Truncation Functors

Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ_≤nA• be the truncated complex

··· → Aⁿ⁻² → Aⁿ⁻¹ → ker(Aⁿ → Aⁿ⁺¹)→ 0 → 0 → ··· , and dually we let τ_≥nA be the complex

··· → 0 → 0 → coker(Aⁿ⁻¹ → Aⁿ) → Aⁿ⁺¹ → Aⁿ⁺² → ···

Note that

H^m(τ_≤nA•) = H^m(A•) if m ≤ n

= 0 if m > n

and that

H^m(τ_≥nA•) = H^m(A•)  if m ≥ n

= 0 if m < n

One checks that τ_≥n (respectively τ_≤n) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D. In fact if D_≤n (respectively D_≥n) is the full subcategory of D whose objects are the complexes A• such that H^m(A•) = 0 for m > n (respectively m < n) then we have additive functors

τ_≤n : D → D_≤n ⊂ D

τ_≥n : D → D_≥n ⊂ D

together with obvious functorial maps

iⁿ_A : τ_≤n A• → A•

jⁿ_A : A• → τ_≥n A•

The preceding iⁿ_A , jⁿ_A induce functorial isomorphisms

Hom_D≤n (B•,τ_≤nA•) →^~ Hom_D(B•, A•) (B• ∈ D_≤n) —– (1)

Hom_D≥n (τ_≥nA•,C•) →^~ Hom_D(A•,C• ) (C• ∈ D_≥n) —– (2)

Bijectivity of (1) means that any map φ : B• → A• (in D) with B• ∈ D_≤n factors uniquely via i_A := iⁿ_A

Given φ, we have a commutative diagram

and since B• ∈ D_≤n, therefore i_B is an isomorphism in D, so we can write

φ = i ◦ (τ_≤nφ ◦ i⁻¹_B),

and thus (1) is surjective.

To prove that (1) is also injective, we assume that i_A ◦ τ_≤n φ = 0 and deduce that τ_≤n φ = 0. The assumption means that there is a commutative diagram in K(A)

where s and s′′ are quasi-isomorphisms, and f/s = τ_≤nφ

Applying the (idempotent) functor τ_≥n, we get a commutative diagram

Since τ_≤ns and τ_≤ns′′ are quasi-isomorphisms, we have

τ_≤nφ = τ_≤n f/τ_≤ns = 0/τ_≤ns′′ = 0

as desired.

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Bernard Cache’s Earth Moves: The Furnishing of Territories (Writing Architecture)

Take the concept of singularity. In mathematics, what is said to be singular is not a given point, but rather a set of points on a given curve. A point is not singular; it becomes singularized on a continuum. And several types of singularity exist, starting with fractures in curves and other bumps in the road. We will discount them at the outset, for singularities that are marked by discontinuity signal events that are exterior to the curvature and are themselves easily identifiable. In the same way, we will eliminate singularities such as backup points [points de rebroussement]. For though they are indeed discontinuous, they refer to a vector that is tangential to the curve and thus trace a symmetrical axis that constitutive of the backup point. Whether it be a reflection of the tan- gential plane or a rebound with respect to the orthogonal plane, the backup point is thus not a basic singularity. It is rather the result of an operation effectuated on any part of the curve. Here again, the singular would be the sign of too noisy, too memorable an event, while what we want to do is to deal with what is most smooth: ordinary continua, sleek and polished.

On one hand there are the extrema, the maximum and minimum on a given curve. And on the other there are those singular points that, in relation to the extrema, figure as in-betweens. These are known as points of inflection. They are different from the extrema in that they are defined only in relation to themselves, whereas the definition of the extrema presupposes the prior choice of an axis or an orientation, that is to say of a vector.

Indeed, a maximum or a minimum is a point where the tangent to the curve is directed perpendicularly to the axis of the ordinates [y-axis]. Any new orientation of the coordinate axes repositions the maxima and the min- ima; they are thus extrinsic singularities. The point of inflection, however, designates a pure event of curvature where the tangent crosses the curve; yet this event does not depend in any way on the orientation of the axes, which is why it can be said that inflection is an intrinsic singularity. On either side of the inflection, we know that there will be a highest point and a lowest point, but we cannot designate them as long as the curve has not been related to the orientation of a vector. Points of inflection are singularities in and of themselves, while they confer an indeterminacy to the rest of the curve. Preceding the vector, inflection makes of each of the points a possible extremum in relation to its inverse: virtual maxima and minima. In this way, inflection represents a totality of possibilities, as well as an openness, a receptiveness, or an anticipation……

