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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeSep 18th 2011
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   Author: Harry Gindi
   Format: MarkdownItexSince some of you have thought much longer and much more intensely about the notion of a ω-fibration, I was wondering if someone could point out the deficiencies in the following proposed definition of such a fibration: Let $p:E\to B$ be a strict ω-functor between strict ω-categories. We will attempt to axiomatize the idea that $p$ is a Grothendieck (non-street) fibration: We say that an $n$-cell $\phi:D_n\to E$ with $s(\phi)=a$ and $t(\phi)=b$ is __cartesian__ if for any $n-1$-cell $c:D_{n-1}\to E$ such that $s(c)=s(a)$ and $t(c)=t(a)$, the induced map $E(c,a)\to lim(B(pc,pa)\to B(pc,pb) \leftarrow E(c,b))$ exhibits $E(c,a)$ as a strict (maybe homotopy?) pullback of strict $\omega$-categories. (I believe that asking for this to be a good homotopy notion will make it so it is well-preserved by the cellular nerve functor, which should, conjecturally, be a right Quillen functor between the natural model structure on strict $\omega$-categories and the CIsinski-Joyal model structure on $\omega$-quasicategories). Then $p$ is a __strict $\omega$-fibration__ if for every $n$-cell $\lambda: s\to p(t)$ in $B$, there exists a cartesian $n$-cell $\lambda'$ of $E$ such that $p(\lambda')=\lambda$.

   Since some of you have thought much longer and much more intensely about the notion of a ω-fibration, I was wondering if someone could point out the deficiencies in the following proposed definition of such a fibration:

   Let $p:E\to B$ be a strict ω-functor between strict ω-categories. We will attempt to axiomatize the idea that $p$ is a Grothendieck (non-street) fibration:

   We say that an $n$-cell $\phi:D_n\to E$ with $s(\phi)=a$ and $t(\phi)=b$ is cartesian if for any $n-1$-cell $c:D_{n-1}\to E$ such that $s(c)=s(a)$ and $t(c)=t(a)$, the induced map $E(c,a)\to lim(B(pc,pa)\to B(pc,pb) \leftarrow E(c,b))$ exhibits $E(c,a)$ as a strict (maybe homotopy?) pullback of strict $\omega$-categories. (I believe that asking for this to be a good homotopy notion will make it so it is well-preserved by the cellular nerve functor, which should, conjecturally, be a right Quillen functor between the natural model structure on strict $\omega$-categories and the CIsinski-Joyal model structure on $\omega$-quasicategories).

   Then $p$ is a strict $\omega$-fibration if for every $n$-cell $\lambda: s\to p(t)$ in $B$, there exists a cartesian $n$-cell $\lambda'$ of $E$ such that $p(\lambda')=\lambda$.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 4th 2011
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   Author: Mike Shulman
   Format: MarkdownItexWell, your definition of cartesianness looks exactly like Definition 1 at [[n-fibration]] interpreted in hom-categories (modulo strictness), and the assertion that cartesian lifts exist for all $n$ looks exactly like the conditions (1) and (2) of Definition 2 there, strictified and unraveled. Thus, if the schematic definition at [[n-fibration]] is correct, then I would guess that the only thing wrong with yours is that it's missing an analogue of condition (3) (precomposition with preserves cartesianness).

   Well, your definition of cartesianness looks exactly like Definition 1 at n-fibration interpreted in hom-categories (modulo strictness), and the assertion that cartesian lifts exist for all $n$ looks exactly like the conditions (1) and (2) of Definition 2 there, strictified and unraveled. Thus, if the schematic definition at n-fibration is correct, then I would guess that the only thing wrong with yours is that it’s missing an analogue of condition (3) (precomposition with preserves cartesianness).

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 4th 2011
   • (edited Oct 4th 2011)
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   Author: Harry Gindi
   Format: MarkdownItexHmm... Do you know of a way to give an analogue of condition (3) similar to how I gave analogues of conditions (1) and (2) that do not depend on some kind of coinduction? Also, in case you're interested, I figured out an easy construction using polygraphs of the strict ω-category P such that morphisms P->C classify ω-equivalences in C. Let $S_0$ be a set with two elements, and then the condition at each stage is to let the new generating set $S_n\rightrightarrows S^*_{n-1}$ simply be the equivalence relation of $x\parallel y$. In fact, this condition for $n$-polygraphs gives the "free contractible n-groupoid" on the set $S_0$. This would allow us to repair the strictness problems with (1) and (2), since we can classify equivalences using $P$ and its suspensions. For the record, I care about these strict notions only because the strict notions are the ones that generalize easily to the homotopical case, where they can be systematically weakened.

   Hmm…

   Do you know of a way to give an analogue of condition (3) similar to how I gave analogues of conditions (1) and (2) that do not depend on some kind of coinduction?

   Also, in case you’re interested, I figured out an easy construction using polygraphs of the strict ω-category P such that morphisms P->C classify ω-equivalences in C. Let $S_0$ be a set with two elements, and then the condition at each stage is to let the new generating set $S_n\rightrightarrows S^*_{n-1}$ simply be the equivalence relation of $x\parallel y$. In fact, this condition for $n$-polygraphs gives the “free contractible n-groupoid” on the set $S_0$. This would allow us to repair the strictness problems with (1) and (2), since we can classify equivalences using $P$ and its suspensions.

   For the record, I care about these strict notions only because the strict notions are the ones that generalize easily to the homotopical case, where they can be systematically weakened.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 4th 2011
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   Author: Mike Shulman
   Format: MarkdownItexIt's always possible to unravel a coinductive definition into a definition that doesn't look coinductive. I could do it for you, but it would be a good exercise in learning to use coinduction... (-:

   It’s always possible to unravel a coinductive definition into a definition that doesn’t look coinductive. I could do it for you, but it would be a good exercise in learning to use coinduction… (-: