Let us focus on the more abstract, elementary definition of a topos and discuss materiality in the categorical context. The materiality of being can, indeed, be defined in a way that makes no material reference to the category of Sets itself.
The stakes between being and materiality are thus reverted. From this point of view, a Grothendieck-topos is not one of sheaves over sets but, instead, it is a topos which is not defined based on a specific geometric morphism E → Sets – a materialization – but rather a one for which such a materialization exists only when the topos itself is already intervened by an explicitly given topos similar to Sets. Therefore, there is no need to start with set-theoretic structures like sieves or Badiou’s ‘generic’ filters.
Strong Postulate, Categorical Version: For a given materialization the situation E is faithful to the atomic situation of truth (Setsγ∗(Ω)op) if the materialization morphism itself is bounded and thus logical.
In particular, this alternative definition suggests that materiality itself is not inevitably a logical question. Therefore, for this definition to make sense, let us look at the question of materiality from a more abstract point of view: what are topoi or ‘places’ of reason that are not necessarily material or where the question of materiality differs from that defined against the ‘Platonic’ world of Sets? Can we deploy the question of materiality without making any reference – direct or sheaf-theoretic – to the question of what the objects ‘consist of’, that is, can we think about materiality without crossing Kant’s categorical limit of the object? Elementary theory suggests that we can.
Elementary Topos: An elementary topos E is a category which
- has finite limits, or equivalently E has so called pull-backs and a terminal object 1,
- is Cartesian closed, which means that for each object X there is an exponential functor (−)X : E → E which is right adjoint to the functor (−) × X, and finally
- axiom of truth E retains an object called the subobject classifier Ω, which is equipped with an arrow 1 →true Ω such that for each monomorphism σ : Y ֒→ X in E, there is a unique classifying map φσ : X → Ω making σ : Y ֒→ X a pull-back of φσ along the arrow true.
Grothendieck-topos: In respect to this categorical definition, a Grothendieck-topos is a topos with the following conditions satisfies:
(1) E has all set-indexed coproducts, and they are disjoint and universal,
(2) equivalence relations in E have universal co-equalisers,
(3) every equivalence relation in E is effective, and every epimorphism in E is a coequaliser,
(4) E has ‘small hom-sets’, i.e. for any two objects X, Y , the morphisms of E from X to Y are parametrized by a set, and finally
(5) E has a set of generators (not necessarily monic in respect to 1 as in the case of locales).
Together the five conditions can be taken as an alternative definition of a Grothendieck-topos. We should still demonstrate that Badiou’s world of T-sets is actually the category of sheaves Shvs (T, J) and that it will, consequentially, hold up to those conditions of a topos listed above. To shift to the categorical setting, one first needs to define a relation between objects. These relations, the so called ‘natural transformations’ we encountered in relation Yoneda lemma, should satisfy conditions Badiou regards as ‘complex arrangements’.
Relation: A relation from the object (A, Idα) to the object (B,Idβ) is a map ρ : A → B such that
Eβ ρ(a) = Eα a and ρ(a / p) = ρ(a) / p.
It is a rather easy consequence of these two pre-suppositions that it respects the order relation ≤ one retains Idα (a, b) ≤ Idβ (ρ(a), ρ(b)) and that if a‡b are two compatible elements, then also ρ(a)‡ρ(b). Thus such a relation itself is compatible with the underlying T-structures.
Given these definitions, regardless of Badiou’s confusion about the structure of the ‘power-object’, it is safe to assume that Badiou has demonstrated that there is at least a category of T-Sets if not yet a topos. Its objects are defined as T-sets situated in the ‘world m’ together with their respective equalization functions Idα. It is obviously Badiou’s ‘diagrammatic’ aim to demonstrate that this category is a topos and, ultimately, to reduce any ‘diagrammatic’ claim of ‘democratic materialism’ to the constituted, non-diagrammatic objects such as T-sets. That is, by showing that the particular set of objects is a categorical makes him assume that every category should take a similar form: a classical mistake of reasoning referred to as affirming the consequent.