**ALEXANDER GROTHENDIECK
& THE CONCEPT OF SPACE**

F. William
Lawvere

Invited address CT Aveiro June
2015

** **

50 years ago the first international category theory meeting
took place in La Jolla, California. In fact, part of that meeting moved to the beach,
where an inspiring talk by Jean-Louis Verdier introduced many of us to a new
class of categories due to Grothendieck, writing on a blackboard that had been
brought to the beach for the purpose. Jon Beck began to draw diagrams in the
sand, and a lively and enthusiastic discussion started among the participants.
Jean-Luis Verdier suggested that these categories embody set theory, but Erwin
Engeler and I expressed doubt, because the description seemed to need a given
external set theory to parameterize families for the required colimits.

There are several important threads from that meeting that
are still flourishing, for example, the theory of enriched categories as
presented by Eilenberg and Kelly, in particular, the role of 'cartesian-closed'
categories in geometry and logic. The thread I tried to capture at La Jolla,
namely, the increasing use of categories and functors as the language of
abstract mathematics, has continued for the last 50 years.

The explicit formulation of the principles of category theory
in my paper is still in need of improved axiomatization. I will be overjoyed
when some young person responds to that need.

The recent disappearance of so many stalwarts from that epoch
underlines the need for coherent and correct history as a guide to the future. I want to continue the search
for such a history, focusing here on the concept of space.

There is not just one concept of space, but several
categories of smoothness. (To avoid misunderstanding, I am not focussing on
Riemannian space or Space Time. Those important additional structures require
spaces as their domains of definition.)
A common feature of spaces
in these more or less smooth categories I have called COHESION, indicating that
the parts of a space 'stick
together' and 'hesitate' to separate (as in American slang "I'll stick
around a bit, until I split").

The great dialectical geometer Hermann Grassmann discerned
the two main contradictions in mathematics to be 'continuous versus discrete'
and 'equality versus inequality'. Because the term 'continuous' has had a
particular mathematical definition for more than a century, I will instead use
'cohesive' for this philosophical concept, but of course I will immediately try
to tame it with mathematical definitions. The dialectics of inequality/equality
have been rather thoroughly made explicit by mathematicians, on at least two
levels:

Hurewicz (1935), Kan (1955) and
Moore (1955), Quillen (1967), Gabriel & Zisman (1967), Heller (1988),
Grothendieck 1983 & 1989), Kan et al
(2004), Maltsiniotis & Cisinski (1999 - to the present), are some of
the major contributions at the level of spaces themselves.

Another level of the transformation
of equality is codified in the notion of exact category introduced by Myles
Tierney in our Halifax Seminar in 1969; he proved that these categories are a
delinearization of Grothendieck's notion of abelian category in the sense that
abelian groups in an exact category form an abelian category. This theory was
expounded in the book by Barr, van Osdol, and Grillet, as well as in Barr's
address to the 1970 ICM. Exact categories embody the special property of
'sheaf-theoretic images' which can be expressed in 'logical' terms: Define the
image of a map X -->Y to be the
smallest subobject of Y through which the map factors. That definition
expresses precisely the rule of inference of existential quantification; but
then to what extent does it express 'actual existence'? In other words, given a
figure Q --> Y of shape Q in the codomain Y, to what extent does it 'come
from' a figure in X via the map, assuming that Q --> Y lies in the image so
defined? Part of the exactness property guarantees that there exists a
covering P --> Q of Q with an
actual figure P --> X mapping to P --> Q --> Y, to the pullback of the
given figure. The other feature of exact categories is even more transparently
about transformation of equality: co-equalizers come from their kernel pairs
and equivalence relations all arise as kernel pairs. It is clear that the
theory of exact categories has wide applicability. The book by Barr, Grillet,
and Van Osdol did much to popularize it, and the work of Carboni and others
very effectively used 'the exact completion' to adjoin appropriate
co-equalizers to non-exact categories. Most of these works postulate the
exactness properties as given conditions on a category, as does Giraud's
characterization of Grothendieck toposes; part of the significance of the
postulation of function spaces and power objects is that the existence of
adjoint functors implies exactness without further postulation.

