Grothendieck’s Abstract Homotopy Theory

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Let E be a Grothendieck topos (think of E as the category, Sh(X), of set valued sheaves on a space X). Within E, we can pick out a subcategory, C, of locally finite, locally constant objects in E. (If X is a space with E = Sh(X), C corresponds to those sheaves whose espace étale is a finite covering space of X.) Picking a base point in X generalises to picking a ‘fibre functor’ F : C → Setsfin, a functor satisfying various conditions implying that it is pro-representable. (If x0 ∈ X is a base point {x0} → X induces a ‘fibre functor’ Sh(X) → Sh{x0} ≅ Sets, by pullback.)

If F is pro-representable by P, then π1(E, F) is defined to be Aut(P), which is a profinite group. Grothendieck proves there is an equivalence of categories C ≃ π1(E) − Setsfin, the category of finite π1(E)-sets. If X is a locally nicely behaved space such as a CW-complex and E = Sh(X), then π1(E) is the profinite completion of π1(X). This profinite completion occurs only because Grothendieck considers locally finite objects. Without this restriction, a covering space Y of X would correspond to a π1(X) – set, Y′, but if Y is a finite covering of X then the homomorphism from π1(X) to the finite group of transformations of Y factors through the profinite completion of π1(X). This is defined by : if G is a group, Gˆ = lim(G/H : H ◅ G, H of finite index) is its profinite completion. This idea of using covering spaces or their analogue in E raises several important points:

a) These are homotopy theoretic results, but no paths are used. The argument involving sheaf theory, the theory of (pro)representable functors, etc., is of a purely categorical nature. This means it is applicable to spaces where the use of paths, and other homotopies is impossible because of bad (or unknown) local properties. Such spaces have been studied within Shape Theory and Strong Shape Theory, although not by using Grothendieck’s fundamental group, nor using sheaf theory.

b) As no paths are used, these methods can also be applied to non-spaces, e.g. locales and possibly to their non-commutative analogues, quantales. For instance, classically one could consider a field k and an algebraic closure K of k and then choose C to be a category of étale algebras over k, in such a way that π1(E) ≅ Gal(K/k), the Galois group of k. It, in fact, leads to a classification theorem for Grothendieck toposes. From this viewpoint, low dimensional homotopy theory is ssen as being part of Galois theory, or vice versa.

c) This underlines the fact that π1(X) classifies covering spaces – but for i > 1, πi(X) does not seem to classify anything other than maps from Si into X!

This is abstract homotopy theory par excellence.

Derived Tensor Product via Resolutions by Complexes of Flat Modules (Part 1)

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Let U be a topological space, O a sheaf of commutative rings, and A the abelian category of (sheaves of) O-modules. The standard theory of the derived tensor product, via resolutions by complexes of flat modules, applies to complexes in D(A).

A complex P ∈ K(A) is q-flat if for every quasi-isomorphism Q1 → Q2 in K(A), the resulting map P ⊗ Q1 → P ⊗ Q2 is also a quasi-isomorphism; or equivalently, if for every exact complex Q ∈ K(A), the complex P ⊗ Q is also exact.

P ∈ K(A) is q-flat iff for each point x ∈ U, the stalk Px is q-flat in K(Ax), where Ax is the category of modules over the ring Ox. (In verifying this statement, note that an exact Ox-complex Qx is the stalk at x of the exact O-complex Q which associates Qx to those open subsets of U which contain x, and 0 to those which don’t.)

For instance, a complex P which vanishes in all degrees but one (say n) is q-flat iff tensoring with the degree n component Pn is an exact functor in the category of O-modules, i.e., Pn is a flat O-module, i.e., for each x ∈ U, Pxn is a flat Ox-module.

A q-flat resolution of an A-complex C is a quasi-isomorphism P → C where P is q-flat. The totality of such resolutions (with variable P and C) is the class of objects of a category, whose morphisms are the obvious ones.

Every A-complex C is the target of a quasi-isomorphism ψC from a q-flat complex PC, which can be constructed to depend functorially on C, and so that PC[1] = PC[1] and ψC[1] = ψC[1].

Every O-module is a quotient of a flat one; in fact there exists a functor P0 from A to its full subcategory of flat O-modules, together with a functorial epimorphism P0(F) ։ F (F ∈ A). Indeed, for any open V ⊂ U let OV be the extension of O|V by zero, (i.e., the sheaf associated to the presheaf taking an open W to O(W) if W ⊂ V and to 0 otherwise), so that OV is flat,its stalk at x ∈ U being Ox if x ∈ V and 0 otherwise. There is a canonical isomorphism

ψ : F (V) → Hom (OV, F) (F ∈ A)

such that ψ(λ) takes 1 ∈ OV(V) to λ. With Oλ := OV for each λ ∈ F(V),

the maps ψ(λ) define an epimorphism, with flat source,

P0(F) := (⊕V openλ∈F(V) Oλ) → F,

and this epimorphism depends functorially on F.

We deduce then, for each F, a functorial flat resolution ··· → P2(F) → P1(F) → P0(F) → F

with P1(F) := P0 (ker(P0(F) → F), etc. Set Pn(F) = 0 if n < 0. Then to a complex C we associate the flat complex P = PC such that Pr := ⊕m−n=r Pn(Cm) and the restriction of the differential Pr → Pr+1 to Pn(Cm) is Pn(Cm → Cm+1) ⊕ (−1)m Pn(Cm) → Pn−1(Cm), together with the natural map of complexes P → C induced by the epimorphisms P0(Cm) → Cm (m ∈ Z). Elementary arguments, with or without spectral sequences, show that for the truncations τ≤mC, the maps Pτ≤m C → τ≤m C are quasi-isomorphisms. Since homology commutes with direct limits, the resulting map

ψC : PC = limm Pτ≤m C → limτ≤m C = C

is a quasi-isomorphism….

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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Badiou’s Diagrammatic Claim of Democratic Materialism Cuts Grothendieck’s Topos. Note Quote.

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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

  1. has finite limits, or equivalently E has so called pull-backs and a terminal object 1,
  2. 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
  3. 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.

Badiou Contra Grothendieck Functorally. Note Quote.

What makes categories historically remarkable and, in particular, what demonstrates that the categorical change is genuine? On the one hand, Badiou fails to show that category theory is not genuine. But, on the other, it is another thing to say that mathematics itself does change, and that the ‘Platonic’ a priori in Badiou’s endeavour is insufficient, which could be demonstrated empirically.

Yet the empirical does not need to stand only in a way opposed to mathematics. Rather, it relates to results that stemmed from and would have been impossible to comprehend without the use of categories. It is only through experience that we are taught the meaning and use of categories. An experience obviously absent from Badiou’s habituation in mathematics.

To contrast, Grothendieck opened up a new regime of algebraic geometry by generalising the notion of a space first scheme-theoretically (with sheaves) and then in terms of groupoids and higher categories. Topos theory became synonymous to the study of categories that would satisfy the so called Giraud’s axioms based on Grothendieck’s geometric machinery. By utilising such tools, Pierre Deligne was able to prove the so called Weil conjectures, mod-p analogues of the famous Riemann hypothesis.

These conjectures – anticipated already by Gauss – concern the so called local ζ-functions that derive from counting the number of points of an algebraic variety over a finite field, an algebraic structure similar to that of for example rational Q or real numbers R but with only a finite number of elements. By representing algebraic varieties in polynomial terms, it is possible to analyse geometric structures analogous to Riemann hypothesis but over finite fields Z/pZ (the whole numbers modulo p). Such ‘discrete’ varieties had previously been excluded from topological and geometric inquiry, while it now occurred that geometry was no longer overshadowed by a need to decide between ‘discrete’ and ‘continuous’ modalities of the subject (that Badiou still separates).

Along with the continuous ones, also discrete variates could then be studied based on Betti numbers, and similarly as what Cohen’s argument made manifest in set-theory, there seemed to occur ‘deeper’, topological precursors that had remained invisible under the classical formalism. In particular, the so called étale-cohomology allowed topological concepts (e.g., neighbourhood) to be studied in the context of algebraic geometry whose classical, Zariski-description was too rigid to allow a meaningful interpretation. Introducing such concepts on the basis of Jean-Pierre Serre’s suggestion, Alexander Grothendieck did revolutionarize the field of geometry, and Pierre Deligne’s proof of the Weil-conjenctures, not to mention Wiles’ work on Fermat’s last theorem that subsequentely followed.

