Grothendieck’s Abstract Homotopy Theory

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 → Sets_fin, 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) − Sets_fin, 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, G^ˆ = 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, π_i(X) does not seem to classify anything other than maps from Sⁱ into X!

This is abstract homotopy theory par excellence.

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Two Conceptions of Morphogenesis – World as a Dense Evolutionary Plasma of Perpetual Differentiation and Innovation. Thought of the Day 57.0

Sanford Kwinter‘s two conceptions of morhpogenesis, of which, one is appropriate to a world capable of sustaining transcendental ontological categories, while the other is inherent in a world of perfect immanence. According to the classical, hylomorphic model, a necessarily limited number of possibilities (forms or images) are reproduced (mirrored in reality) over a substratum, in a linear time-line. The insufficiency of such a model, however, is evident in its inability to find a place for novelty. Something either is or is not possible. This model cannot account for new possibilities and it fails to confront the inevitable imperfections and degradations evident in all of its realizations. It is indeed the inevitability of corruption and imperfection inherent in classical creation that points to the second mode of morphogenesis. This mode is dependent on an understanding of the world as a ceaseless pullulation and unfolding, a dense evolutionary plasma of perpetual differentiation and innovation. In this world forms are not carried over from some transcendent realm, but instead singularities and events emerge from within a rich plasma through the continual and dynamic interaction of forces. The morphogenetic process at work in such a world is not one whereby an active subject realizes forms from a set of transcendent possibilities, but rather one in which virtualities are actualized through the constant movement inherent in the very forces that compose the world. Virtuality is understood as the free difference or singularity, not yet combined with other differences into a complex ensemble or salient form. It is of course this immanentist description of the world and its attendant mode of morphogenesis that are viable. There is no threshold beneath which classical objects, states, or relations cease to have meaning yet beyond which they are endowed with a full pedigree and privileged status. Indeed, it is the nature of real time to ensure a constant production of innovation and change in all conditions. This is evidenced precisely by the imperfections introduced in an act of realizing a form. The classical mode of morphogenesis, then, has to be understood as a false model which is imposed on what is actually a rich, perpetually transforming universe. But the sort of novelty which the enactment of the classical model produces, a novelty which from its own perspective must be construed as a defect is not a primary concern if the novelty is registered as having emerged from a complex collision of forces. Above all, it is a novelty uncontaminated by procrustean notions of subjectivity and creation.