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 x_{0} ∈ X is a base point {x_{0}} → X induces a ‘fibre functor’ Sh(X) → Sh{x_{0}} ≅ Sets, by pullback.)

If F is pro-representable by P, then π_{1}(E, F) is defined to be Aut(P), which is a profinite group. Grothendieck proves there is an equivalence of categories C ≃ π_{1}(E) − Sets_{fin}, the category of finite π_{1}(E)-sets. If X is a locally nicely behaved space such as a CW-complex and E = Sh(X), then π_{1}(E) is the profinite completion of π_{1}(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 π_{1}(X) – set, Y′, but if Y is a finite covering of X then the homomorphism from π_{1}(X) to the finite group of transformations of Y factors through the profinite completion of π_{1}(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 π_{1}(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 π_{1}(X) classifies covering spaces – but for i > 1, π_{i}(X) does not seem to classify anything other than maps from S^{i} into X!

This is abstract homotopy theory par excellence.

Aside: Cab u describe https://en.m.wikipedia.org/wiki/Bose–Einstein_condensate. In more layman terms. ? What Is meant by ‘new kind of matter ‘?

These weren’t discovered till the 90s and that too in the laboratory. A Bose-Einstein condensate is a group of atoms cooled to within a hair of absolute zero. When they reach that temperature the atoms are hardly moving relative to each other; they have almost no free energy to do so. At that point, the atoms begin to clump together, and enter the same energy states. They become identical, from a physical point of view, and the whole group starts behaving as though it were a single atom. The atoms fall into the same quantum states, and can’t be distinguished from one another. At that point the atoms start obeying what are called Bose-Einstein statistics, which are usually applied to particles you can’t tell apart, such as photons. What the two found was that ordinarily, atoms have to have certain energies — in fact one of the fundamentals of quantum mechanics is that the energy of an atom or other subatomic particle can’t be arbitrary. This is why electrons, for example, have discrete “orbitals” that they have to occupy, and why they give off photons of specific wavelengths when they drop from one orbital, or energy level, to another. But cool the atoms to within billionths of a degree of absolute zero and some atoms begin to fall into the same energy level, becoming indistinguishable. These superstorms are termed ‘new’ kind of matter.

Thx so much. Interesting