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.

# Day: June 2, 2017

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

# 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 Q_{1} → Q_{2} in K(A), the resulting map P ⊗ Q_{1} → P ⊗ Q_{2} 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 P^{n} is an exact functor in the category of O-modules, i.e., P^{n} 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 P_{0} from A to its full subcategory of flat O-modules, together with a functorial epimorphism P_{0}(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,

P_{0}(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 ··· → P_{2}(F) → P_{1}(F) → P_{0}(F) → F

with P_{1}(F) := P_{0} (ker(P_{0}(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 P_{0}(C^{m}) → C^{m} (m ∈ Z). Elementary arguments, with or without spectral sequences, show that for the truncations τ_{≤m}C, 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….