Higher algebra and higher geometry are like air: they are everywhere -- even if often you may only notice it when a storm comes up. It's the sea rising:
I can illustrate the second approach with the same image of a nut to be opened. The first analogy which came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months – when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!
A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.
(quote form Alexander Grothendieck, Récoltes et semailles, 1985–1987, pp. 552-3-1)
I don’t want you to think all this is theory for the sake of it, or rather for the sake of itself. It’s theory for the sake of other theory.
(quote from Jacob Lurie, ICM talk 2010)
I have started to compile some commented lists of applications of higher algebra and higher geometry in these nLab entries:
These entries all could and eventually deserve to be expanded much more. The limiting factor is not the number and scope of examples. If you ask for something more specific, maybe it inspires me (or somebody reading this here) to go and add more to these nLab entries.
