointness, the two downward ones expressing the idea of connected components and the other one expressing the idea of points, whereas the upward ones express the minimal completeness of the topos in the fact that any discrete space gives rise to opposite discrete and codiscrete spaces between which any space with those points sits.
Here it is crucial that the term discrete be understood as ‘semi-discrete’, cohesion being relative. For example, in the original context of algebraic geometry, $P$ would typically be the opposite of the category of finite field extensions, with generated topos being Boolean, but not the category of abstract sets, except in the case of algebraically closed ground field. The epimorphicity of the map from points to components says intuitively that for a space $X$, the extent to which its component set is non-empty, is the extent to which its point set is non-empty; however, ‘set’ means an object in the lower topos and by basic internal logic, internal existence means actual existence only locally, so that the point of $X$ which is asserted to exist is not necessarily over the terminal object $1$, but rather over a finite extension field.
For innumerable reasons, it is important that the left most functor of the quartet, the set of components, should preserve finite products. This may not be true if we take $P$ too small, for example, only the terminal object $1$, attempting to use abstract sets as the lower base topos. This is one reason why I have relativized discrete to semi-discrete. The reason for emphasizing the right most functor of the quartet, namely the codiscrete space, may not be obvious, but becomes more significant if we hope to obtain a strong version of the Nullstellensatz by considering an intermediate category I of infinitesimal motions between $P$ and $A$, thus expressing some of the functors of the quartet as composites. Because of the relative completeness of toposes, the inverse limit of algebras implicit in the analog of Birkhoff’s construction can be interpreted as simply the function algebra of the subspace $sk(X)$ of $X$ obtained as the union of all infinitesimal subspaces. But there is then also the dual inclusion of this intermediate topos into the upper one, assigning to $X$ a space $cosk(X)$. A typical mathematical problem could be stylized as whether a formal function $sk(X) \to R$ can be extended to $X$ without changing $R$, or dually, whether a formal path $R \to cosk(X)$ can be lifted to $X$. The strong Nullstellensatz becomes rather a property of particular objects $R$, typically objects of $A$, but not of all objects of $C$, namely that $R$ ‘perceives’ the inclusion $sk(X) \to X$ as epimorphic (meaning that the induced map of function spaces $R^X \to R^{sk(X)}$ is monomorphic.)
Bill Lawvere (who is following this thread from the shadows) sent me another e-mail to post "in response to the concern expressed in some of the comments about the adjointness of $\pi_0$" (presumably on another answer):
The functor including (semi-)discrete spaces into general smooth or cohesive
spaces is indeed right adjoint to the functor assigning the (semi-)discrete space
of suitably connected components to any smooth or cohesive space.
This tends to be generally true when the two categories in question are (‘gros’)
toposes constructed by the usual methods and connected by induced functors and their adjoints.
Here the semidiscrete spaces are indeed more structured than the totally discrete
abstract sets of Cantor but NOT in the sense of the ‘zero-dimensional spaces’ that
seem to arise if one mistakenly adheres to the idea that there is a default notion of
space determined entirely by Sierpinski-valued funtions. Rather the inside of a space should instead be analyzed by the covariantly associated system of figures and incidence relations, as I explain in my Palermo paper on Volterra’s Functionals.
The duality is not at all a naïve metaphysical one, as illustrated dialectically by Liouville’s well-known theorem that projective space is not determined by its meager algebra of functions even though functions are a key tool in analyzing the whole topos of spaces in which it sits. Hurewicz did not submit to the default when he invented k-spaces in the late 40’s in response to the needs of analysis and of algebraic topology for map spaces satisfying the exponential laws.
It is often casually asserted (for example in Wikipedia) that schemes are topological spaces. That borders on disinformation because the functor from schemes to topological spaces does not even preserve products (hence not group objects, etc.). That is why it was such a tour de force when Grothendieck proved that fibered products even exist for schemes (on the basis of the old definition). In a colloquium talk here at Buffalo in 1973 Grothendieck forcefully advocated that the definition of scheme based on locally ringed spaces be abandoned as basic in favor of a definition based on simple gros topos constructions from which the prime ideals, open set lattices, petite sheaf of functions, etc could be recovered whenever helpful by geometrically intuitive means, but not as baggage in the working definition. (I had already reached that conclusion myself in 1966 discussions with Gabriel, which immediately suggested that similar methods could be applicable in the smooth case as well, as Grothendieck had begun to do in embryonic form already in 1960 in the analytic case.)
Geometric intuition, in the roughly ‘topological’ sense advocated by Grothendieck in
his plea for Tame Topology, applies to directly in any category with suitable properties, even as general as ‘extensive’ categories. For example, the opposite of
the category of Boolean algebras relates directly to the ‘algebraic geometry’ of
the topos of presheaves on finite non-empty sets, whose manifold uses have been unjustly neglected. The ideal-theoretic techniques developed by Noether, Krull,
Gelfand, Stone, Jacobson, et al should be used when appropriate for calculation
but should not be permitted to obscure the geometric intuition.
