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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeNov 8th 2017
   • (edited Nov 8th 2017)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexIn the case of strict ω-categories, there is an ostensible generalization of the comma construction, where, given maps $f:A \to B \leftarrow C:g$ we let $f\downarrow_n g$ denote the limit of the cospan $A^{D^{n-1}} \times_{B^{\partial D^{n-1}}} C^{D^{n-1}} \to B^{D^{n-1}} \times_{B^{\partial D^{n-1}}} B^{D^{n-1}} \leftarrow B^{D^n}$. The objects of this strict ω-category are pairs of $n-1$-cells $a\in A$, $c \in C$ whose images under $f$ resp. $g$ are parallel, together with an $n$-cell $\gamma: f(a) \to g(c)$, and the arrows are obvious arrows forming commuting diagrams. I think this construction might be interesting since we have a natural equipment of pairs of projections $\pi_{s,t}: f \downarrow_n g \to f \downarrow_{n-1} g$ for each $n$. In particular, this should be interesting if we want to define cartesian fibrations of ω-categories globally, when, given an isofibration $p:E\to B$ we ask for the existence of a left adjoint right inverse to the induced maps $E^{D^n} \to p\downarrow_n B$ for each $n\in \mathbb{N}$, and moreover we require that these are compatible with the projections (some kind of Beck-Chevalley condition, I think, and what this should mean intuitively, is that every cartesian $n$-cell lies between cartesian $n-1$-cells). Is this kind of idea covered anywhere?

   In the case of strict ω-categories, there is an ostensible generalization of the comma construction, where, given maps $f:A \to B \leftarrow C:g$ we let

   $f\downarrow_n g$ denote the limit of the cospan

   $A^{D^{n-1}} \times_{B^{\partial D^{n-1}}} C^{D^{n-1}} \to B^{D^{n-1}} \times_{B^{\partial D^{n-1}}} B^{D^{n-1}} \leftarrow B^{D^n}$.

   The objects of this strict ω-category are pairs of $n-1$-cells $a\in A$, $c \in C$ whose images under $f$ resp. $g$ are parallel, together with an $n$-cell $\gamma: f(a) \to g(c)$, and the arrows are obvious arrows forming commuting diagrams.

   I think this construction might be interesting since we have a natural equipment of pairs of projections $\pi_{s,t}: f \downarrow_n g \to f \downarrow_{n-1} g$ for each $n$.

   In particular, this should be interesting if we want to define cartesian fibrations of ω-categories globally, when, given an isofibration $p:E\to B$

   we ask for the existence of a left adjoint right inverse to the induced maps $E^{D^n} \to p\downarrow_n B$ for each $n\in \mathbb{N}$, and moreover we require that these are compatible with the projections (some kind of Beck-Chevalley condition, I think, and what this should mean intuitively, is that every cartesian $n$-cell lies between cartesian $n-1$-cells).

   Is this kind of idea covered anywhere?

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeNov 12th 2017
   • (edited Nov 12th 2017)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexI found something in Gray's old papers through Mike's old notes on 3-toposes, and the correct construction uses the (op)lax Gray cotensor but is related.

   I found something in Gray’s old papers through Mike’s old notes on 3-toposes, and the correct construction uses the (op)lax Gray cotensor but is related.