Unformed Bodies Without Organs (BwO) and Protevi’s Version of Autopoiesis Spreading Rhizomatically. Thought of the Day 32.0

Protevi’s interpretation of autopoietic organisation as equivalent to the virtual, unformed, unorganised BwO is in many ways radical. For many, the theory of autopoiesis is a ‘closed’ system theory in contrast to virtuality which signals the third wave cybernetics of open systems. One’s position on this issue, though distinctions are indeed ‘fuzzy’, dictate the descriptives of discourse. The preference here favours the catalysis of human-machinic interplay as it veers towards the transductive and transversal. But these terms of fluidity should remain fluid. Despite a nearly universal theoretical disavowal of the Cartesian paradigm, is it still problematic to surrender the Enlightenment’s legacy of the liberal humanist subject? To surrender the notion of identity, of self and other as individually determined? Does the plausibility of the posthuman send silent shivers down the vertebrae of the elitist homo sapien? Are realities constructed from an always already individual being or is it that, “autonomous will is merely the story consciousness tells itself to explain results that actually come about through chaotic dynamics and emergent structures”? To in any way grasp the dimension of the collective through collaborative practice, a path must be traversed through the (trans)individual. The path explored here is selective. It begins with Bergson and spreads rhizomatically.

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Conjuncted: Non-Performing Assets and Indian Banking – Unfolding Crisis

The banking crisis is indeed getting ‘Alarming’ now. With Credit Suisse citing the NPAs as 8.4 per cent of the GDP and the latest figures toppling the country from its so-called vantage position of fastest growing major economy and bracketing the GDP growth at 6.1 per cent, we are actually faced with a negative growth considering banking sector in the country as the vanguard of India’s economic prowess. The ratio of NPAs to GDP measures the potential losses in relation to the size of the economy. This is especially useful in cross-country comparisons, given that countries are at different levels of GDP. The problem with this measure is that it does not indicate whether banks are able to handle the NPAs with their own resources or their own capital opening up possibilities for either capital infusion, which the Government would have to raise exponentially, or raising funds from the capital market. A more commonly used metric is the NPA to loans ratio. This shows the fraction of bank loans that has turned bad. One shortcoming of this measure is that it suggests the problem can be solved through denominator management – growing the loan books of banks to make the NPA ratio smaller. This growing out of the problem approach is not a reliable one. It depends on the overall economic environment and on the demand for credit. It assumes that the source of the NPA problem is external to the banking system and not in the weaknesses of the lending processes. If the demand for credit is slow it becomes difficult for the banks to grow their loan books. Even if the demand for credit were high, it would be difficult for the NPA-ridden banks to grow. Prolonged NPA episodes erode banks‘ capital and constrain their ability to grow their loan books. 

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

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 P_x is q-flat in K(A_x), where A_x is the category of modules over the ring O_x. (In verifying this statement, note that an exact O_x-complex Q_x is the stalk at x of the exact O-complex Q which associates Q_x 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, P^x_n is a flat O_x-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 ψ_C from a q-flat complex P_C, which can be constructed to depend functorially on C, and so that P_C[1] = P_C[1] and ψ_C[1] = ψ_C[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 O_V 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 O_V is flat,its stalk at x ∈ U being O_x if x ∈ V and 0 otherwise. There is a canonical isomorphism

ψ : F (V) → Hom (O_V, F) (F ∈ A)

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

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

P₀(F) := (⊕_{V open} ⊕_λ∈F(V) O_λ) → 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 P_n(F) = 0 if n < 0. Then to a complex C we associate the flat complex P = P_C such that P^r := ⊕_m−n=r P_n(C^m) and the restriction of the differential P^r → P^r+1 to P_n(C^m) is P_n(C^m → C^m+1) ⊕ (−1)^m P_n(C^m) → P_n−1(C^m), together with the natural map of complexes P → C induced by the epimorphisms P₀(C^m) → C^m (m ∈ Z). Elementary arguments, with or without spectral sequences, show that for the truncations τ_≤mC, the maps P_{τ≤m C} → τ_≤m C are quasi-isomorphisms. Since homology commutes with direct limits, the resulting map

ψ_C : P_C = lim_→m P_{τ≤m C} → lim_→τ≤m C = C

is a quasi-isomorphism….