Alright, here's an old thread of mine, again from a now locked account, about chromatic homotopy theory:

Maybe we should talk about chromatic homotopy theory on here for a second....

I'll assume you know some category theory and some topology.

So first of all, let's suppose you believe it's worth your time to try to classify topological spaces by homotopy type.

I'll assume you know some category theory and some topology.

So first of all, let's suppose you believe it's worth your time to try to classify topological spaces by homotopy type.

This is basically impossible, but it's a noble goal. To make it a little bit easier, we'll limit ourselves to based spaces that can be built by attaching cells.

These are sometimes called CW complexes, or cell complexes.

These are sometimes called CW complexes, or cell complexes.

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.

Hey @FakeUnicode how come I can write ℕ with no serifs, but if I write n∈ℕ it has serifs?

Wait now they both have no serifs wtf is going on. n∈ℕ

I guess typing just ℕ first sets some setting for the whole tweet.

A derived algebra dictionary:

Sets → Spaces/Simplicial Sets/Homotopy types

Pointed Sets→Pointed versions of the above

Set Bijection→Homotopy equivalence

Categories→Model categories/∞-categories

Sets → Spaces/Simplicial Sets/Homotopy types

Pointed Sets→Pointed versions of the above

Set Bijection→Homotopy equivalence

Categories→Model categories/∞-categories

Monoids→𝔸_∞-spaces

Groups→𝔸_∞-spaces whose connected components form a group

Abelian groups→Spectra

Rings→𝔸_∞-spectra

Commutative rings→𝔼_∞-spectra

Abelian categories→Stable model categories/∞-categories

Groups→𝔸_∞-spaces whose connected components form a group

Abelian groups→Spectra

Rings→𝔸_∞-spectra

Commutative rings→𝔼_∞-spectra

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

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