Grothendieck’s crucial insight drew on his observation that if morphisms of varieties were considered by their ‘adjoint’ field of functions, it was possible to consider geometric morphisms as equivalent to algebraic ones. The algebraic category was restrictive, however, because field-morphisms are always monomorphisms which makes geometric morphisms: to generalize the notion of a neighbourhood to algebraic category he needed to embed algebraic fields into a larger category of rings. While a traditional Kuratowski covering space is locally ‘split’ – as mathematicians call it – the same was not true for the dual category of fields. In other words, the category of fields did not have an operator analogous to pull-backs (fibre products) unless considered as being embedded within rings from which pull-backs have a co-dual expressed by the tensor operator ⊗. Grothendieck thus realized he could replace ‘incorporeal’ or contained neighborhoods U ֒→ X by a more relational description: as maps U → X that are not necessarily monic, but which correspond to ring-morphisms instead.

Topos theory applies similar insight but not in the context of only specific varieties but for the entire theory of sets instead. Ultimately, Lawvere and Tierney realized the importance of these ideas to the concept of classification and truth in general. Classification of elements between two sets comes down to a question: does this element belong to a given set or not? In category of S ets this question calls for a binary answer: true or false. But not in a general topos in which the composition of the subobject-classifier is more geometric.

Indeed, Lawvere and Tierney then considered this characteristc map ‘either/or’ as a categorical relationship instead without referring to its ‘contents’. It was the structural form of this morphism (which they called ‘true’) and as contrasted with other relationships that marked the beginning of geometric logic. They thus rephrased the binary complete Heyting algebra of classical truth with the categorical version Ω defined as an object, which satisfies a specific pull-back condition. The crux of topos theory was then the so called Freyd–Mitchell embedding theorem which effectively guaranteed the explicit set of elementary axioms so as to formalize topos theory. The Freyd–Mitchell embedding theorem says that every abelian category is a full subcategory of a category of modules over some ring R and that the embedding is an exact functor. It is easy to see that not every abelian category is equivalent to RMod for some ring R. The reason is that RMod has all small limits and colimits. But for instance the category of finitely generated R-modules is an abelian category but lacks these properties.

But to understand its significance as a link between geometry and language, it is useful to see how the characteristic map (either/or) behaves in set theory. In particular, by expressing truth in this way, it became possible to reduce Axiom of Comprehension, which states that any suitable formal condition λ gives rise to a peculiar set {x ∈ λ}, to a rather elementary statement regarding adjoint functors.

At the same time, many mathematical structures became expressible not only as general topoi but in terms of a more specific class of Grothendieck-topoi. There, too, the ‘way of doing mathematics’ is different in the sense that the object-classifier is categorically defined and there is no empty set (initial object) but mathematics starts from the terminal object 1 instead. However, there is a material way to express the ‘difference’ such topoi make in terms of set theory: for every such a topos there is a sheaf-form enabling it to be expressed as a category of sheaves S etsC for a category C with a specific Grothendieck-topology.

Badiou’s Materiality as Incorporeal Ontology. Note Quote.

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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 × XUeq 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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Symmetry: Mirror of a Manifold is the Opposite of its Fundamental Poincaré ∞-groupoid

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Given a set X = {a,b,c,..} such as the natural numbers N = {0,1,…,p,…}, there is a standard procedure that amounts to regard X as a category with only identity morphisms. This is the discrete functor that takes X to the category denoted by Disc(X) where the hom-sets are given by Hom(a,b) = ∅ if a ≠ b, and Hom(a,b) = {Ida} = 1 if a = b. Disc(X) is in fact a groupoid.

But in category theory, there is also a procedure called opposite or dual, that takes a general category C to its opposite Cop. Let us call Cop the reflection of C by the mirror functor (−)op.

Now the problem is that if we restrict this procedure to categories such as Disc(X), there is no way to distinguish Disc(X) from Disc(X)op. And this is what we mean by sets don’t show symmetries. In the program of Voevodsky, we can interpret this by saying that:

The identity type is not good for sets, instead we should use the Equivalence type. But to get this, we need to move to from sets to Kan complexes i.e., ∞-groupoids.

