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Atrium > Mathematics, Physics & Philosophy: Base change for straightening

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- CommentRowNumber1.
- CommentAuthorHarry Gindi
- CommentTimeJul 21st 2017
- (edited Jul 21st 2017)
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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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">X \to S</annotation></semantics></math>$ a map of simplicial sets, a simplicial functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>C</mi><mo stretchy="false">[</mo><mi>S</mi><mo stretchy="false">]</mo><mo>→</mo><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\phi:C[S] \to C^{op}</annotation></semantics></math>$ for a simplicially-enriched category $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ , and we define

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi> <mrow><mi>X</mi><mo>,</mo><mi>ϕ</mi></mrow></msub><mo>=</mo><mi>C</mi><mo stretchy="false">[</mo><msup><mi>X</mi> <mo>▹</mo></msup><mo stretchy="false">]</mo><msub><mo lspace="0.16667em" rspace="0.16667em">∐</mo> <mrow><mi>C</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></msub><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">M_{X,\phi} = C[X^\triangleright] \coprod_{C[X]} C^{op}</annotation></semantics></math>$

Then we define $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Str</mi> <mi>ϕ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mi>sSet</mi><mo stretchy="false">/</mo><mi>X</mi><mo>→</mo><msup><mi>sSet</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">Str_{\phi} (X) : sSet/X \to sSet^C</annotation></semantics></math>$ by the formula $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Str</mi> <mi>ϕ</mi></msub><mi>X</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Map</mi> <mrow><msub><mi>M</mi> <mrow><mi>X</mi><mo>,</mo><mi>ϕ</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Str_{\phi}X(C) = Map_{M_{X,\phi}}( C , p )</annotation></semantics></math>$ , where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ is the adjoined cone point.

2.2.1.1 states that given a simplicial functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\pi: C \to C'</annotation></semantics></math>$ , there is a natural isomorphism between $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi> <mo>!</mo></msub><msub><mi>Str</mi> <mi>ϕ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_! Str_{\phi}(X)</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Str</mi> <mrow><msup><mi>π</mi> <mi>op</mi></msup><mo>∘</mo><mi>ϕ</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Str_{\pi^{op} \circ \phi}(X)</annotation></semantics></math>$ where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">\pi_!</annotation></semantics></math>$ is left adjoint to the pullback $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>π</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\pi^*</annotation></semantics></math>$ induced by $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>$ 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:

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi> <mrow><mi>X</mi><mo>,</mo><mi>ϕ</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>π</mi> <mo>*</mo></msup><msub><mi>M</mi> <mrow><mi>X</mi><mo>,</mo><msup><mi>π</mi> <mi>op</mi></msup><mo>∘</mo><mi>ϕ</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_{X,\phi}(-,p) \to \pi^* M_{X, \pi^{op} \circ \phi} ( - , p)</annotation></semantics></math>$

It seems like morally everything should work because $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi> <mrow><mi>X</mi><mo>,</mo><msup><mi>π</mi> <mo lspace="0em" rspace="0.16667em">op</mo></msup><mo>∘</mo><mi>ϕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">M_{X,\pi^{\op} \circ \phi}</annotation></semantics></math>$ is a pushout, and the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">\pi_!</annotation></semantics></math>$ should mean pushing forward that mapping object by change of base, but I’m stuck.
- CommentRowNumber2.
- CommentAuthorDavidRoberts
- CommentTimeJul 21st 2017
- PermaLink
Author: DavidRoberts
Format: MarkdownItexWelcome back, Harry!

Welcome back, Harry!
- CommentRowNumber3.
- CommentAuthorHarry Gindi
- CommentTimeJul 21st 2017
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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.
- CommentRowNumber4.
- CommentAuthorDavidRoberts
- CommentTimeJul 22nd 2017
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Author: DavidRoberts
Format: MarkdownItexSorry to not have any mathematical comments on your question

Sorry to not have any mathematical comments on your question
- 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?
- 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.
- 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/
- 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.

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Atrium > Mathematics, Physics & Philosophy: Base change for straightening