Okay so here's a thread on operads from my old account. I hope people find it useful.
So what's an operad? The basic idea is that it's a blueprint for certain kinds of algebraic structure. Other ways to control algebraic structure are monads, algebraic and Lawvere theories, and just collections of commutative diagrams. But operads are distinct from these.
Although there is of course non-trivial overlap, e.g. you can always get a monad from an operad (but not vice-versa). The thing that operads really do well is control operations, i.e. given some object X in a category, an operad classifies a bunch of...
Just a reminder that the problem with the academic job market is not that there are "too many PhDs." The problem is that universities are working really hard to eliminate permanent, stable employment for their faculty and thereby reducing the quality of undergraduate education.
Those things that everyone hates Wal-Mart for doing, like breaking apart unions and finding clever ways of avoiding giving people benefits or job stability? Yeah, that's also universities.
The number of undergraduate students in the US has massively increased over the past several decades. It stands to reason that we should need a lot more professors to teach them, and expose them to active scholarship and research.
Sets → Spaces/Simplicial Sets/Homotopy types
Pointed Sets→Pointed versions of the above
Set Bijection→Homotopy equivalence
Groups→𝔸_∞-spaces whose connected components form a group
Abelian categories→Stable model categories/∞-categories
Integers ℤ→Sphere spectrum 𝕊
Classical fields→Classical fields + Morava K-theories
Modules over a ring R→Module spectra over the Eilenberg MacLane spectrum HR
Ideals of a ring R→Bousfield localizations of the ∞-category of HR-modules