Guide Topoi (collected papers)

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The setting of the Neue Nationalgalerie in West Berlin was arguably the most labyrinthine place of them all in the era, easy to associate with walls and no exits. The context of the Cold War division of Berlin, in which traditional museums were located in East Berlin, called for a Western response. Here, a modernist Gesamtkunstwerk —in form and content a stark contrast to state socialism—placed close to the Berlin Wall in the Kulturforum ensemble of cultural institutions, was a profound statement.

Exhibition view of Piet Mondrian, Neue Nationalgalerie The building design was based on a two-story structure of an underground hall for the permanent collection and an overground hall for special exhibitions, under the giant gridded steel roof. This was a delineation of space that made it as much a Kunsthalle for changing exhibitions as a traditional, collection-based museum. The enclosed lower floor appeared as a 58 x 58 meter base for the twentieth century and the extrovert upper floor with its glass outer walls a showcase for modern art. In the 8. The experimental totality of the exhibition space, which exceeded the conventional proportions of the gallery space, made an overwhelming impression on the contemporary viewer, who often experienced it all as a total work of art.

The building presented itself as an autonomous artwork and a monument to modernism, rather than as a functioning museum machine, as Buddensieg has stated. This image provides a stage—almost transparent—on which the homelessness and nihilism so central to the experience of modernity can be enacted as both a crisis and an opportunity for constructive self-fashioning.


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The museum of modern art in Berlin was the epitome of an international movement to create a new kind of museum defined in the vision of modernity. In that case the exhibition pioneered the modernist ideal of white walls and single-row hanging, and introduced international art such as French impressionism into the national museum. Werner Haftmann in front of Neue Nationalgalerie, Berlin, If the exhibition is an immanent rendition of the contemporary condition, this contemporaneity must be commented upon.

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Between contemporary fears and utopian futures, it was up to oneself to find a way through the labyrinth of modern art. However, it is also important to compare with the less-remembered events, like the Darmstadt exhibition, to show the potentialities of this seminal era in modern exhibition-making and draw parallels the newly built environments like Neue Nationalgalerie, which might test the reach of the concept labyrinthine, but also relate to the ideas about modern art and its presentation.

Idee und Institution , ed. Martin Schneckenburger Munich: Bruckmann, , Sie ist der erste Modellfall von Weltkultur. Knud W.

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Translation by the author. Tauris , 37— Darmstadt: , 8. Darmstadt: , 77—97, Die Kunst des Mir wurde die enge Beziehung zwischen Architektur und Zivilisation klar. Und dass sie — in ihrer Vollendung — ein Ausdruck der innersten Struktur ihrer Zeit sein kann. Paula Marincola London: Reaktion Books, , This is ultimately because a -algebra is closed under all Boolean operations, whereas a topology is only closed under union and intersection.

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Example 1 Discrete case A simple case arises when is a discrete space which is at most countable. If we assign a set to each , with a singleton if.

One then sets , with the obvious restriction maps, giving rise to a stochastic set. Thus, a local element of can be viewed as a map on that takes values in for each.

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Conversely, it is not difficult to see that any stochastic set over an at most countable discrete probability space is of this form up to isomorphism. In this case, one can think of as a bundle of sets over each point of positive probability in the base space.

Note that we permit some of the to be empty, thus it can be possible for to be empty whilst for some strict subevents of to be non-empty. This is analogous to how it is possible for a sheaf to have local sections but no global sections. As such, the space of global elements does not completely determine the stochastic set ; one sometimes needs to localise to an event in order to see the full structure of such a set. Thus it is important to distinguish between a stochastic set and its space of global elements.

Example 2 Measurable spaces as stochastic sets Returning now to a general base space , any deterministic measurable space gives rise to a stochastic set , with being defined as in previous discussion as the measurable functions from to modulo almost everywhere equivalence in particular, a singleton set when is null , with the usual restriction maps. The constraint of measurability on the maps , together with the quotienting by almost sure equivalence, means that is now more complicated than a plain Cartesian product of fibres, but this still serves as a useful first approximation to what is for the purposes of developing intuition.

Indeed, the measurability constraint is so weak as compared for instance to topological or smooth constraints in other contexts, such as sheaves of continuous or smooth sections of bundles that the intuition of essentially independent fibres is quite an accurate one, at least if one avoids consideration of an uncountable number of objects simultaneously.

