The development of modern physics in the first half of the 20th century was closely related to the development of differential geometry, first via Riemannian geometry in Einstein’s theory of gravity and then later via Cartan geometry in Yang-Mills’s theory of gauge fields. But, as highlighted by Grothendieck in the second half of the 20th century and as witnessed by a multitude of modern developments, a more natural mathematical description of many phenomena in geometry is obtained by refining from traditional geometric spaces to more refined kinds of spaces known as “stacks”.
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Our main motivation to consider sheaves and stacks is to provide a nonperturbative framework in which we can do physics. Much of gauge theory is done in perturbation theory, but in fact non-perturbative effects such as Dirac monopoles and Yang-Mills instantons play a crucial role in fundamental physics [5]. The language of stacks is the natural language for these phenomena.
locality principle + gauge principle = stack principlehttps://ncatlab.org/schreiber/files/SchreiberTrento14.pdf
The idea of using stacks goes back to a manuscript titled Pursuing Stacks by Alexander Grothendieck in 1983. For some more information, see https://ncatlab.org/nlab/show/Pursuing+Stacks