The idea of an opposition between a category of cohesive
spaces and a category of anti-cohesive sets also applies in particular to
Cantor's description of the relation between a category of Mengen and a
sub-category of Kardinalen. In fact, it appears that in general the discrete is
a co-reflective subcategory of the cohesive, with the co-reflection extracting,
as an Aristotelian arithmos, the Cantor 'cardinal of X' or the Hausdorff
'points of X'. (The sets of 'lauter Einsen' have isomorphisms, as do the
objects in any category; the issue here, however, is not to pass to isomorphism
classes, but simply to extract the underlying discrete aspect of each given
space/ Menge.) The Grassmann dialectic develops further. The discrete
subcategory is the negation of an identical subcategory at the opposite end,
with the same functor as reflection. That is, the same category has insertions
as two opposite sub-categories, the one illustrating that the 'lauter Einsen'
are totally distinct, but the other demonstrating that they are nearly
indistinguishable. More precisely,
joint work with Matias Menni has shown that under very general assumptions the
co-discrete inclusion consists of the Boolean sheaves that any topos has.
However, for a category of spaces there is an additional adjoint to the
sheafification. This indicates a non-trivial restriction on that cohesive
category, namely the existence of that additional Cantor adjoint. Such
restrictions serve as axioms for cohesion, which is our proposed
characterization of 'Categories of Space'.

My use of concepts such as the Boolean sub-category reveals
that I am convinced that categories of space are most effectively modeled as
appropriate toposes. One of the two axioms for toposes, namely the existence of
function spaces (the feature that has been called 'cartesian closed' since the
Eilenberg & Kelly contribution 50 years ago) had been recognized as
fundamental by Hadamard and Volterra at the time of the 1897 ICM in Zurich. That this property is essential was
emphasized by Grothendieck in 1957 in his Tohoku paper. These and many other
reasons point to this operation as central to all branches of mathematics. In
order to achieve the function space property (in models of cohesion that are
constructed as categories of structures in a discrete base), the fundamental
structure needs to have the nature of figures and incidence relations, rather
than of algebras of functions (which can be recovered by naturality). My 1997
Palermo paper attempted to explain this necessity. That paper, like the 1965
paper by Eilenberg & Kelly, and like publications by Steenrod, Kelley, and
Brown, mentioned as an important example the k-spaces based on using compact
spaces as figure types. However, none of us authors mentioned the actual origin
of the k-spaces, of which I learned later on the phone from David Gale (when
following up his 1950 publication in the Proceedings of the AMS). Namely, the
notion of k-space was introduced
by Witold Hurewicz in his Princeton lectures in the late 1940's. In fact, in
the early 1940's, Hurewicz had emphasized the need, which led to the partial
solution by Ralph Fox in 1945 for the case of convergent sequences as figures.
Hurewicz did not speak explicitly in terms of categories, but in the
exponential laws that he demanded one immediately discerns the feature of
adjointness.

It is striking that Hurewicz, who in 1935 had initiated
fundamental advances in the study of the Grassmann transformation of inequality
into equality, made also fundamental contributions to the development of the
other Grassmann transformation between continuity and discreteness. Important
were his well-known contributions to dimension theory (which already made use
of function spaces in 1941), but also his less cited contribution to the
recognition of the fundamental role of function spaces in general. Had it not
been for the temptation of the pyramid at Uxmal, he would have shown us more of
the relation between the two Grassmann principles.

From functional analysis to derivateurs, Alexander
Grothendieck's work has immensely illuminated that relation.

Hausdorff's great book 'Mengenlehre' was actually about
topology (which is an important product of the study of cohesion), illustrating
again the opposition and mutual transformation between cohesion and
discreteness, as approached in his work about chaos under the pseudonym Paul Mongre'.

