I donāt even mean āhow involves the concepts areā, I literally mean āitās a shitton of dataā. Without even explaining what any of those things mean:
A spectrum, naively, involves an infinite sequence of topological spaces together with structure maps from a suspension of previous to the next one (most versions of āspectrumā that we use nowadays require specifying even more data)
A ring spectrum is something that people have been trying to define for like 30 years. Naively, a ring spectrum is a spectrum - I.e. an infinite sequence of topological spaces with structure maps - with an additional binary operation which we call smash product , and a unit, which satisfy the usual associativity and unit a little diagrams up to homotopy
An e infinity ring spectrum, again, naively, is a ring spectrum (infinite sequence of topological spaces plus a binary operation plus a unit) such that the multiplication is coherently homotopy commutative: so thereās a usual commutativity diagram commutative up to homotopy, PLUS when we take three-fold product, there are two ways to group and permute factosa, each of which gives us a homotopy (remember: we require commutativity up ho homotopy), and we ask those homotopies to be homotopical, via a new homotopy. Now when we combine 4 factors and we ask that all the homotopies between homotopies between homotopies are also homotopical etc ad infimum. In conclusion: infinitely many spaces with infinitely many homotopies with infinitely many coherence conditions
So thatās an e infinity ring spectrum. Now a sheaf. To define a sheaf, you first need to define a topology on a category. This requires specifying a certain kind of collection of morphisms into C for every object C in the category.
Now, a sheaf of STUFF on a category means assigning to every object of the category an instance of STUFF (plus some conditions). So thatāsā¦.a lot of STUFF. In particular, a sheaf of e infinity ring spectra requiresspecifying a separate e infinity ring spectrum (and all the data that comes with it) for every object in our category of interest.
How big is the category of interest? Well, for one, itās a stack, and a stack is sort of like a sheaf (see above) of groupoids (so now to every object in the base weāre assigning a whole category) +descent data, which requires specifying a covering family for each object of the underlying category, and morphisms that behave a certain way on restrictions.
The stack in question is the moduli stack of (a certain kind of) elliptic curves which you can think of a āspaceā where each point is an elliptic curve, and we can describe it as a āsheafā on your favorite category of schemes, where to every scheme we assign the groupoid of (a certain kind of) elliptic curves over it
So now we take this gigantic object and to every single point in it we assign another gigantic object, and now considering this assignment as another gigantic object, we assign YET ANOTHER gigantic object to every single point in the new object, and require all of this stuff interact nicely
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