Extensions of Galois's Theory For Groups: Galois Groupoids, Categories and Functors
Galois has developed a powerful, fundamental algebraic theory in mathematics that provides very efficient computations for certain algebraic problems by utilizing the algebraic concept of groups, which is now known as the theory of Galois groups; such computations were not possible before, and also in many cases are much more effective than the 'direct' calculations without using groups. To begin with, Alexander Grothendieck stated in his proposal: "Thus, the group of Galois is realized as the automorphism group of a concrete, pro-finite group which respects certain structures that are essential to this group." This fundamental, Galois group theory in mathematics has been considerably expanded, at first to groupoids- as proposed in Alexander Grothendieck's Esquisse d' un Programme (EdP)- and now already partially carried out for groupoids; the latter are now further developed beyond groupoids to categories by several groups of mathematicians. Here, we shall focus only on the well-established and fully validated extensions of Galois' theory. Thus, EdP also proposed and anticipated, along previous Alexander Grothendieck's IHÉS seminars (SGA1 to SGA4) held in the 1960s, the development of even more powerful extensions of the original Galois's theory for groups by utilizing categories, functors and natural transformations, as well as further expansion of the manifold of ideas presented in Alexander Grothendieck's Descent Theory. The notion of motive has also been pursued actively. This was developed into the motivic Galois group, Grothendieck topology and Grothendieck category . Such developments were recently extended in algebraic topology via representable functors and the fundamental groupoid functor.
Read more about this topic: Esquisse D'un Programme, Abstract of Grothendieck's Programme
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