Example 3 Extended Hilbert modules This example is the one that motivated this post for me. Suppose that one has an extension of the base space , thus we have a measurable factor map such that the pushforward of the measure by is equal to. Then we have a conditional expectation operator , defined as the adjoint of the pullback map. As is well known, the conditional expectation operator also extends to a contraction ; by monotone convergence we may also extend to a map from measurable functions from to the extended non-negative reals , to measurable functions from to.

This is an extended version of the Hilbert module , which is defined similarly except that is required to lie in ; this is a Hilbert module over which is of particular importance in the Furstenberg-Zimmer structure theory of measure-preserving systems.


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We can then define the stochastic set by setting. In the case that are standard Borel spaces, one can disintegrate as an integral of probability measures supported in the fibre , in which case this stochastic set can be viewed as having fibres though if is not discrete, there are still some measurability conditions in on the local and global elements that need to be imposed. However, I am interested in the case when are not standard Borel spaces in fact, I will take them to be algebraic probability spaces, as defined in this previous post , in which case disintegrations are not available.

However, it appears that the stochastic analysis developed in this blog post can serve as a substitute for the tool of disintegration in this context. We make the remark that if is a stochastic set and are events that are equivalent up to null events, then one can identify with through their common restriction to , with the restriction maps now being bijections. As such, the notion of a stochastic set does not require the full structure of a concrete probability space ; one could also have defined the notion using only the abstract -algebra consisting of modulo null events as the base space, or equivalently one could define stochastic sets over the algebraic probability spaces defined in this previous post.

However, we will stick with the classical formalism of concrete probability spaces here so as to keep the notation reasonably familiar. As a corollary of the above observation, we see that if the base space has total measure , then all stochastic sets are trivial they are just points. Exercise 2 If is a stochastic set, show that there exists an event with the property that for any event , is non-empty if and only if is contained in modulo null events.

In particular, is unique up to null events. Hint: consider the numbers for ranging over all events with non-empty, and form a maximising sequence for these numbers.

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Then use all three axioms of a stochastic set. One can now start take many of the fundamental objects, operations, and results in set theory and, hence, in most other categories of mathematics and establish analogues relative to a finite measure space. Implicitly, what we will be doing in the next few paragraphs is endowing the category of stochastic sets with the structure of an elementary topos. However, to keep things reasonably concrete, we will not explicitly emphasise the topos-theoretic formalism here, although it is certainly lurking in the background.

Firstly, we define a stochastic function between two stochastic sets to be a collection of maps for each which form a natural transformation in the sense that for all and nested events. In the case when is discrete and at most countable and after deleting all null points , a stochastic function is nothing more than a collection of functions for each , with the function then being a direct sum of the factor functions :. Thus in the discrete, at most countable setting, at least stochastic functions do not mix together information from different states in a sample space; the value of at depends only on the value of at.

One can now form the stochastic set of functions from to , by setting for any event to be the set of local stochastic functions of the localisations of to ; this is a stochastic set if we use the obvious restriction maps. In the case when is discrete and at most countable, the fibre at a point of positive measure is simply the set of functions from to.

In a similar spirit, we say that one stochastic set is a stochastic subset of another , and write , if we have a stochastic inclusion map, thus for all events , with the restriction maps being compatible. We can then define the power set of a stochastic set by setting for any event to be the set of all stochastic subsets of relative to ; it is easy to see that is a stochastic set with the obvious restriction maps one can also identify with in the obvious fashion. Again, when is discrete and at most countable, the fibre of at a point of positive measure is simply the deterministic power set.

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Note that if is a stochastic function and is a stochastic subset of , then the inverse image , defined by setting for any event to be the set of those with , is a stochastic subset of. In particular, given a -ary relation , the inverse image is a stochastic subset of , which by abuse of notation we denote as. In a similar spirit, if is a stochastic subset of and is a stochastic function, we can define the image by setting to be the set of those with ; one easily verifies that this is a stochastic subset of.

Remark 2 One should caution that in the definition of the subset relation , it is important that for all events , not just the global event ; in particular, just because a stochastic set has no global sections, does not mean that it is contained in the stochastic empty set. Now we discuss Boolean operations on stochastic subsets of a given stochastic set.

Given two stochastic subsets of , the stochastic intersection is defined by setting to be the set of that lie in both and :.