Bernard Cache Earth Moves The Furnishing of Territories

The Womb of Cosmogony. Thought of the Day 30.0

Nowhere and by no people was speculation allowed to range beyond those manifested gods. The boundless and infinite UNITY remained with every nation a virgin forbidden soil, untrodden by man’s thought, untouched by fruitless speculation. The only reference made to it was the brief conception of its diastolic and systolic property, of its periodical expansion or dilatation, and contraction. In the Universe with all its incalculable myriads of systems and worlds disappearing and re-appearing in eternity, the anthropomorphised powers, or gods, their Souls, had to disappear from view with their bodies: — “The breath returning to the eternal bosom which exhales and inhales them,” says our Catechism. . . . In every Cosmogony, behind and higher than the creative deity, there is a superior deity, a planner, an Architect, of whom the Creator is but the executive agent. And still higher, over and around, withinand without, there is the UNKNOWABLE and the unknown, the Source and Cause of all these Emanations. – The Secret Doctrine

Many are the names in the ancient literatures which have been given to the Womb of Being from which all issues, in which all forever is, and into the spiritual and divine reaches of which all ultimately returns, whether infinitesimal entity or macrocosmic spacial unit.

The Tibetans called this ineffable mystery Tong-pa-nnid, the unfathomable Abyss of the spiritual realms. The Buddhists of the Mahayana school describe it as Sunyata or the Emptiness, simply because no human imagination can figurate to itself the incomprehensible Fullness which it is. In the Eddas of ancient Scandinavia the Boundless was called by the suggestive term Ginnungagap – a word meaning yawning or uncircumscribed void. The Hebrew Bible states that the earth was formless and void, and darkness was upon the face of Tehom, the Deep, the Abyss of Waters, and therefore the great Deep of kosmic Space. It has the identical significance of the Womb of Space as envisioned by other peoples. In the Chaldaeo-Jewish Qabbalah the same idea is conveyed by the term ‘Eyn (or Ain) Soph, without bounds. In the Babylonian accounts of Genesis, it is Mummu Tiamatu which stands for the Great Sea or Deep. The archaic Chaldaean cosmology speaks of the Abyss under the name of Ab Soo, the Father or source of knowledge, and in primitive Magianism it was Zervan Akarana — in its original meaning of Boundless Spirit instead of the later connotation of Boundless Time.

In the Chinese cosmogony, Tsi-tsai, the Self-Existent, is the Unknown Darkness, the root of the Wuliang-sheu, Boundless Age. The wu wei of Lao-tse, often mistranslated as passivity and nonaction, imbodies a similar conception. In the sacred scriptures of the Quiches of Guatemala, the Popol Vuh or “Book of the Azure Veil,” reference is made to the “void which was the immensity of the Heavens,” and to the “Great Sea of Space.” The ancient Egyptians spoke of the Endless Deep; the same idea also is imbodied in the Celi-Ced of archaic Druidism, Ced being spoken of as the “Black Virgin” — Chaos — a state of matter prior to manvantaric differentiation.

The Orphic Mysteries taught of the Thrice-Unknown Darkness or Chronos, about which nothing could be predicated except its timeless Duration. With the Gnostic schools, as for instance with Valentinus, it was Bythos, the Deep. In Greece, the school of Democritus and Epicurus postulated To Kenon, the Void; the same idea was later voiced by Leucippus and Diagoras. But the two most common terms in Greek philosophy for the Boundless were Apeiron, as used by Plato, Anaximander and Anaximenes, and Apeiria, as used by Anaxagoras and Aristotle. Both words had the significance of frontierless expansion, that which has no circumscribing bounds.

The earliest conception of Chaos was that almost unthinkable condition of kosmic space or kosmic expanse, which to human minds is infinite and vacant extension of primordial Aether, a stage before the formation of manifested worlds, and out of which everything that later existed was born, including gods and men and all the celestial hosts. We see here a faithful echo of the archaic esoteric philosophy, because among the Greeks Chaos was the kosmic mother of Erebos and Nyx, Darkness and Night — two aspects of the same primordial kosmic stage. Erebos was the spiritual or active side corresponding to Brahman in Hindu philosophy, and Nyx the passive side corresponding to pradhana or mulaprakriti, both meaning root-nature. Then from Erebos and Nyx as dual were born Aether and Hemera, Spirit and Day — Spirit being here again in this succeeding stage the active side, and Day the passive aspect, the substantial or vehicular side. The idea was that just as in the Day of Brahma of Hindu cosmogony things spring into active manifested existence, so in the kosmic Day of the Greeks things spring from elemental substance into manifested light and activity, because of the indwelling urge of the kosmic Spirit.