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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 30th 2010
   • (edited Nov 30th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[tmf]] a section that gives an outline of the proof strategy for how to compute the homotopy groups of the $tmf$-spectrum from global sections of the $E_\infty$-structure sheaf on the moduli stack of elliptic curves. A point which I wanted to emphasize is that 1. The problem of constructing $tmf$ as global sections of an $\infty$-structure sheaf has a tautological solution: take the underlying space to be $Spec tmf$. 1. From this tautological but useless solution one gets to the one that is used for actual computations by one single crucial fact: In the $\infty$-topos over the $\infty$-site of formal duals of $E_\infty$-rings, the dual $Spec M U$ of the Thom spectrum, is a [[well-supported object]]. the terminal morphism $$ Spec M U \to *$$ in the $\infty$-topos is an effective epimorphism, hence a covering of the point. Using this we can pull back the tautological solution of the problem to the cover and then compute there. This is what actually happens in practice: the decategorification of the pullback of $Spec tmf$ to $Spec M U$ is the moduli stack of elliptic curves. And it is a happy coincidence that despite this drastic decategorification, there is still enough information left to compute $\mathcal{O} Spec tmf$ on that.

   added to tmf a section that gives an outline of the proof strategy for how to compute the homotopy groups of the $tmf$-spectrum from global sections of the $E_\infty$-structure sheaf on the moduli stack of elliptic curves.

   A point which I wanted to emphasize is that

   1. The problem of constructing $tmf$ as global sections of an $\infty$-structure sheaf has a tautological solution: take the underlying space to be $Spec tmf$.

   2. From this tautological but useless solution one gets to the one that is used for actual computations by one single crucial fact:

      In the $\infty$-topos over the $\infty$-site of formal duals of $E_\infty$-rings, the dual $Spec M U$ of the Thom spectrum, is a well-supported object. the terminal morphism

      $$Spec M U \to *$$

      in the $\infty$-topos is an effective epimorphism, hence a covering of the point.

   Using this we can pull back the tautological solution of the problem to the cover and then compute there. This is what actually happens in practice: the decategorification of the pullback of $Spec tmf$ to $Spec M U$ is the moduli stack of elliptic curves. And it is a happy coincidence that despite this drastic decategorification, there is still enough information left to compute $\mathcal{O} Spec tmf$ on that.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 7th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have considerably expanded the [idea-section](http://ncatlab.org/nlab/show/tmf#idea) at _[[tmf]]_. Also I started some notes at _[Definition and construction -- Decomposition via Arithmetic fracture squares](http://ncatlab.org/nlab/show/tmf#DecomopositionViaArithmeticSquares)_, which is however very much stubby still.

   I have considerably expanded the idea-section at tmf. Also I started some notes at Definition and construction – Decomposition via Arithmetic fracture squares, which is however very much stubby still.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 9th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave added to _[[tmf]]_ a section _[Maps to K-theory and to Tate K-theory](http://ncatlab.org/nlab/show/tmf#MapToTateKTheory)_. Also I have split the "Definition and Construction"-section into a Definition-section and a Construcion-section and added some actual (though basic) content to the Definition section (the Construction-section remains very piecemeal, naturally but nevertheless woefully).

   Have added to tmf a section Maps to K-theory and to Tate K-theory.

   Also I have split the “Definition and Construction”-section into a Definition-section and a Construcion-section and added some actual (though basic) content to the Definition section (the Construction-section remains very piecemeal, naturally but nevertheless woefully).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 14th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a list of the [low degree homotopy groups of tmf](http://ncatlab.org/nlab/show/tmf#HomotopyGroups)

   added a list of the low degree homotopy groups of tmf

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 20th 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexCorrected the indexing on the table in #4 (started at 1 instead of 0)

   Corrected the indexing on the table in #4 (started at 1 instead of 0)