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

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 (Sets^γ∗(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 φ_σ : X → Ω making σ : Y ֒→ X a pull-back of φ_σ 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, Id_α) to the object (B,Id_β) is a map ρ : A → B such that

E_β ρ(a) = E_α 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 Id_α (a, b) ≤ Id_β (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 Id_α. 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.

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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 ets^C for a category C with a specific Grothendieck-topology.

Egyptology

The ancient Egyptians conceived man and kosmos to be dual: firstly, the High God or Divine Mind arose out of the Primeval Waters of space at the beginning of manifestation; secondly, the material aspect expressing what is in the Divine Mind must be in a process of ever-becoming. In other words, the kosmos consists of body and soul. Man emanated in the image of divinity is similarly dual and his evolutionary goal is a fully conscious return to the Divine Mind.

Space, symbolized by the Primeval Waters, contains the seeds and possibilities of all living things in their quiescent state. At the right moment for awakenment, all will take up forms in accordance with inherent qualities. Or to express it in another way: the Word uttered by the Divine Mind calls manifested life to begin once more.

Growth is effected through a succession of lives, a concept that is found in texts and implied in symbolism. Herodotus, the Greek historian (5th century B.C.), wrote that

the Egyptians were the first to teach that the human soul is immortal, and at the death of the body enters into some other living thing then coming to birth; and after passing through all creatures of land, sea, and air (which cycle it completes in three thousand years) it enters once more into a human body, at birth.

The theory of reincarnation is often ascribed to Pythagoras, since he spent some time in Egypt studying its philosophy and, according to Herodotus, “adopted this opinion as if it were his own.”

Margaret A. Murray, who worked with Flinders Petrie, illustrates the Egyptian belief by referring to the ka-names of three kings (The ka-name relates to the vital essence of an individual); the first two of the twelfth dynasty: that of Amonemhat I means “He who repeats births,” Senusert I: “He whose births live,” and the ka-name of Setekhy I of the nineteenth dynasty was “Repeater of births.” (The Splendour That Was Egypt)

Reincarnation has been connected with the rites of Osiris, one of the Mysteries or cycles of initiation perpetuated in Egypt. The concept of transformation as recorded in the Egyptian texts has been interpreted in various ways. De Briere expresses it in astronomical terms: “The sensitive soul re-entered by the gate of the gods, or the Capricorn, into the Amenthe, the watery heavens, where it dwelt always with pleasure; until, descending by the gate of men, or the Cancer, it came to animate a new body.” Herodotus writes of transmigration, i.e., that the soul passes through various animals before being reborn in human form. This refers not to the human soul but to the molecules, atoms, and other components that clothe it. They gravitate to vehicles similar in qualities to their former host’s, drawn magnetically to the new milieu by the imprint made by the human soul, whether it be fine or gross. It is quite clear from the Book of the Dead and other texts that the soul itself after death undergoes experiences in the Duat (Dwat) or Underworld, the realm and condition between heaven and earth, or beneath the earth, supposedly traversed by the sun from sunset to sunrise.

The evolution of consciousness is symbolized by the Solar Barque moving through the Duat. In this context the “hours” of travel represent stages of development. Bika Reed states that at a certain “hour” the individual meets the “Rebel in the Soul,”  that is, at the “hour of spiritual transformation.” And translating from the scroll Reed gives: “the soul warns, only if a man is allowed to continue evolving, can the intellect reach the heart.”

Not only does the scripture deal with rituals assumed to apply to after-death conditions — in some respects similar to the Book of the Dead — but also it seems quite patently a ritual connected with initiation from one level of self-becoming to another. Nevertheless the picture that emerges is that of the “deceased” or candidate for initiation reaching a fork offering two paths called “The Two Paths of Liberation” and, while each may take the neophyte to the abode of the Akhu (the “Blessed”) — a name for the gods, and also for the successful initiates — they involve different experiences. One path, passing over land and water, is that of Osiris or cyclic nature and involves many incarnations. The other way leads through fire in a direct or shortened passage along the route of Horus who in many texts symbolizes the divine spark in the heart.

