Let $F: \mathcal{E}\to \mathcal{C}$ be a cocartesian fibration of ordinary categories satisfying certain conditions so that we can take the animation of $F$. How to show that $\mathrm{Ani}(F)$ is a cocartesian fibration of $\infty$-categories ?
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2$\begingroup$ The animation of F is the induced functor between the infinity-categories of simplicial sheaves? $\endgroup$– David Roberts ♦Commented Apr 28 at 5:19
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4$\begingroup$ I think it would help if there were a little bit of context. There are a lot of people that know about this stuff, but they don't use the terminology "animated." You won't even tell us what conditions the categories need to satisfy to be "animated!" I'll try not to be cranky about new terminology, but it's not great when it gets hard to even communicate because people have completely abandoned the usual terms. $\endgroup$– Jonathan BeardsleyCommented Apr 28 at 7:03
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$\begingroup$ @JonathanBeardsley Thanks, I am going to add more context. $\endgroup$– Nguyễn Xuân BáchCommented Apr 28 at 7:10
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1$\begingroup$ I think saying 'anima' for infinity-groupoids/homotopy types is borderline fair game, but the verb 'animate' here hides a lot of subtlety, so that's a bit too much. $\endgroup$– David Roberts ♦Commented Apr 28 at 9:23
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1$\begingroup$ This terminology is explained in the first page of section 5.1.4 of Purity for Flat Cohomology by Cesnavicius and Scholze. $\endgroup$– GabrielCommented Oct 13 at 16:35
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