A remarkable aspect of the continuous/discrete dialectic is
that the abstract sets of 'lauter Einsen', abstracted from the cohesion of
spaces, can reciprocally act as the basis, via specific diagrams in their
category, for structures constituting models of all sorts of mathematical
objects, including in particular the spaces themselves. As a criterion for the adequacy
of our axioms, Myles Tierney and I insisted on the proof of the Grothendieck
constructions of sites and sheaves. That proof was published by Radu Diaconescu
in 1975 as a necessary preliminary to his proof of change of base for toposes.
That is, for a geometric morphism E-->U satisfying a boundedness condition,
E can be reconstructed, by a zigzag of three geometric morphisms, from a site
internal to U: the first leg is the local homeomorphism given by the slice
topos over an object of U that parametrizes the objects of an internal
category, the second is given by the left-exact comonad that adjoins the
'presheaf' action of the maps of that internal category, and the third is the
full inclusion of sheaves for a localness operator. (Each of the three is a
special case of a distinct important general closure property of the class of
toposes.) For such a 'U- Topos' E, the U itself can be any elementary topos,
re-inforcing Grothendieck's observation concerning the ubiquity of the powerful
principle of relativization; it need not be an inaccessible universe, as in
Grothendieck's original SGA4 examples; nor need it be the discrete part of a
cohesive topos, as emphasized here. For each topos U there is the 2-category
Top/U of U-toposes; indeed, varying U may simplify the treatment of certain
problems.

The 1959 Warsaw lecture by Saunders Mac Lane in effect
introduced the idea of enriched category, in its special case of 'locally small
category'. As a reflection of the Bernays class/set distinction, the belief developed
that categories that are not locally small are 'illegitimate'. I suggest the following alternate point of view.

Within the metacategory of categories, there are monoidal
closed categories and hence other categories enriched in them. This shows the
need for the existence of an actual category called the category of small sets,
within the cartesian closed metacategory of all actual categories. The functor
category of any two actual categories should also be actual, although of course
properties like local finiteness will not be preserved. Potential categories
(corresponding to subcategories of that metacategory) may or may not have
actual categories that represent them up to equivalence. One of the main goals
of abstract mathematics is to illustrate and use the mutual transformation
between space and quantity. The spaces and quantities of primary interest are
'small', so it is reasonable to define small sets to mean those satisfying the
Banach-Isbell duality and to postulate that there is an actual category U
representing that notion of smallness. This postulate now seems to be one of
the reasonable amendments to my 1965 La Jolla attempt to summarize in axioms
the key useful features of a metacategory of categories. So functor categories
of actual categories may not have small hom sets, but they are actual and thus
subject to all the properties of actual categories in general.

What I have said so far has been profoundly influenced by the
work of Alexander Grothendieck. Let me now touch on his contributions specific
to the problem of Space as I have outlined it. It is often said that he
invented toposes as domains for cohomology and that they were a
'generalization' of topological spaces.
But already in 1960 he was defining and using categories in complex
geometry that were toposes even if not explicity so called. His famous Me'daille
de Chocolat exercise (in SGA4) is, as I told him, a key to the whole theory and
application of toposes; he agreed, obviously pleased that someone had noticed.
There he explains a version (in terms of sites) of the relationship between the
gros topos of a space and a petit topos of the same space; the spaces in question are taken from a
category of spaces which could only itself be the gros topos of a point. It is
still an ongoing exercise to clarify the qualitative distinction between the
kinds of toposes that appear as 'gros' or as 'petit' in this kind of situation,
that is, between categories that represent a general determination of cohesion
and categories that consist of variable sets as parametrized by some sort of
generalized space. The generalized spaces would include the e'tale spaces
discovered by Grothendieck.

What was the undesirable feature, of the earlier Dieudonne'-Grothendieck foundation of schemes, that Grothendieck so
emphatically rejected in his 1973 Buffalo colloquium lecture?

The contravariant structure had already been seen by Hurewicz
and others to be problematic, but in the notion of local ringed space, that
structure was further dissected in two interacting components, open sets and
sheaves of local rings. In hindsight, problems could have already been
discerned from Serge Lang's 1960 review of EGA. There, Lang is enthusiastic
about the fact that so many classical concepts can be subsumed under base change;
he is also enthusiastic about Grothendieck's virtuoso proof that such fibered
products actually exist. Indeed from the point of view of us less able
calculators, a virtuoso was required to take the separated ingredients and
re-assemble them into similar ingredients for a product scheme; in particular,
the underlying topological space does not underly the product scheme.

What was the nature of Grothendieck's
solution?

