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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 21st 2017
   • (edited Jul 21st 2017)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexReading back through HTT (Hi by the way), and going back over the straightening construction. Proposition 2.2.1.1 states that some properties are obvious. They don't appear to be obvious. First, recall the situation: We have $X \to S$ a map of simplicial sets, a simplicial functor $\phi:C[S] \to C^{op}$ for a simplicially-enriched category $C$, and we define $M_{X,\phi} = C[X^\triangleright] \coprod_{C[X]} C^{op}$ Then we define $Str_{\phi} (X) : sSet/X \to sSet^C$ by the formula $Str_{\phi}X(C) = Map_{M_{X,\phi}}( C , p )$, where $p$ is the adjoined cone point. 2.2.1.1 states that given a simplicial functor $\pi: C \to C'$, there is a natural isomorphism between $\pi_! Str_{\phi}(X)$ and $Str_{\pi^{op} \circ \phi}(X)$ where $\pi_!$ is left adjoint to the pullback $\pi^*$ induced by $\pi$ on the simplicial categories of simplicial op-presheaves. There is an obvious morphism in one direction arising from the pushout and the fact that the cone point is pushed out again to a cone point: $M_{X,\phi}(-,p) \to \pi^* M_{X, \pi^{op} \circ \phi} ( - , p)$ It seems like morally everything should work because $M_{X,\pi^{\op} \circ \phi}$ is a pushout, and the $\pi_!$ should mean pushing forward that mapping object by change of base, but I'm stuck.

   Reading back through HTT (Hi by the way), and going back over the straightening construction. Proposition 2.2.1.1 states that some properties are obvious. They don’t appear to be obvious.

   First, recall the situation: We have $X \to S$ a map of simplicial sets, a simplicial functor $\phi:C[S] \to C^{op}$ for a simplicially-enriched category $C$, and we define

   $M_{X,\phi} = C[X^\triangleright] \coprod_{C[X]} C^{op}$

   Then we define $Str_{\phi} (X) : sSet/X \to sSet^C$ by the formula $Str_{\phi}X(C) = Map_{M_{X,\phi}}( C , p )$, where $p$ is the adjoined cone point.

   2.2.1.1 states that given a simplicial functor $\pi: C \to C'$, there is a natural isomorphism between $\pi_! Str_{\phi}(X)$ and $Str_{\pi^{op} \circ \phi}(X)$ where $\pi_!$ is left adjoint to the pullback $\pi^*$ induced by $\pi$ on the simplicial categories of simplicial op-presheaves.

   There is an obvious morphism in one direction arising from the pushout and the fact that the cone point is pushed out again to a cone point:

   $M_{X,\phi}(-,p) \to \pi^* M_{X, \pi^{op} \circ \phi} ( - , p)$

   It seems like morally everything should work because $M_{X,\pi^{\op} \circ \phi}$ is a pushout, and the $\pi_!$ should mean pushing forward that mapping object by change of base, but I’m stuck.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 21st 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexWelcome back, Harry!

   Welcome back, Harry!

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 21st 2017
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexGood to be back I guess. I have a grad school interview in two weeks; let's hope it goes alright.

   Good to be back I guess. I have a grad school interview in two weeks; let’s hope it goes alright.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 22nd 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexSorry to not have any mathematical comments on your question

   Sorry to not have any mathematical comments on your question

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 22nd 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexDid you notice that Emily [just blogged](https://golem.ph.utexas.edu/category/2017/07/what_is_the_comprehension_cons.html) about a generalization of the straightening/unstraightening construction?

   Did you notice that Emily just blogged about a generalization of the straightening/unstraightening construction?

6. • CommentRowNumber6.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 22nd 2017
   • (edited Jul 22nd 2017)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexNo, complete fluke. I picked HTT back up as a refresher for the interview haha, also going through atiyah-macdonald, lang, hartshorne, and I even cracked open ETCS for an hour. I'll check that out. I think Lurie did a pretty mediocre job of actually working out all of these proofs in HTT, but it was until now the only place where straightening/unstraightening was proven. I am still interested in the answer. I spent a while trying to find the inverse of that map and trying to unravel the shriek functor to no avail.

   No, complete fluke. I picked HTT back up as a refresher for the interview haha, also going through atiyah-macdonald, lang, hartshorne, and I even cracked open ETCS for an hour.

   I’ll check that out. I think Lurie did a pretty mediocre job of actually working out all of these proofs in HTT, but it was until now the only place where straightening/unstraightening was proven.

   I am still interested in the answer. I spent a while trying to find the inverse of that map and trying to unravel the shriek functor to no avail.

7. • CommentRowNumber7.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 26th 2017
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexAhahahaha I found it. I remembered asking this question years ago, and I thought I remembered Urs answering it. Here we go. https://nforum.ncatlab.org/discussion/2442/fully-formal-proof-of-htt-proposition-2211/

   Ahahahaha I found it. I remembered asking this question years ago, and I thought I remembered Urs answering it.

   Here we go.

   https://nforum.ncatlab.org/discussion/2442/fully-formal-proof-of-htt-proposition-2211/

8. • CommentRowNumber8.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 26th 2017
   • (edited Jul 26th 2017)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexI guess I'm going to run headlong into another proof later on which is the one that Urs gave involving another weird cone construction. Edit: Found all of my old topics for the nasty bits of chapter 2. I might compile them into a little companion guide for HTT.

   I guess I’m going to run headlong into another proof later on which is the one that Urs gave involving another weird cone construction.

   Edit: Found all of my old topics for the nasty bits of chapter 2.

   I might compile them into a little companion guide for HTT.