The notion of a Kan complex is an abstraction of the combinatorial structure found in the singular simplicial complex of a topological space. There the existence of retractions of any geometric simplex to any of its horns – simplices missing one face and their interior – means that all horns in the singular complex can be filled with genuine simplices, the Kan filler condition.

At the same time, the notion of a Kan complex is an abstraction of the structure found in the nerve of a groupoid, the Duskin nerve of a 2-groupoid and generally the nerves of n-groupoids ∀ n ≤ ∞ n. In other words, Kan complexes constitute a geometric model for ∞-groupoids/homotopy types which is based on the shape given by the simplex category. Thus Kan complexes serve to support homotopy theory.

So far we’ve used set theory with this lack of symmetries, as foundations for mathematics. Grothendieck has seen this when he moved from sheaves of sets, to sheaves of groupoid (stacks), because he wanted to allow objects to have symmetries (automorphisms). If we look at the Giraud-Grothendieck picture on nonabelian cohomology, then what happens is an extension of coefficients U : Set ֒→ Cat. We should consider first the comma category Cat ↓ U, whose objects are functors C → Disc(X). And then we should consider the full subcategory consisting of functors C → Disc(X) that are equivalences of categories. This will force C to be a groupoid, that looks like a set. And we call such C → Disc(X) a Quillen-Segal U-object.

This category of Quillen-Segal objects should be called the category of sets with symmetries. Following Grothendieck’s point of view, we’ve denoted by CatU[Set] the comma category, and think of it as categories with coefficients or coordinates in sets. This terminology is justified by the fact that the functor U : Set ֒→ Cat is a morphism of (higher) topos, that defines a geometric point in Cat. The category of set with symmetries is like the homotopy neighborhood of this point, similar to a one-point going to a disc or any contractible object. The advantage of the Quillen-Segal formalism is the presence of a Quillen model structure on CatU[Set] such that the fibrant objects are Quillen-Segal objects.

In standard terminology this means that if we embed a set X in Cat as Disc(X), and take an ‘projective resolution’ of it, then we get an equivalence of groupoids P → Disc(X), and P has symmetries. Concretely what happens is just a factorization of the identity (type) Id : Disc(X) → Disc(X) as a cofibration followed by a trivial fibration:

Disc(X)  ֒→ P → Disc(X)

This process of embedding Set ֒→ QS{CatU[Set]} is a minimal homotopy enhancement. The idea is that there is no good notion of homotopy (weak equivalence) in Set, but there are at least two notions in Cat: equivalences of categories and the equivalences of classifying spaces. This last class of weak equivalences is what happens with mirror phenomenons. The mirror of a manifold should be the opposite of its fundamental Poincaré ∞-groupoid.

Grothendieck Sheaves and Topologies Within Monetary Value Measures. Part 2

A contravariant functor ρ : Cop → Set is  called a presheaf for a category C. By definition, a monetary value measure is a presheaf. The name presheaf suggests that it is related to another concept sheaves, which is a quite important concept in some classical branches in mathematics such as algebraic topology. For a given set, a topology defined on it provides a criteria to distinguish continuous functions from given functions on the set. In a similar way, there is a concept called a Grothendieck topology defined on a given category that gives a criteria to distinguish good presheaves (=sheaves) from given presheaves on the category. In both cases, a Grothendieck topology can be seen as a vehicle to identify good functions (presheaves) among general functions (presheaves).

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On the other hand, if we have a set of functions that we want to make continuous, we can find the weakest topology that makes the functions continuous. In a similar way, if we have a set of presheaves that we want to make good, it is known that we can pick a Grothendieck topology with which the presheaves become sheaves. Since a monetary value measure is a presheaf, if we have a set of good monetary value measures (= the monetary value measures that satisfy a given set of axioms), we may find a Grothendieck topology with which the monetary value measures become sheaves. Now suppose we have a weak topology that makes given functions continuous. This, however, does not imply the fact that any continuous function w.r.t. the topology is contained in the originally given functions. Similarly, Suppose that we have a Grothendieck topology that makes all monetary value measures satisfying a given set of axioms sheaves. It, however, does not mean that any sheaf w.r.t. the Grothendieck topology satisfies the given set of axioms.

What are Grothendieck topologies and sheaves?

