Badiou criticises the proper form of intuition associated with multiplicities such as space and time. However, his own ’intuitions’ are constrained by set theory. His intuition is therefore as ‘transitory’ as is the ontology in terms of which it is expressed. Following this constrained line of reasoning, however, let me now discuss how Badiou encounters the question of ‘atoms’ and materiality: in terms of the so called ‘atomic’ T-sets.
If topos theory designates the subobject-classifier Ω relationally, the external, set-theoretic T-form reduces the classificatory question again into the incorporeal framework. There is a set-theoretical, explicit order-structure (T,<) contra the more abstract relation 1 → Ω pertinent to categorical topos theory. Atoms then appear in terms of this operator <: the ‘transcendental grading’ that provides the ‘unity through which all the manifold given in an intuition is united in a concept of the object’.
Formally, in terms of an external Heyting algebra this comes down to an entity (A,Id) where A is a set and Id : A → T is a function satisfying specific conditions.
Equaliser: First, there is an ‘equaliser’ to which Badiou refers as the ‘identity’ Id : A × A → T satisfies two conditions:
1) symmetry: Id(x, y) = Id(y, x) and
2) transitivity: Id(x, y) ∧ Id(y, z) ≤ Id(x, z).
They guarantee that the resulting ‘quasi-object’ is objective in the sense of being distinguished from the gaze of the ‘subject’: ‘the differences in degree of appearance are not prescribed by the exteriority of the gaze’.
This analogous ‘identity’-function actually relates to the structural equalization-procedure as appears in category theory. Identities can be structurally understood as equivalence-relations. Given two arrows X ⇒ Y , an equaliser (which always exists in a topos, given the existence of the subobject classifier Ω) is an object Z → X such that both induced maps Z → Y are the same. Given a topos-theoretic object X and U, pairs of elements of X over U can be compared or ‘equivalized’ by a morphism XU × XU →eq ΩU structurally ‘internalising’ the synthetic notion of ‘equality’ between two U-elements. Now it is possible to formulate the cumbersome notion of the ‘atom of appearing’.
An atom is a function a : A → T defined on a T -set (A, Id) so that
(A1) a(x) ∧ Id(x, y) ≤ a(y) and
(A2) a(x) ∧ a(y) ≤ Id(x, y).
As expressed in Badiou’s own vocabulary, an atom can be defined as an ‘object-component which, intuitively, has at most one element in the following sense: if there is an element of A about which it can be said that it belongs absolutely to the component, then there is only one. This means that every other element that belongs to the component absolutely is identical, within appearing, to the first’. These two properties in the definition of an atom is highly motivated by the theory of T-sets (or Ω-sets in the standard terminology of topological logic). A map A → T satisfying the first inequality is usually thought as a ‘subobject’ of A, or formally a T-subset of A. The idea is that, given a T-subset B ⊂ A, we can consider the function
IdB(x) := a(x) = Σ{Id(x,y) | y ∈ B}
and it is easy to verify that the first condition is satisfied. In the opposite direction, for a map a satisfying the first condition, the subset
B = {x | a(x) = Ex := Id(x, x)}
is clearly a T-subset of A.
The second condition states that the subobject a : A → T is a singleton. This concept stems from the topos-theoretic internalization of the singleton-function {·} : a → {a} which determines a particular class of T-subsets of A that correspond to the atomic T-subsets. For example, in the case of an ordinary set S and an element s ∈ S the singleton {s} ⊂ S is a particular, atomic type of subset of S.
The question of ‘elements’ incorporated by an object can thus be expressed externally in Badiou’s local theory but ‘internally’ in any elementary topos. For the same reason, there are two ways for an element to be ‘atomic’: in the first sense an ‘element depends solely on the pure (mathematical) thinking of the multiple’, whereas the second sense relates it ‘to its transcendental indexing’. In topos theory, the distinction is slightly more cumbersome. Badiou still requires a further definition in order to state the ‘postulate of materialism’.
An atom a : A → T is real if if there exists an element x ∈ T so that a(y) = Id(x,y) ∀ y ∈ A.
This definition gives rise to the postulate inherent to Badiou’s understanding of ‘democratic materialism’.
Postulate of Materialism: In a T-set (A,Id), every atom of appearance is real.
What the postulate designates is that there really needs to exist s ∈ A for every suitable subset that structurally (read categorically) appears to serve same relations as the singleton {s}. In other words, what ‘appears’ materially, according to the postulate, has to ‘be’ in the set-theoretic, incorporeal sense of ‘ontology’. Topos theoretically this formulation relates to the so called axiom of support generators (SG), which states that the terminal object 1 of the underlying topos is a generator. This means that the so called global elements, elements of the form 1 → X, are enough to determine any particular object X.
Thus, it is this specific condition (support generators) that is assumed by Badiou’s notion of the ‘unity’ or ‘constitution’ of ‘objects’. In particular this makes him cross the line – the one that Kant drew when he asked Quid juris? or ’Haven’t you crossed the limit?’ as Badiou translates.
But even without assuming the postulate itself, that is, when considering a weaker category of T-sets not required to fulfill the postulate of atomism, the category of quasi-T -sets has a functor taking any quasi-T-set A into the corresponding quasi-T-set of singletons SA by x → {x}, where SA ⊂ PA and PA is the quasi-T-set of all quasi-T-subsets, that is, all maps T → A satisfying the first one of the two conditions of an atom designated by Badiou. It can then be shown that, in fact, SA itself is a sheaf whose all atoms are ‘real’ and which then is a proper T-set satisfying the ‘postulate of materialism’. In fact, the category of T-Sets is equivalent to the category of T-sheaves Shvs(T, J). In the language of T-sets, the ‘postulate of materialism’ thus comes down to designating an equality between A and its completed set of singletons SA.
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