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1. • CommentRowNumber1.
   • CommentAuthorMirco Richter
   • CommentTimeFeb 21st 2013
   • (edited Feb 21st 2013)
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   Author: Mirco Richter
   Format: MarkdownItexThis is more or less a warm up on my previous post on that topic. The point is, that I got no response the the core question: In [[over-(infinity,1)-category]], what is written under 'idea', it doesn't seem to make sense in some pictures on (oo,1)-categories. Let me copy/paste and comment it to make my concern clear: >For $C$ an [[(∞,1)-category]] and $X \in C$ an [[object]], the _over-$(\infty,1)$-category_ or _slice $(\infty,1)$-category_ $C_{/X}$ is the $(\infty,1)$-category whose [[objects]] are [[morphism]] $p : Y \to X$ in $C$, whose morphisms $\eta : p_1 \to p_2$ are [[2-morphisms]] [...] such that a diagram commutes. Lets say our (oo,1)-category has objects and simplicial sets as homs. Then a 'morphism' $p : Y \to X$ is the image of a simplicial set map $p: \Delta[0] \to hom(Y,X)$. But here's the problem: If we have two different such morphisms $p_1:Y_1 \to X$ and $p_2:Y_2 \to X$ for $Y_1 \neq Y_2$, then what exactly is a two morphism $\eta:p_1 \to p_2$ here? There are the '2-morphisms' $\eta_1: \Delta[1] \to hom(Y_1,X)$ and $\eta_2: \Delta[1] \to hom(Y_2,X)$, but that is different from something like $\eta:p_1 \to p_2$. I stuck, because in the simplicial enriched picture a two morphism is not something like $\eta: hom(Y_1,X) \to hom(Y_2,X)$,so the diagram doesn't make sense here.

   This is more or less a warm up on my previous post on that topic. The point is, that I got no response the the core question:

   In over-(infinity,1)-category, what is written under ’idea’, it doesn’t seem to make sense in some pictures on (oo,1)-categories. Let me copy/paste and comment it to make my concern clear:

   For $C$ an (∞,1)-category and $X \in C$ an object, the over-$(\infty,1)$-category or slice $(\infty,1)$-category $C_{/X}$ is the $(\infty,1)$-category whose objects are morphism $p : Y \to X$ in $C$, whose morphisms $\eta : p_1 \to p_2$ are 2-morphisms

   […] such that a diagram commutes.

   Lets say our (oo,1)-category has objects and simplicial sets as homs. Then a ’morphism’ $p : Y \to X$ is the image of a simplicial set map $p: \Delta[0] \to hom(Y,X)$. But here’s the problem:

   If we have two different such morphisms $p_1:Y_1 \to X$ and $p_2:Y_2 \to X$ for $Y_1 \neq Y_2$, then what exactly is a two morphism $\eta:p_1 \to p_2$ here?

   There are the ’2-morphisms’ $\eta_1: \Delta[1] \to hom(Y_1,X)$ and $\eta_2: \Delta[1] \to hom(Y_2,X)$, but that is different from something like $\eta:p_1 \to p_2$.

   I stuck, because in the simplicial enriched picture a two morphism is not something like $\eta: hom(Y_1,X) \to hom(Y_2,X)$,so the diagram doesn’t make sense here.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 21st 2013
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   Author: Mike Shulman
   Format: MarkdownItexA morphism from $p_1:Y_1 \to X$ to $p_2:Y_2 \to X$ in $C_{/X}$ is a morphism $q:Y_1\to Y_2$ in $C$ together with a homotopy $\Delta[1] \to hom(Y_1,X)$ between $p_1$ and $p_2\circ q$.

   A morphism from $p_1:Y_1 \to X$ to $p_2:Y_2 \to X$ in $C_{/X}$ is a morphism $q:Y_1\to Y_2$ in $C$ together with a homotopy $\Delta[1] \to hom(Y_1,X)$ between $p_1$ and $p_2\circ q$.

3. • CommentRowNumber3.
   • CommentAuthorMirco Richter
   • CommentTimeFeb 21st 2013
   • (edited Feb 21st 2013)
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   Author: Mirco Richter
   Format: MarkdownItexThanks Mike. Is there a paper or something, where I can look for this in a bigger context? Comes a bit from nowhere to me...

   Thanks Mike. Is there a paper or something, where I can look for this in a bigger context? Comes a bit from nowhere to me…

4. • CommentRowNumber4.
   • CommentAuthorMirco Richter
   • CommentTimeFeb 21st 2013
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   Author: Mirco Richter
   Format: MarkdownItexAh ok... Have to look in more detail on this tomorrow, but this is a consequence of HTT, prop. 5.5.5.12., I think In case that's true, no other reference needed.

   Ah ok… Have to look in more detail on this tomorrow, but this is a consequence of HTT, prop. 5.5.5.12., I think

   In case that’s true, no other reference needed.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 21st 2013
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   Author: Tim_Porter
   Format: MarkdownItexJean-Marc Cordier and I, way back, used something on the lines that Mike suggests, but I forget where. Even further back, lost in the mists of time, Keith Hardie, Heiner Kamps and myself wrote a little note that showed how a homotopy coherent slice category definition corresponded to inverting the homotopy equivalences over X, but HTT is likely to have all the necessary details.

