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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeFeb 4th 2016
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   Author: Fosco
   Format: MarkdownItexHas anybody studied some generalizations of the notion of derivator, like for example 1. 2-Functors $Cat^op\to \mathcal{K}$, where $\mathcal{K}$ is any 2-category playing the role of $CAT$? 2. 2-Functors $(Cat^op)^n \to CAT$ for some $n\ge 2$, such that when restricted on a single component they are derivators? Do these animals show up in nature?

   Has anybody studied some generalizations of the notion of derivator, like for example

   1. 2-Functors $Cat^op\to \mathcal{K}$, where $\mathcal{K}$ is any 2-category playing the role of $CAT$?
   2. 2-Functors $(Cat^op)^n \to CAT$ for some $n\ge 2$, such that when restricted on a single component they are derivators?

   Do these animals show up in nature?

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 4th 2016
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   Author: DavidRoberts
   Format: MarkdownItexThe things in 1. might show up in Riehl and Verity's work on model-independent $(\infty,1)$-category theory using 2-categories.

   The things in 1. might show up in Riehl and Verity’s work on model-independent $(\infty,1)$-category theory using 2-categories.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 5th 2016
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   Author: Mike Shulman
   Format: MarkdownItexFor (1), if they don't, they certainly could. Let $E$ be an $(\infty,1)$-topos, then any internal $(\infty,1)$-category $C$ in $E$ has cotensors $C^A$ for any $A\in Cat$, which have homotopy categories $Ho(C^A)$ that are internal 1-categories in $E$. So we should get a "derivator" $Cat^{op} \to 1Cat(E)$. For (2) I can't think offhand of any examples that don't factor through $\prod : Cat^n \to Cat$; what leads you to ask?

   For (1), if they don’t, they certainly could. Let $E$ be an $(\infty,1)$-topos, then any internal $(\infty,1)$-category $C$ in $E$ has cotensors $C^A$ for any $A\in Cat$, which have homotopy categories $Ho(C^A)$ that are internal 1-categories in $E$. So we should get a “derivator” $Cat^{op} \to 1Cat(E)$.

   For (2) I can’t think offhand of any examples that don’t factor through $\prod : Cat^n \to Cat$; what leads you to ask?