17 Oct, 7 tweets, 1 min read
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
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
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
Some remarks:
-in the derived setting there are countably infinite "degrees of commutativity" between associative, 𝔸_∞, and commutative, 𝔼_∞. Setting 𝔸_∞ to 𝔼₁, then we denote the intermediate cases by 𝔼_n. There are maps 𝔼_n→𝔼_{n+1} such that colim(𝔼_n)≃𝔼_∞.
-the Eilenberg-MacLane functor H:Abelian groups→Spectra is lax symmetric monoidal, so it takes (commutative) monoids to (commutative) monoids, but not strongly, since the unit ℤ∈Ab does not go to 𝕊, the unit in Spectra, and H(A⊗B) is not equivalent to HA⊗HB.
-the way to "get back" to Set from Spectra or Spaces is to take homotopy groups, but taking homotopy groups does not often preserve desired algebraic structure. This difference is often mediated by a spectral sequence.
-if you follow the pattern of commutative = 𝔼_∞ and say that abelian groups should be 𝔼_∞-spaces, you won't be wrong exactly, but you'll be missing a lot of stuff because spectra can have negative homotopy groups, but spaces cannot

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