1.0 Let U ∈ χ

1.1 ↓ U := {V ∈ χ | V ⊂ U}

1.2 A sieve on U is a set I ⊂↓ U such that (∀ V ∈ ↓ U)(∀ W ∈ ↓ U)[W ⊂ V ∈ I ⇒ W ∈ I]

1.3 For a sieve I on U and V ⊂ U in χ, I ↓ V := I ∩ ↓ V

1.4 A family of I is an element X ∈ ∏V∈I L(V), or X = (XV)V∈I

1.5 A family X = (XV)V∈I is called a P-martingale if (∀ V ∈ I)(∀ W ∈ I)[W ⊂ V ⇒ EP[XV | W] = XW]

A sieve on U is considered as a kind of a time domain.

2.0 Ξ : χop → Set is a contravariant functor such that for iVU : V → U in χΞ(U) is the set of all sieves on U, and that Ξ(iVU)(I) = I ↓ V for I ∈ Ξ(U)

2.1 A Grothendieck topology on χ is a sub-factor J → Ξ satisfying the following conditions

2.2 (∀ U ∈ χ) ↓ (U ∈ J(U))

2.3 (∀ V ∈ χ)(∀I ∈ J(U))(∀K ∈ Ξ(U))[(∀ V ∈ I)K ↓ V ∈ J(V) ⇒ K ∈ J(U)]

This way sieve I J-covers U if I ∈ J(U)

U is considered as a time horizon of a time domain I if it is covered by I.

3.0 Theorem Let {Ja | a ∈ A} be a collection of Grothendieck topologies on χ. Then the sub-functor J → Ξ defined by J(U) := a∈A Ja(U) is a Grothendieck topology.

4.0 Let Ψ ∈ Setχop be a monetary value measure, and I be a sieve on U ∈ χ

4.1 A family X = (XV)V∈I is called Ψ-matching if (∀ V ∈ I)(∀ W ∈ I)ΨV∧WV = ΨV∧WW (XW)

4.2 A random variable X ̄∈ L(U) be a Ψ-amalgamation for a family X = (XV)V ∈ I if (∀ V ∈ I)ΨVU(X ̄) = XV

5.0 Let Ψ ∈ Setχop be a monetary value measure. I be a sieve on U ∈ χ and X = (XV)V∈I be a family that has Ψ-amalgamation. Then X is Ψ-matching. 

Let X ̄∈  L(U) be a Ψ-amalgamation. Then for every V ∈ I, XV = ΨVU(X ̄). Therefore, for every V, W ∈ I ΨV∧WV (XV) = XV∧W = ΨV∧WW (XW)

6.0 Ψ ∈ Setχop be a monetary value measure. I be a sieve on U ∈ χ and X = (XV)V∈I be a Ψ-matching family.

6.1 For V, W ∈ I, if W ⊂ V, we have ΨWV(XV) = XW

ΨWV(XV) = ΨV∧WV (XV) = ΨV∧WW (XW) = ΨWW(XW) = (XW)

6.2 If U ∈ I, XU is the unique Ψ-amalgamation for X. By 6.1, XU is the unique Ψ-amalgamation for X. Now, let X ̄∈ L(U) be another Ψ-amalgamation for X. Then for every V ∈ I, XV = ΨVU(X ̄). Put V := U. Then we have XU = ΨUU(X ̄) = 1U(X ̄) = X ̄.

7.0 Let J be a Grothendieck topology on Ψ ∈ Setχop. A monetary value measure Ψ ∈ Setχop is called a sheaf if for any U ∈ χ. Any covering sieve I ∈ J(U) and any Ψ-matching family X = (XV)V∈I b, X has a unique Ψ-amalgamation. Now, we will try to find a Grothendieck topology for which a given class of monetary value measures specified by a given set of (extra) axioms are sheaves.

Let us consider a sieve I on U ∈ χ as a subfunctor I → HomX(-,U), that is a contravariant functor I : χop → Set defined by

I(V) := {iVU} if V ∈ I

:= Φ if V ∉ I

for V ∈ χ

8.0 Let M ⊂ Setχop be the collection of all monetary value measures satisfying a given set of axioms. Then, there exists a Grothendieck topology for which all monetary value measures in M sheaves, where the topology is largest among topologies rep resenting the axioms. Let this topology be denoted by JM

Let J:= ∩Ψ∈MJΨ

Then, it is the largest Grothendieck topology for which every monetary value measure M is a sheaf……..