In the Corpus Hermeticum, Thoth — Tehuti — was the Mind of the Deity, whom the Alexandrian Greeks identified with Hermes. For example, one of the chief books in the Hermetica is the Poimandres treatise, or Pymander. The early trinity Atum-Ptah-Thoth was rendered into Greek as theos (god) — demiourgos or demourgos-nous (Demiurge or Demiurgic Mind) — nous and logos (Mind and Word). The text states that Thoth, after planning and engineering the kosmos, unites himself with the Demiurgic Mind. There are other expressions proving that the Poimandres text is a Hellenized version of Egyptian doctrine. An important concept therein is that of “making-new-again.” The treatise claims that all animal and vegetable forms contain in themselves “the seed of again-becoming” — a clear reference to reimbodiment — “every birth of flesh ensouled . . . shall of necessity renew itself.” G. R. S. Mead interprets this as palingenesis or reincarnation — “the renewal on the karmic wheel of birth-and-death.” (Thrice-Greatest Hermes)

The Corpus Hermeticum or Books of Hermes are believed by some scholars to have been borrowed from Christian texts, but their concepts are definitely ancient Egyptian in origin, translated into Alexandrian Greek, and Latin.

Looking at Walter Scott’s translation of Poimandres, it states that “At the dissolution of your material body, you first yield up the body itself to be changed,” and it will be absorbed by nature. The rest of the individual’s components return to “their own sources, becoming parts of the universe, and entering into fresh combinations to do other work.” After this, the real or inner man “mounts upward through the structure of the heavens,” leaving off in each of the seven zones certain energies and related substances. The first zone is that of the Moon; the second, the planet Mercury; the third, Venus; fourth, the Sun; fifth, Mars; sixth, Jupiter; and seventh, Saturn. “Having been stripped of all that was wrought upon him” in his previous descent into incarnation on Earth, he ascends to the highest sphere, “being now possessed of his own proper power.” Finally, he enters into divinity. “This is the Good; this is the consummation, for those who have got gnosis.” (According to Scott, gnosis in this context means not only knowledge of divinity but also the relationship between man’s real self and the godhead.)

Further on, the Poimandres explains that the mind and soul can be conjoined only by means of an earth-body, because the mind by itself cannot do so, and an earthly body would not be able to endure

the presence of that mighty and immortal being, nor could so great a power submit to contact with a body defiled by passion. And so the mind takes to itself the soul for a wrap

In Hermetica, Isis to Horus, there is the statement:

. . . . For there are [in the world above, two gods] who are attendants of the Providence that governs all. One of them is Keeper of souls; the other is Conductor of souls. The Keeper is he that has in his charge the unembodied souls; the Conductor is he that sends down to earth the souls that are from time to time embodied, and assigns to them their several places. And both he that keeps watch over the souls, and he that sends them forth, act in accordance with God’s will.

There are many texts using the term “transformations” and a good commentary on the concept by R. T. Rundle Clark follows:

In order to reach the heights of the sky the soul had to undergo those transformations which the High God had gone through as he developed from a spirit in the Primeval Waters to his final position as Sun God . . .” — Myth-And-Symbol-In-Ancient-Egypt

This would appear to mean that in entering upon physical manifestation human souls follow the path of the divine and spiritual artificers of the universe.

There is reason to believe that the after-death adventures met with by the soul through the Duat or Underworld were also undergone by a neophyte during initiation. If the trial ends in success, the awakened human being thereafter speaks with the authority of direct experience. In the most ancient days of Egypt, such an initiate was called a “Son of the Sun” for he embodied the solar splendour. For the rest of mankind, the way is slower, progressing certainly, but more gradually, through many lives. The ultimate achievement is the same: to radiate the highest qualities of the spiritual element locked within the aspiring soul.