Background and genesis of the theory of moles

This page broadly presents the mathematical idea of moles. This is a branch of category theory, and it has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but the context can be indicated. This will be done partly in terms of historical development, but also to some degree by giving an account of, admittedly different, attitudes towards category theory.

Grothendieck School

During the latter part of the 1950s, the fundamentals of algebraic geometry were being rewritten; and it is here that the origins of the concept of moles must be found. At the time, Weil's conjectures were an exceptional motivation for research. As we now know, the route to proof of it, and other advances, lies in the construction of etal cohomology.

With the advantage of hindsight, it can be said that algebraic geometry had been struggling with two problems, for a long time. The first had to do with points: in the old days of projective geometry it was clear that the absence of enough points on an algebraic manifold was a barrier to obtaining a good geometric theory (in which there was something as a compact variety). There was also the difficulty, which became apparent as soon as topology took shape in the first half of the twentieth century, that the topology of algebraic varieties had too few open sets.

The point question was nearing resolution around 1950 when Grothendieck took an approach that (appealing to Yoneda's lemma) ended it - naturally at a cost, that each manifold or more general scheme had to become a functor. However, it was not possible to add open sets. The way to advance was another.

The definition of moles first appeared somewhat obliquely, in or about 1960. The general problems of the so-called 'descent' in algebraic geometry they were considered, in the same period when the fundamental group was generalized to the context of algebraic geometry (as a profinite group). to the light of further work, 'descent' it is part of the theory of comonads; here we can see the way in which the Grothendieck school diverges in its approach from the 'pure' of category, a subject that will be important for the understanding of how the concept of topos was treated later.

There was perhaps a more direct route available: the concept of an abelian category had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of bundles of abelian groups, and of modules. An abelian category must be closed under certain categorical-theoretic operations - using this kind of definition one can focus entirely on the structure, without saying anything about the nature of the objects involved. This type of definition can perhaps be traced back to the concept of lattice. It was an acceptable question, around 1957, about a similar purely categorical-theoretic characterization, about categories of bundles of sets.

This definition was eventually given (angl.), around 1962, by Grothendieck and Verdier (see Verdier's Bourbaki seminar: Situs Analysis). The characterization was made by means of categories 'with sufficient colimits', and applied to what is now called a Grothendieck topos. The theory was completed, establishing that Grothendieck's moles were a category of bundles, where now the word 'bundle' means 'bundle'. it had acquired an extended meaning with respect to Grothendieck's idea of topology. (also called a site).

From pure category theory to categorical logic

The current definition of moles goes back to William Lawvere. While the timing closely follows that described above, as a matter of history, the attitude is different, and the definition is more inclusive. That is, there are examples of moles that are not Grothendieck moles. And, what's more, these may be of interest to a number of logical disciplines.

Lawvere's definition selects the subobject classifier as a central role in the theory of moles. In the usual category of sets, this is the set of two elements of Boolean truth values, true and false. It is almost tautological to say that the subsets of a given set X are the same as (as good as) the functions of X to any given set of two elements: set the first element and make a subset Y correspond to the function that sends Y to it and its complement in X to the other element.

Now, sub-object classifiers can be found within bundle theory. Still tautologically, though certainly more abstractly, for a topological space X there is a direct description of a bundle in X which plays such a role with respect to all bundles of sets in X. Indeed, in terms of the space associated with a bundle it is described as the union of disjoint copies of each of the open sets U of X. This maps to X by an obvious local homeomorphism: as the stack of all open subsets of X projecting onto it. the stem for x in X has a point for each U that contains x; so that this stem looks like the graph of the membership relation.

Lawvere then formulated axioms for a topos that assume a classifier of subobjects, and some boundary conditions (to make the category Cartesian-closed, at least). For a time this notion of moles was called elementary moles.

Once the idea of a connection to logic was formulated, there were several developments that tested the new theory:

• models of the theory of sets showing the independence of the hypothesis of the continuum
• recognition of the connection with the idea of the intuiistic logic about the existential quantifier
• combining these, the discussion of the intuiistic theory of real numbers, by models of beam.

Position of the theory of moles

There was a certain irony in that through David Hilbert's far-reaching program a natural home for the central ideas of intuitionistic logic was found: Hilbert had detested, not cordially, the Brouwer school. Existence as local existence in the bundle-theoretical sense is a good idea. On the other hand, Brouwer's long efforts on species, as he called the intuitionist theory of reals, are probably deprived of meaning beyond the historical one.

Later work on etal cohomology has tended to suggest that the complete, general theory of topos is not required. On the other hand, other sites are used, and Grothendieck moles have taken their place within homological algebra.

Lawvere's program was to write higher-order logic in terms of category theory. That this can be done cleanly is demonstrated by Lambek and Scott's treatment of the book. What results is essentially an intuitionist (ie constructivist) theory, and its content is clarified by the existence of free moles. This is set theory, in a broad sense, but also something that belongs to the realm of pure syntax. The structure of its subobject classifier is that of a Heyting algebra. To get a more classical set theory one needs to move up to Boolean algebra, a return to the case of two Boolean truth values. In that book, the talk is about constructivist mathematics; but in fact this can be read as foundational computing (although it is not mentioned). If one wishes to discuss ensemble operations, such as the formation of the image (range) of a function, in topos it is guaranteed to be able to express this, in an entirely constructive way.

It also made pointless topology more accessible, where the concept of local isolates some of the more accessible insights found by treating moles as a meaningful generalization of topological space. The motto is 'the points come later'. The point of view was prepared in Peter Johnstone's Stone Spaces, what has been called by a leader in the field of computing 'a treatise on extensionality'. The extensional is treated in mathematics as "environment" - is not something mathematicians really hope to have a theory about. Perhaps this is the reason why the theory of topos has been treated as an oddity; it goes beyond what the traditionally geometric mode of thought admits. The needs for in-depth theories of the intentional, eg untyped lambda calculus, have been resolved within denotational semantics. The Topos theory has been considered for a long time, as a possible base theory in this area.

Summary

The concept of moles appeared in algebraic geometry, as a consequence of combining the concept of bundle and lock under categorical operations. Playing a certain definite role in cohomology theories.

Subsequent developments associated with logic and are more interdisciplinary. They include examples based on homotopy theory. It involved relationships between category theory and mathematical logic, and also (as a high-level, organizational discussion) between category theory and type-theoretic theoretical computing. Granted the general vision of Saunders McLane on the ubiquity of concepts, they thus receive a defined status and as an effective use it is enough to remember etal cohomology.