Promenade 13

Toposes or The Double Bed

The new perspective and language introduced by the use of Leray's concepts of sheaves has led us to consider every kind of "space" and "variety" in a new light.These did not however have anything to say about the concept of space itself, and was content if it enabled us to refine our understanding of the already traditional and familiar "spaces". At the same time it was recognized that this way of looking at space was insufficient for taking into account the "topological invariants" which were most essential for expression the "form" of these "abstract algebraic varieties" ( such as those which figure in the Weil Conjectures), let alone that of general "schemes" ( for the most part the classical varieties). For the desired "marriage" of "Number and Magnitude" one would have a rather narrow bed, one in which at most one of the future spouses ( for example, the bride) could accomodate herself for better or worse, but never both at the same time! The "new principle" that needed to be found so that the marriage announced by the guardian spirits could be consummated, was simply that missing spacious bed, though nobody at the time suspected it.

This "double bed" arrived ( as from the wave of a magic wand) with the idea of the topos. This idea encapsulates, in a single topological intuition, both the traditional topological spaces, incarnation of the world of the continuous quantity, and the so-called "spaces" ( or "varieties") of the unrepentant abstract algebraic geometers, and a huge number of other sorts of structures which until that moment had appeared to belong irrevocably to the "arithmetic world" of "discontinuous" or "discrete" aggregates.

It was certainly the sheaf perspective that was my sure and quiet guide, the right key ( hardly secret) to lead me without detours nor procrastination towards the nuptial chamber and its vast conjugal bed. A bed so enormous in fact ( like a vast, deep and peaceful stream) in which

"Tous Les Chevaux du Roi
Y pourraient boire ensemble"

- as the old ballad that you must surely have heard or sung at one point tells us.

And he who was the first to sing it was he who has best savored the secret beauty and passive force of the topos, better than any of my clever students and former friends ....

It was the same key, both in the initial and provisional approach via the convenient, yet unintrinsic, concept of a "site" , as with the topos. I will now attempt to describe the topos concept.

Consider the set formed by all sheaves over a (given) topological space or, if you like, the formidable arsenal of all the "rulers" that can be used in taking measurements on it. (*)


(*)( For the mathematician): properly speaking, one is speaking of sheaves of ensembles, not the Abelian sheaves introduced by Leray as generalized coefficients in the formation of "cohomology groups" I believe that I'm the first person to have worked systematically with sheaves of ensembles ( starting in 1955 at the University of Kansas, with my article "A general theory of fibre spaces with structure sheaf")

We will treat this "ensemble", or "arsenal" as one equipped with a structure that may be considered "self-evident", one that crops up "in front of one's nose": that is to say, a Categorical structure. ( Let not the non-mathematical reader trouble himself if he's unaware of the technical meanings of these terms, which will not be needed for what follows).

It functions as a kind of "superstructure of measurement", called the "Category of Sheaves" ( over the given space), which henceforth shall be taken to incoorporate all that is most essential about that space. This is in all respects a lawful procedure, ( in terms of "mathematical common sense") because it turns out that one can "reconstitute" in all respects, the topological space(**) by means of the associated "category of sheaves" ( or "arsenal" of measuring instruments)


(**) (For the mathematical reader) Strictly speaking, this is only true for so-called "tame" spaces. However these include virtually all of the spaces one has to deal with, notably the "separable spaces" so dear to functional analysts.

( The verification of this is a simple exercise- once someone thinks to pose the question, naturally) One needs nothing more ( if one feels the need for one reason or another), henceforth one can drop the initial space and only hold onto its associated "category" ( or its "arsenal"), which ought to be considered as the most complete incarnation of the "topological (or spatial) structure" which it exemplifies

As is often the case in mathematics, we've succeeded ( thanks to the crucial notion of a "sheaf" or "cohomological ruler") to express a certain idea ( that of a "space" in this instance), in terms of another one ( that of the "category"). Each time the discovery of such a translation from one notion (representing one kind of situation) to another (which corresponds to a different situation) enriches our understanding of both notions, owing to the unanticipated confluence of specific intuitions which relate first to one then to the other. Thus we see that a situation said to have a "topological" character ( embodied in some given space) has been translated into a situation whose character is "algebraic" (embodied in the category); or, if you wish, "continuity" ( as present in the space) finds itself "translated" or "expressed" by a categorical structure of an "algebraic" character, ( which until then had been understood only in terms of something "discrete" or "discontinuous".)

Yet there is more here. The first idea, that of the space, was perceived by us as a "maximal" thing - a notion already so general that one could hardly envisage any kind of "rational" extension to it. On the contrary, it has turned out that, on the other side of the mirror(*)


(*) The"mirror" refered to, as in Alice in Wonderland, is that which yields as the "image" of a space placed in front of it, the associated "category", considered as a kind of "double" of the space , on the "other side of the mirror(*)

these "categories", ( or "arsenals") one ends up with in dealing with topological spaces, are of a very particular character. Their collection of traits is in fact highly specific.(**), and tend to join up in patchwork combinations of an unbelievably simple nature- those which on can obtain by taking as one's point of departure the reduction of a space to a single point.


(**) (For the mathematical reader) We're speaking about primarily the properties which I introduced into Category Theory under the name of "exact characteristics", ( along with the categorical notions of general projective and inductive limits). See " On several points of homological algebra", Tohoku Math Journal, 1957 (p. 119-221))

Having said this, a "space defined in the new way" ( or topos) one that generalizes the traditional topological space, can be simply described as a "category" which, without necessarily deriving from an ordinary space, nevertheless possesses all of the good properties ( explicitely designated once and for all, naturally) of the "sheaf category".


This therefore is the new idea. Its appearance may perhaps be understood in the light of the observation, a childlike one at that, that what really counts in a topological space is neither its "points" nor its subsets of points.(*), nor the proximity relations between them, rather it is the sheaves on that space, and the category that they produce.
(*)Thus, one can actually construct "enormous" topoi with only a single point, or without any points at all!
All that I've done was to draw out the ultimate consequences of the initial notion of Leray - and by doing so, lead the way .

As even the idea of sheaves (due to Leray), or that of schemes, as with all grand ideas that overthrow the established vision of things, the idea of the topos had everything one could hope to cause a disturbance, primarily through its "self-evident" naturalness, through its simplicity ( at the limit naive, simple-minded, "infantile")- through that special quality which so often makes us cry out: "Oh, that's all there is to it!", in a tone mixing betrayal with envy, that innuendo of the "extravagant", the "frivolous", that one reserves for all things that are unsettling by their unforseen simplicity, causing us to recall, perhaps, the long buried days of our infancy....

Promenade 14

Mutability of the concept of space- or Breath and Faith


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