In a topos of set-valued functors on the category of
finitely-presentable algebras, each space X has, thanks to Yoneda, an 'inside'
whose objects are (in general singular) figures of representable shapes, with
incidence relations given by commutative triangles. This can be viewed as a
discretely-opfibered category, but such is equivalent to a set-valued functor.
[I disagree with the term 'functor of points' for this, because it is a functor
whose actual values include all the figures of X. Of course, 'points' of some
other space associated to X may represent figures in X, but for X itself the
points of it are just the restriction of X to the category of finite field
extensions. That category generates the Boolean part of the big topos. The
usual definition of point is unwieldy because it amounts to taking the
non-exact direct limit of that restricted points functor. In general, this
Boolean topos is much better suited than the category of abstract sets to serve
as 'base topos' in the case of non-algebraically closed ground field.
Conflating 'figures in X' with 'points of X' has a sort of science fiction air
he probably did not intend. Volterra called them 'elements'.] A better version
of the 'underlying topological space' is internal to the Barr-Boole-Galois
topos where the actual points functor lands; this choice is also necessary for
a product preserving components functor.

Between the Galois site for the Boolean part and the site for
the whole category of spaces, there is the category of algebras that are
finite-dimensional over the ground field; because the corresponding
representable spaces are the domains of the crucial infinitesimal figures in X,
we may call it a Leibniz site. The importance of these figures was emphasized
by Grothendieck and his colleagues in connection with tangent bundles, e'tale
maps, and so on. Two strong properties of this category in relation to the much
bigger category are the following: The general figure shapes Y from the big
site have the Birkhoff property relative to the inclusions L(X) --> X of the
Leibniz core of any X; namely Y perceives these inclusions as epimorphisms in
the sense that an infinitesimal map L(X)-->Y can be integrated in at most
one way to a map X --> Y. (This means that the algebra of Y-valued
functions on X is a subalgebra of a product of special very small algebras.)
The other strong property (which has traces in Euler) is that any subtopos of
the whole which contains the Leibniz objects will contain all the objects Y of
the big (affine) site; this follows from the fact that there are enough
infinitesimal function spaces to generate that big site, for example, the line
is a retract of the self-exponential of the dual numbers domain. One can easily
extract the subcategory of locally affine spaces i.e. algebraic schemes.

The above outline of a topos of Grothendieck algebraic spaces
over a base field seems to work as well for a base rig. It was shown in detail
to work for smooth geometry by Wraith, Kock, Reyes, Moerdijk, Bunge, Dubuc,
Gago, Lavendhomme, and others. Some version is likely to work also for analytic
geometry.

Indeed, the field of complex analysis/geometry is much
advanced since 1960 and should have many topos implications and clarifications.
For example, the relation between the Grauert direct image theorem as a
relativization of its special case by Cartan-Serre should be clarified by
explicit topos-relativization. When I proposed that to Grothendieck, he allowed
that it is interesting, but pleaded insufficient expertise in logic to carry
out a proof. More recently, the study of Brady-hyperbolic spaces has a very
strong topos flavor that has not yet been made explicit (as far as I know).

Grothendieck made an important contribution to what he called
'tame topology'. He gave no general definition, but urged (as I had in my 1977
Milan lectures) the discovery of suitable categories that would not contain
certain old pathologies that come up again in cohomology. In my view objects
such as space-filling curves should have led to a 'criticism of foundations'
more central than the so-called paradoxes; however, they were apparently simply
tolerated for many decades, with the resignation that complication is
inevitable. But Grothendieck boldly proposed using accumulated knowledge to
construct less pathological categories that would still suffice for
mathematical work. He arrived at a proposal involving piecewise real-analytic
functions. Meanwhile, logicians including Wilkie, Pillay, MacIntyre, and van
den Dries, had been pursuing a related problem of Tarski, phrased in terms of
decidability. They solved it in 1986, also finding 'piecewise real analytic' to
be a key ingredient, although by no means the only one. These logicians came to
recognize Grothendieck's work as being related to their own. It is to be
expected that conversely the work of their o-minimal school will illuminate the
deeper study of cohesive space.

The work of Grothendieck illuminated and advanced the work of
Cantor, Grassmann, Volterra, Hausdorff, Hurewicz, Galois, Kan, Eilenberg &
Mac Lane and inspired our whole community; it will continue to inspire and guide
the work of future generations.