Badiou’s Vain Platonizing, or How the World is a Topos? Note Quote.

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As regards the ‘logical completeness of the world’, we need to show that Badiou’s world of T-sets does indeed give rise to a topos.

Badiou’s world consisting of T-Sets – in other words pairs (A, Id) where Id : A × A → T satisfies the particular conditions in respect to the complete Heyting algebra structure of T—is ‘logically closed’, that is, it is an elementary topos. It thus encloses not only pull-backs but also the exponential functor. These make it possible for it to internalize a Badiou’s infinity arguments that operate on the power-functor and which can then be expressed from insde the situation despite its existential status.

We need to demonstrate that Badiou’s world is a topos. Rather than beginning from Badiou’s formalism of T -sets, we refer to the standard mathematical literature based on which T-sets can be regarded as sheaves over the particular Grothendieck-topology on the category T: there is a categorical equivalence between T-sets satisfying the ‘postulate of materialism’ and S hvs(T,J). The complications Badiou was caught up with while seeking to ‘Platonize’ the existence of a topos thus largely go in vain. We only need to show that Shvs(T,J) is a topos.

Consider the adjoint sheaf functor that always exists for the category of presheaves

Idα : SetsCop → Shvs(Cop,J)

, where J is the canonical topology. It then amounts to an equivalence of categories. Thus it suffices to replace this category by the one consisting of presheaves SetsTop. This argument works for any category C rather than the specific category related to an external complete Heyting algebra T. In the category of Sets define YX as the set of functions X → Y. Then in the category of presheaves,

SetsCopYX(U) ≅ Hom(hU,YX) ≅ Hom(hU × X,Y)

, where hU is the representable sheaf hU(V) = Hom(V,U). The adjunction on the right side needs to be shown to exist for all sheaves – not just the representable ones. The proof then follows by an argument based on categorically defined limits, which has an existence. It can also be verified directly that the presheaf YX is actually a sheaf. Finally, for the existence of the subobject-classifier ΩSetsCop, it can be defined as

ΩSetsCop(U) ≅ Hom(hU,Ω) ≅ {sub-presheaves of hU} ≅ {sieves on U}, or alternatively, for the category of proper sheaves Shvs(C,J), as

ΩShvs(C,J)(U) = {closed sieves on U}

Here it is worth reminding ourselves that the topology on T is defined by a basis K(p) = {Θ ⊂ T | ΣΘ = p}. Therefore, in the case of T-sets satisfying the strong ‘postulate of materialism’, Ω(p) consists of all sieves S (downward dense subsets) of T bounded by relation ΣS ≤ p. These sieves are further required to be closed. A sieve S with an envelope ΣS = s is closed if for any other r ≤ s, ie. for all r ≤ s, one has the implication

frs(S) ∈ J(r) ⇒ frs ∈ S

, where frs : r → s is the unique arrow in the poset category. In particular, since ΣS = s for the topology whose basis consists of territories on s, we have the equation 1s(S) = fss(S) = S ∈ J(s). Now the condition that the sieve is closed implies 1s ∈ S. This is only possible when S is the maximal sieve on s—namely it consists of all arrows r → s for r ≤ s. In such a case S itself is closed. Therefore, in this particular case

Ω(p)={↓(s)|s ≤ p} = {hs | s ≤ p}

This is indeed a sheaf whose all amalgamations are ‘real’ in the sense of Badiou’s postulate of materialism. Thus it retains a suitable T-structure. Let us assume now that we are given an object A, which is basically a functor and thus a T-graded family of subsets A(p). For there to exist a sub-functor B ֒→ A comes down to stating that B(p) ⊂ A(p) for each p ∈ T. For each q ≤ p, we also have an injection B(q) ֒→ B(p) compatible (through the subset-representation with respect to A) with the injections A(q) ֒→ B(q). For any given x ∈ A(p), we can now consider the set

φp(x) = {q | q ≤ p and x q ∈ B(q)}

This is a sieve on p because of the compatibility condition for injections, and it is furthermore closed since the map x → Σφp(x) is in fact an atom and thus retains a real representative b ∈ B. Then it turns out that φp(x) =↓ (Eb). We now possess a transformation of functors φ : A → Ω which is natural (diagrammatically compatible). But in such a case we know that B ֒→ A is in turn the pull-back along φ of the arrow true, which is equivalent to the category of T-Sets.

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