History of topos theory

Source: Wikipedia, the free encyclopedia.

This article gives some very general background to the

mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. [citation needed
]

In the school of Grothendieck

During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of étale cohomology.

With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems for a long time. The first was to do with its points: back in the days of projective geometry it was clear that the absence of 'enough' points on an algebraic variety was a barrier to having a good geometric theory (in which it was somewhat like a compact manifold). There was also the difficulty, that was clear as soon as topology took form in the first half of the twentieth century, that the topology of algebraic varieties had 'too few' open sets.

The question of points was close to resolution by 1950; Alexander Grothendieck took a sweeping step (invoking the Yoneda lemma) that disposed of it—naturally at a cost, that every variety or more general scheme should become a functor. It wasn't possible to add open sets, though. The way forward was otherwise.

The topos definition first appeared somewhat obliquely, in or about 1960. General problems of so-called '

comonads
; here we can see one way in which the Grothendieck school bifurcates in its approach from the 'pure' category theorists, a theme that is important for the understanding of how the topos concept was later treated.

There was perhaps a more direct route available: the

Tôhoku paper
).

Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and

Grothendieck topos. The theory was rounded out by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning, since it involved a Grothendieck topology
.

The idea of a Grothendieck topology (also known as a site) has been characterised by

flat cohomology and crystalline cohomology). At this point—about 1964—the developments powered by algebraic geometry had largely run their course. The 'open set' discussion had effectively been summed up in the conclusion that varieties had a rich enough site of open sets in unramified covers of their (ordinary) Zariski-open sets
.

From pure category theory to categorical logic

Further information: categorical logic

The current definition of topos goes back to William Lawvere and Myles Tierney. While the timing follows closely on from that described above, as a matter of history, the attitude is different, and the definition is more inclusive. That is, there are examples of toposes that are not a Grothendieck topos. What is more, these may be of interest for a number of logical disciplines.

Lawvere and Tierney's definition picks out the central role in topos theory of the

truth-values
, true and false. It is almost tautologous to say that the subsets of a given set X are the same as (just as good as) the functions on X to any such given two-element set: fix the 'first' element and make a subset Y correspond to the function sending Y there and its complement in X to the other element.

Now sub-object classifiers can be found in sheaf theory. Still tautologously, though certainly more abstractly, for a topological space X there is a direct description of a sheaf on X that plays the role with respect to all sheaves of sets on X. Its set of sections over an open set U of X is just the set of open subsets of U. The space associated with a sheaf, for it, is more difficult to describe.

Lawvere and Tierney therefore formulated axioms for a topos that assumed a sub-object classifier, and some limit conditions (to make a

cartesian-closed category
, at least). For a while this notion of topos was called 'elementary topos'.

Once the idea of a connection with logic was formulated, there were several developments 'testing' the new theory:

Position of topos theory

There was some irony that in the pushing through of

Kripke–Joyal semantics
, is a good match. On the other hand Brouwer's long efforts on 'species', as he called the intuitionistic theory of reals, are presumably in some way subsumed and deprived of status beyond the historical. There is a theory of the real numbers in each topos, and so no one master intuitionist theory.

The later work on étale cohomology has tended to suggest that the full, general topos theory isn't required. On the other hand, other sites are used, and the Grothendieck topos has taken its place within homological algebra.

The Lawvere programme was to write

constructive mathematics; but in fact this can be read as foundational computer science
(which is not mentioned). If one wants to discuss set-theoretic operations, such as the formation of the image (range) of a function, a topos is guaranteed to be able to express this, entirely constructively.

It also produced a more accessible spin-off in

locale concept isolates some insights found by treating topos as a significant development of topological space. The slogan is 'points come later': this brings discussion full circle on this page. The point of view is written up in Peter Johnstone's Stone Spaces, which has been called by a leader in the field of computer science 'a treatise on extensionality'. The extensional is treated in mathematics as ambient—it is not something about which mathematicians really expect to have a theory. Perhaps this is why topos theory has been treated as an oddity; it goes beyond what the traditionally geometric way of thinking allows. The needs of thoroughly intensional theories such as untyped lambda calculus have been met in denotational semantics
. Topos theory has long looked like a possible 'master theory' in this area.

Summary

The topos concept arose in algebraic geometry, as a consequence of combining the concept of sheaf and closure under categorical operations. It plays a certain definite role in cohomology theories. A 'killer application' is étale cohomology.

The subsequent developments associated with logic are more interdisciplinary. They include examples drawing on homotopy theory (classifying toposes). They involve links between category theory and mathematical logic, and also (as a high-level, organisational discussion) between category theory and theoretical computer science based on type theory. Granted the general view of Saunders Mac Lane about ubiquity of concepts, this gives them a definite status. The use of toposes as unifying bridges in mathematics has been pioneered by Olivia Caramello in her 2017 book.[1]

References

  1. ISBN 9780198758914
    .
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