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Now I'm going to try to talk a bit about algebra in spectra, or "derived algebra," or "brave new algebra," or "higher algebra," or whatever you want, mainly at the request of @joyo_fresh. This is basically a continuation of this thread:

So what we came away with was that there was this new category of things called "spectra" which admitted interesting functors from pointed spaces (this Σ^∞ functor) and from abelian groups (the Eilenberg Mac Lane functor H).

And what's more, there were other spectra that didn't come from abelian groups or spaces at all, but which still give us cohomology theories, like complex K-theory, cobordism theories, elliptic cohomology, and algebraic K-theory of rings.

Then what we want to do now is ask: how much "algebra" can I do in this setting? Because, in a genralization of the "smash product" of pointed spaces, it turns out that Spectra is a symmetric monoidal (∞-)category!

The first thing to note is that the functor H:Ab⭢Spectra is lax symmetric monoidal. In other words, it's not the case that H(A⊗B)≃H(A)⊗H(B), but it is true that H of a (commutative) monoid in Ab is a (commutative) monoid in Spectra.

As a result, every Eilenberg MacLane spectrum HR, for R a ring, is a monoid object in Spectra, and we'd call it a "ring spectrum." One terminological note: monoids in Ab are rings, so monoids in Spectra are ring spectra, but HR is not, for instance, a RING object in Ab.

But, again, lots of spectra which aren't Eilenberg MacLane spectra are ring spectra. In particular the topological K-theory spectra KU and KO are both commutative rings, as well MG, where MG is the cobordism theory associated to G oriented bordisms.

It also happens that the sphere spectrum 𝕊=Σ^∞(S^0)={S^0, S¹, S²,...} is a commutative ring spectrum. In fact, 𝕊 is the monoidal unit in Spectra, so sometimes people call Spectra 𝕊-modules, and ring spectra 𝕊-algebras.

Really there's an enormous amount to say about all of this, and about a thousand different directions one could take a thread on the topic, so I don't think I'll say a whole lot more. But just a few quick hits:

1. John Rognes has generalized classical Galois theory to the setting of ring spectra, and this has proven really useful both theoretically and computationally. In particular, one can use non-discrete groups for Galois groups, and do Galois descent and Galois cohomology!

2. Lurie and others have written a whole lot about doing algebraic geometry in this setting. In particular, there is a really useful notion of "spectral scheme," and many classical AG concepts have been developed in this framework.

3. The whole world of chromatic homotopy theory can be thought of as investigating the "prime ideals" of the sphere spectrum 𝕊, which turn out to be the usual prime ideals of ℤ, plus for each prime p∊ℤ, a new list of prime ideals K(n) called the Morava K-theories.

(see this older thread I wrote for more on chromatic homotopy theory:

)3.5: One caveat though: the general theory of "ideals" DOES NOT WORK in spectra. It doesn't make sense to talk about ideals or prime ideals. What we can talk about is localizing and thick subcategories (and boy howdy we sure do talk about em...).

4. Some people feel that working in spectra some kind of approach to working over the "field with one element." In particular, since 𝕊 is the monoidal unit in spectra instead of Hℤ, there's a ring map 𝕊⭢Hℤ, and we can "descend" along it. So 𝕊 is a base which is below ℤ!

4 cont'd: This sort of thing is somewhat wrapped up in ideas and writing of Jack Morava as well as Katia Consani and Alain Connes. See here for a little more about it: alainconnes.org/docs/gammasets…

5. There are also two major extensions of the theory of derived algebra: motivic homotopy theory, which (very roughly) does the whole story about spectra again but uses schemes instead of spaces and recovers the Weil cohomology theories (instead of the Eilenberg-Steenrod ones).

5 cont'd: And there's equivariant homotopy theory, which tries to redo the whole story of spectra but in the category of G-spaces, i.e. spaces with an action of a group. Both of these things have been and continue to be very active areas of research.

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On this MLK Day eve, I want you all to see this & know this as well as the “I Have A Dream” speech. Speechless at his wisdom and truth. "No other ethnic group has been a slave on American soil. And then color became a stigma." Transcribed remainder below.

“America freed slaves in 1863, but gave them no land or nothing to get started on. Emancipation for the negro was really freedom to hunger. Freedom to the winds & rains of heaven. Freedom w/o food to eat or land to cultivate. Therefore it was freedom & famine at the same time.”

