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nLab > Latest Changes: model structure for complete Segal spaces

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeFeb 16th 2010
- (edited Oct 23rd 2012)
- PermaLink
Author: Urs
Format: MarkdownI expanded [[complete Segal space]] a little bit and started [[model structure for complete Segal spaces]]

I expanded complete Segal space a little bit and started model structure for complete Segal spaces
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeMar 12th 2012
- PermaLink
Author: Urs
Format: MarkdownItexI added to the _Definition_-section at _[[model structure for complete Segal spaces]]_ the explicit statement of its nature as a left Bousfield localization of the Reedy model structure. In the course of this I reorganized the section somewhat. Hoping to come back to this entry later to prettify it more.

I added to the Definition-section at model structure for complete Segal spaces the explicit statement of its nature as a left Bousfield localization of the Reedy model structure. In the course of this I reorganized the section somewhat.

Hoping to come back to this entry later to prettify it more.
- CommentRowNumber3.
- CommentAuthorMike Shulman
- CommentTimeApr 3rd 2012
- PermaLink
Author: Mike Shulman
Format: MarkdownItexIn the section on "relation to quasi-categories", the page [[model structure for complete Segal spaces]] defines the adjunction $t_! \dashv t^!$ using the functor $$t(k,l) = \Delta^k \times Ex^\infty \Delta^l.$$ However, Joyal and Tierney define this adjunction using instead $$t(k,l) = \Delta^k \times (\Delta^l)'.$$ where $(\Delta^l)'$ is the nerve of the groupoid freely generated by $[l]$. Are these the same for some reason?

In the section on “relation to quasi-categories”, the page model structure for complete Segal spaces defines the adjunction $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>t</mi> <mo>!</mo></msup></mrow><annotation encoding="application/x-tex">t_! \dashv t^!</annotation></semantics></math>$ using the functor
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo>×</mo><msup><mi>Ex</mi> <mn>∞</mn></msup><msup><mi>Δ</mi> <mi>l</mi></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">t(k,l) = \Delta^k \times Ex^\infty \Delta^l.</annotation></semantics></math>$
However, Joyal and Tierney define this adjunction using instead
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo>×</mo><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>l</mi></msup><mo stretchy="false">)</mo><mo>′</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">t(k,l) = \Delta^k \times (\Delta^l)'.</annotation></semantics></math>$
where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>l</mi></msup><mo stretchy="false">)</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">(\Delta^l)'</annotation></semantics></math>$ is the nerve of the groupoid freely generated by $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>l</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[l]</annotation></semantics></math>$ . Are these the same for some reason?
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeApr 3rd 2012
- PermaLink
Author: Urs
Format: MarkdownItexThis is wrong notation for that groupoid. I am fixing it now. Thanks.

This is wrong notation for that groupoid. I am fixing it now. Thanks.
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeApr 3rd 2012
- (edited Apr 3rd 2012)
- PermaLink
Author: Urs
Format: MarkdownItexI have changed the notation to "$\Delta_J[n]$" (for J-T's " $(\Delta^n)'$ ") to harmonize with the notation of the more recent discussion at _[[model structure for dendroidal complete Segal spaces]]_ [here](http://ncatlab.org/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces#RelationToDendroidalSets).

I have changed the notation to “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Δ</mi> <mi>J</mi></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta_J[n]</annotation></semantics></math>$ ” (for J-T’s ” $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">(\Delta^n)'</annotation></semantics></math>$ “) to harmonize with the notation of the more recent discussion at model structure for dendroidal complete Segal spaces here.
- CommentRowNumber6.
- CommentAuthorMike Shulman
- CommentTimeApr 3rd 2012
- PermaLink
Author: Mike Shulman
Format: MarkdownItexGreat, thanks!

Great, thanks!
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeOct 23rd 2012
- PermaLink
Author: Urs
Format: MarkdownI have touched _[[model structure for complete Segal spaces]]_, added some hpyerlinks, some more proposition environments.

I have touched model structure for complete Segal spaces, added some hpyerlinks, some more proposition environments.

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nLab > Latest Changes: model structure for complete Segal spaces