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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 30th 2012
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThere is a standard embedding $\Delta \hookrightarrow \Theta$. Suppose we were to find a functor that "freely oriented" not only the simplices, but also freely oriented any object of $\Theta$ and whose restriction to $\Delta$ is actually Street's oriental functor. Since this can be thought of as replacing all commutative diagrams with diagrams that are "commutative" up to a non-invertible cell of appropriate height, and since $\Theta$ is dense in $\omega$-cat, shouldn't we be able to extend this to all $\omega$-categories. It seems like the functor applied to a strict $\omega$-category $X$ gives a new strict $\omega$-category $OX$ such that $NorLax(X,Y)=Hom(OX,Y)$, where NorLax means normalized lax $\omega$-functors, but I'm not sure. Moreover, I saw that the strict $\omega$-category of lax $\omega$-functors $Lax(X,Y)$ from $X$ to $Y$ has been defined as oriental-weighted limit $Lim_\Delta^O(Hom(CX,Y))$, where $CX$ denotes the constant simplicial object at $X$. Basically, I found this functor while I was messing around with trying to find a cellular Dold-Kan correspondence, and I'm wondering if it could serve any purpose.

   There is a standard embedding $\Delta \hookrightarrow \Theta$. Suppose we were to find a functor that “freely oriented” not only the simplices, but also freely oriented any object of $\Theta$ and whose restriction to $\Delta$ is actually Street’s oriental functor. Since this can be thought of as replacing all commutative diagrams with diagrams that are “commutative” up to a non-invertible cell of appropriate height, and since $\Theta$ is dense in $\omega$-cat, shouldn’t we be able to extend this to all $\omega$-categories. It seems like the functor applied to a strict $\omega$-category $X$ gives a new strict $\omega$-category $OX$ such that $NorLax(X,Y)=Hom(OX,Y)$, where NorLax means normalized lax $\omega$-functors, but I’m not sure. Moreover, I saw that the strict $\omega$-category of lax $\omega$-functors $Lax(X,Y)$ from $X$ to $Y$ has been defined as oriental-weighted limit $Lim_\Delta^O(Hom(CX,Y))$, where $CX$ denotes the constant simplicial object at $X$.

   Basically, I found this functor while I was messing around with trying to find a cellular Dold-Kan correspondence, and I’m wondering if it could serve any purpose.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 30th 2012
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   Author: DavidRoberts
   Format: MarkdownItexThere is a famous conjecture about zeroes of the zeta function. Suppose we were to find a proof of this conjecture... (etc) Sorry, it just sounded funny because I guessed the punchline after the second sentence. Personally I think it is very interesting and could be used for ...something, I just don't know what.

   There is a famous conjecture about zeroes of the zeta function. Suppose we were to find a proof of this conjecture… (etc)

   Sorry, it just sounded funny because I guessed the punchline after the second sentence. Personally I think it is very interesting and could be used for …something, I just don’t know what.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 30th 2012
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   Author: Harry Gindi
   Format: MarkdownItex@David: No, I mean that I actually gave a cute little construction for this functor, and moreover, the formula that you get ends up being "the only thing that it could possibly be". However, on its own without a corresponding Alexander-Whitney map, it's not really useful for me. I was hoping that maybe someone else had wondered about generalized orientals (I have heard that Benabou once thought about a version of the 2-truncated orientals that can be applied to arbitrary strict 2-categories). I was also wondering if anyone had thought of Θ-indexed analogue of descent, but that was sort of a fishing expedition.

   @David: No, I mean that I actually gave a cute little construction for this functor, and moreover, the formula that you get ends up being “the only thing that it could possibly be”. However, on its own without a corresponding Alexander-Whitney map, it’s not really useful for me. I was hoping that maybe someone else had wondered about generalized orientals (I have heard that Benabou once thought about a version of the 2-truncated orientals that can be applied to arbitrary strict 2-categories).

   I was also wondering if anyone had thought of Θ-indexed analogue of descent, but that was sort of a fishing expedition.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 30th 2012
   • (edited May 30th 2012)
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   Author: zskoda
   Format: MarkdownItexIt is a good question. Maybe Joyal or Cisinski would know something about this. They have collaborated on $n$-categorical descent a couple of year ago or so, and some $\Theta$-machinery is presumably involved.

   It is a good question. Maybe Joyal or Cisinski would know something about this. They have collaborated on $n$-categorical descent a couple of year ago or so, and some $\Theta$-machinery is presumably involved.