F. William Lawvere - What are Foundations of Geometry and Algebra? (2013)

Keynote lecture at the Fifty Years of Functorial Semantics conference, Union College, October 2013.
http://www.math.union.edu/~niefiels/13conference/Web/

Transcript: http://www.math.union.edu/~niefiels/13conference/Web/Slides/Fifty_Years_of_Functorial_Semantics.pdf

Abstract
From observation of, and participation in, the ongoing actual practice of Mathematics, Decisive Abstract
General Relations (DAGRs) can be extracted; when they are made explicit, these DAGRs become a
guide to further rational practice of mathematics. The worry that these DAGRs may turn out to be as
numerous as the specific mathematical facts themselves is overcome by viewing the ensemble of DAGRs as
a ’Foundation’, expressed as a single algebraic system whose current description can be finitely-presented.
The category of categories (as a cartesian closed category with an object of small discrete categories) aims
to serve as such a Foundation. One basic DAGR is the contrast between space and quantity, and especially
the relation between the two that is expressed by the role of spaces as domains of variation for intensively
and extensively variable quantity; in that way, the foundational aspects of cohesive space and variable
quantity inherently includes also the conceptual basis for analysis, both for functional analysis and for
the transformation from continuous cohesion to combinatorial semi-discreteness via abstract homotopy
theory. Function spaces embody a pervasive DAGR.
The year 1960 was a turning point. Kan, Isbell, Grothendieck and Yoneda had further developed the
Eilenberg-Mac Lane Theory of Naturality. Their work implicitly pointed towards such a Foundation as
a foreseeable goal. Although the work of those four great mathematicians was still unknown to me, I
had independently traversed a sufficient fragment of a similar path to encourage me to become a student
of Professor Eilenberg. As I slowly became aware of the importance of those earlier developments, I
attempted to participate in the realization of a Foundation in the sense described above, first through
concentration on the particular docrine known as Universal Algebra, making explicit the fibered category
whose base consists of abstract generals (called theories) and whose fibers are concrete generals (known
as algebraic categories). The term ’Functorial Semantics’ simply refers to the fact that in such a fibered
category, any interpretation T
0 → T of theories induces a map in the opposite direction between the two
categories of concrete meanings; this is a direct generalization of the previously observed cases of linear
algebra, where the abstract generals are rings and the fibers consist of modules, and of group theory
where the dialectic between abstract groups and their actions had long been fundamental in practice.
This kind of fibration is special, because the objects T in the base are themselves categories, as I had
noticed after first rediscovering the notion of clone, but then rejecting the latter on the basis of the
principle that, to compare two things, one must first make sure that they are in the same category; when
the two are (a) a theory and (b) a background category in which it is to be interpreted, comparisons being
models., the category of categories with products serves. Left adjoints to the re-interpretation functors
between fibers exist in this particular doctrine of general concepts, unifying a large number of classical
and new constructions of algebra. Isbell conjugacy can provide a first approximation to the general space
vs quantity pseudo-duality, because recent developments (KIGY) had shown that also spaces themselves
are determined by categories (of figures and incidence relations inside them).
My 1963 thesis clearly explains that presentations (having a signature consisting of names for generators
and another signature consisting of names for equational axioms) constitute one important source of
theories. This syntactical left adjoint directly generalizes the presentations known from elimination
theory in linear algebra and from word problems in group theory. No one would confuse rings and groups
themselves with their various syntactical presentations, but previous foundations of algebra had underemphasized
the existence of another important method for constructing examples, namely the Algebraic
Structure functor. Being a left adjoint , it can be calculated as a colimit over finite graphs. Fundamental
examples, like cohomology operations as studied by the heroes of the 50’s, show that typically an abstract
general (such as an isometry group) arises by naturality; to find a syntactical presentation for it may then
be an important question. This extraction, by naturality from a particular family of cases, provides much
finer invariants, and as a process bears a profound resemblance to the basic extraction of abstract generals
from experience.

Видео F. William Lawvere - What are Foundations of Geometry and Algebra? (2013) канала Matt Earnshaw