"When white Americans tell the negro to lift himself up by his bootstraps, they don’t look at the legacy of slavery & segregation. We ought to do all we can to lift ourselves up, but it’s a cruel jest to say to a bootless man he ought to lift himself up by his own bootstraps.”

Just met an amazing man in Endcliffe Park, Sheffield. Tony Foulds was an 8-yr-old playing in the park when a US plane crashed in Feb 1944. He has diligently maintained the memorial ever since. He was planting new flowers. Almost 75 yrs of service. What a man. I’m in bits 😢👏🏻

He’s been sweeping the leaves away all morning and has replanted and cleaned the whole thing. He re-told the story of the day so vividly and knew the names of all 10 who died. “They are like family to me even though I never knew them” 👏🏻👏🏻👏🏻

Right... I’ve been back to talk to Tony. He doesn’t want a medal. All he wants is a fly-past on the 75th anniversary on the 22nd Feb 2019. Can anyone help? Who has contacts in the Royal Air Force or Red Arrows?

New #QAnon Thread

6/30/18

Patriots FIGHT! WWG1WGA!

#QArmy #WWG1WGA #Trump #potus #PatriotsFight #WeThePeople #GreatAwakening #DrainTheSwamp @realDonaldTrump @POTUS

6/30/18

Patriots FIGHT! WWG1WGA!

#QArmy #WWG1WGA #Trump #potus #PatriotsFight #WeThePeople #GreatAwakening #DrainTheSwamp @realDonaldTrump @POTUS

New #QAnon 1645.

Play LOUD.

Be PROUD!

FIGHT for what is RIGHT!

FIGHT for FREEDOM!

They will not win.

Divide they try.

Fail they will.

TOGETHER.

WWG1WGA!

#QArmy #WWG1WGA #Trump #potus #PatriotsFight #GreatAwakening #DrainTheSwamp @realDonaldTrump

Play LOUD.

Be PROUD!

FIGHT for what is RIGHT!

FIGHT for FREEDOM!

They will not win.

Divide they try.

Fail they will.

TOGETHER.

WWG1WGA!

#QArmy #WWG1WGA #Trump #potus #PatriotsFight #GreatAwakening #DrainTheSwamp @realDonaldTrump

New #QAnon 1646.

POWER TO THE PEOPLE.

THEY WANT YOU DIVIDED.

#QArmy #WWG1WGA #WeThePeople #GreatAwakening #Trump #potus #PatriotsFight #DrainTheSwamp @realDonaldTrump

POWER TO THE PEOPLE.

THEY WANT YOU DIVIDED.

#QArmy #WWG1WGA #WeThePeople #GreatAwakening #Trump #potus #PatriotsFight #DrainTheSwamp @realDonaldTrump

I've started planning for my #totalmunisummer Every line, end to end. 74 days from end of school year to beginning. @peterhartlaub @hknightsf

Tomorrow is the first day of summer vacation, and also the first day of #totalmunisummer. We will ride from Dogpatch to the Marina on the 22 Fillmore, hit the Marina branch of @SFPublicLibrary and come back to 4th and King on the 30 Stockton. @sfmta_muni will the ride be smooth?

#totalmunisummer is kicking off! Our first stop id is 13410 and our first coach is number 5476. Destination: Bay St.

#New PA cong. map increases D chance of winning the House majority by 4%. They only need a 6.8% generic ballot margin now (down from 7.7) to be favored.

Diff. will be closer to 10% in Nov when uncertainty from polls is ⬇️. Redraw is a very big deal.

Diff. will be closer to 10% in Nov when uncertainty from polls is ⬇️. Redraw is a very big deal.

One major reason the new PA map is such a big gain for Democrats is that it solidifies, not just tips, PA districts in Philly burbs. Also creates one additional vulnerable R district. (Easily seen in this map made by @JMilesColeman)

My forecasting model will have new PA map in the near future. This is not just a simple switch; it’s a coding redesign, restructuring of input, & drawing new geographic files. Takes a while (for me, a student).

In the meantime, just add 3-4% to the model: bit.ly/2018_house

In the meantime, just add 3-4% to the model: bit.ly/2018_house