   Jean-Marc Cordier and I, way back, used something on the lines that Mike suggests, but I forget where. Even further back, lost in the mists of time, Keith Hardie, Heiner Kamps and myself wrote a little note that showed how a homotopy coherent slice category definition corresponded to inverting the homotopy equivalences over X, but HTT is likely to have all the necessary details.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 21st 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexI don't think that Mike "suggested" anything. He just repeated the definition.

   I don’t think that Mike “suggested” anything. He just repeated the definition.

7. • CommentRowNumber7.
   • CommentAuthorKarol Szumiło
   • CommentTimeFeb 21st 2013
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   Author: Karol Szumiło
   Format: MarkdownItex> Jean-Marc Cordier and I, way back, used something on the lines that Mike suggests, but I forget where. Was this written in the language of quasicategories or simplicial categories? Is there a reference for slices of (fibrant) simplicial categories?

   Jean-Marc Cordier and I, way back, used something on the lines that Mike suggests, but I forget where.

   Was this written in the language of quasicategories or simplicial categories? Is there a reference for slices of (fibrant) simplicial categories?

8. • CommentRowNumber8.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 21st 2013
   • (edited Feb 21st 2013)
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   Author: Tim_Porter
   Format: MarkdownItexWe had some difficulty sorting that one out (for simplicially enriched categories) and I don't know that the eventual published version in the TAMS had it in. @Urs: `suggest': To bring or call to mind by logic or association; evoke.

   We had some difficulty sorting that one out (for simplicially enriched categories) and I don’t know that the eventual published version in the TAMS had it in.

   @Urs: ‘suggest’: To bring or call to mind by logic or association; evoke.

9. • CommentRowNumber9.
   • CommentAuthorMirco Richter
   • CommentTimeFeb 22nd 2013
   • (edited Feb 22nd 2013)
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   Author: Mirco Richter
   Format: MarkdownItex@Urs/Mike: What definition? ... I ask because the definition then likely contains more then what Mike said. More on the other higher k-morphisms and the simplicial structure of the hom in the slice, I guess.

   @Urs/Mike: What definition? … I ask because the definition then likely contains more then what Mike said. More on the other higher k-morphisms and the simplicial structure of the hom in the slice, I guess.

10. • CommentRowNumber10.
    • CommentAuthorMirco Richter
    • CommentTimeFeb 22nd 2013
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    Author: Mirco Richter
    Format: MarkdownItex@Tim: What had it in? (Don't know what TAMS means)

    @Tim: What had it in? (Don’t know what TAMS means)

11. • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 22nd 2013
    • (edited Feb 22nd 2013)
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    Author: Tim_Porter
    Format: MarkdownItex@Mirco: Sorry. TAMS = Transactions Amer. Math. Soc. More precisely Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54. but as I said I have not got a copy with me, so am not sure that that idea found its way into the final version (or for that matter that we got it `right'!) Have a look at that and also the other papers we wrote (Cordier and myself) at about that time. The methods are still useful, although sometimes they may seem less model category theoretic than is the current preferred flavour. (The work for that paper was initially done on a walk in 1984, in a beautiful bit of countryside near the sea, looking across to the mountains of North Wales. It then took us 15 years to work it up to the full detail of the paper. These things take time to mature sometimes! :-)) (Edit: Slightly later. I found a copy of the preprint form. I have checked and it looks like we did not give the definition but mentioned the difficulty of doing so (with our tools at that time). Checking in Lurie, he seems to use the join operation to define the hom in a slice, and starngely th join is around in a hidden way in that TAMS paper, BTW the reference I found in Lurie is earlier in the text than the one you indicated.)

    @Mirco: Sorry. TAMS = Transactions Amer. Math. Soc. More precisely

    Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.

    but as I said I have not got a copy with me, so am not sure that that idea found its way into the final version (or for that matter that we got it ‘right’!) Have a look at that and also the other papers we wrote (Cordier and myself) at about that time. The methods are still useful, although sometimes they may seem less model category theoretic than is the current preferred flavour. (The work for that paper was initially done on a walk in 1984, in a beautiful bit of countryside near the sea, looking across to the mountains of North Wales. It then took us 15 years to work it up to the full detail of the paper. These things take time to mature sometimes! :-))

    (Edit: Slightly later. I found a copy of the preprint form. I have checked and it looks like we did not give the definition but mentioned the difficulty of doing so (with our tools at that time). Checking in Lurie, he seems to use the join operation to define the hom in a slice, and starngely th join is around in a hidden way in that TAMS paper, BTW the reference I found in Lurie is earlier in the text than the one you indicated.)

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 22nd 2013
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    Author: Mike Shulman
    Format: MarkdownItexI wasn't giving a formal definition, or referring to one; I was trying to explain what the *Idea* section would mean if you chose to talk about simplicially enriched categories.

    I wasn’t giving a formal definition, or referring to one; I was trying to explain what the Idea section would mean if you chose to talk about simplicially enriched categories.