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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeOct 6th 2012
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   Author: Mike Shulman
   Format: MarkdownItexDo we have a page about the left adjoint of $N:Cat \to SSet$? What is it called?

   Do we have a page about the left adjoint of $N:Cat \to SSet$? What is it called?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 6th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexDon't know that we do. Joyal has been calling it the "fundamental category" functor, denoted $\tau_1: Set^{\Delta^{op}} \to Cat$, in his Notes on Quasi-categories. We also have that terminology, but for directed spaces. I'd think we could pretty harmlessly borrow that terminology again for your context.

   Don’t know that we do. Joyal has been calling it the “fundamental category” functor, denoted $\tau_1: Set^{\Delta^{op}} \to Cat$, in his Notes on Quasi-categories. We also have that terminology, but for directed spaces. I’d think we could pretty harmlessly borrow that terminology again for your context.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 7th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI suppose that's as good as anything, thanks. I added a section to [[fundamental category]].

   I suppose that’s as good as anything, thanks. I added a section to fundamental category.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeOct 7th 2012
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   Author: Tim_Porter
   Format: MarkdownItexI have seen it called fundamental category in other sources than Joyal's work, so would support that here.

   I have seen it called fundamental category in other sources than Joyal’s work, so would support that here.

5. • CommentRowNumber5.
   • CommentAuthorEmily Riehl
   • CommentTimeOct 10th 2012
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   Author: Emily Riehl
   Format: MarkdownItexFor what it's worth I like calling it the ``homotopy category'' functor and writing $h$, particularly when you restrict the adjunction to the full subcategory of quasi-categories. I think Lurie also uses this convention.

   For what it’s worth I like calling it the “homotopy category” functor and writing $h$, particularly when you restrict the adjunction to the full subcategory of quasi-categories. I think Lurie also uses this convention.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 10th 2012
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   Author: Urs
   Format: MarkdownItex> particularly when you restrict the adjunction to the full subcategory of quasi-categories. I like to enforce that clause and write: $$ \array{ QuasiCat &&\hookrightarrow&& sSet \\ & {}_{\mathllap{Ho}}\searrow && \swarrow_{\mathrlap{\tau_1}} \\ && Cat } \,. $$ I have added that to the entry.

   particularly when you restrict the adjunction to the full subcategory of quasi-categories.

   I like to enforce that clause and write:

   $$\array{ QuasiCat &&\hookrightarrow&& sSet \\ & {}_{\mathllap{Ho}}\searrow && \swarrow_{\mathrlap{\tau_1}} \\ && Cat } \,.$$

   I have added that to the entry.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 11th 2012
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   Author: Mike Shulman
   Format: MarkdownItexYeah, it makes the most sense to me to restrict the usage of "homotopy category" to simplicial sets that are quasicategories. Although I could see an argument going the other way, I guess.

   Yeah, it makes the most sense to me to restrict the usage of “homotopy category” to simplicial sets that are quasicategories. Although I could see an argument going the other way, I guess.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeOct 11th 2012
   • (edited Oct 11th 2012)
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   Author: Urs
   Format: MarkdownItexTo me the thing is that the simplicial set could also be modelling a higher category in a different way, such that $\tau_1$ would _not_ compute the homotopy category. For instance it could be the thought of as the nerve of an $n$-category for $n \gt 1$.

   To me the thing is that the simplicial set could also be modelling a higher category in a different way, such that $\tau_1$ would not compute the homotopy category. For instance it could be the thought of as the nerve of an $n$-category for $n \gt 1$.