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In HTT, Lurie's proof of the $(\infty,1)$-Yoneda lemma uses the standard enriched Yoneda lemma and simplicial sets as a consequence of his larger commitment to doing higher category theory via quasi-categories. I haven't read much on other higher categorical perspectives, (e.g. Segal categories, complete Segal spaces, etc.) but other forms of the Yoneda lemma can surely be stated and proved in these other settings.

Given this, here's my question: Does anyone know of a 'model independent' proof of $(\infty,1)$-Yoneda lemma, appealing only to the desired universal properties of $\infty$-categories and $\infty$-groupoids?

Follow up question: If there isn't such a proof, why is that the case?

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You might be interested in this paper

https://arxiv.org/abs/1506.05500

(and the previous ones of the series that set up the basic theory)

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  • $\begingroup$ I was looking for this paper to answer this question - but I see you got there first! $\endgroup$ Dec 1, 2017 at 10:43
  • $\begingroup$ Sometimes you just have to be fast :) $\endgroup$
    – fosco
    Dec 1, 2017 at 21:01
  • $\begingroup$ I'll apply a fast fourier transform to myself next time ... :)! $\endgroup$ Dec 2, 2017 at 3:39
  • $\begingroup$ Just be sure it's categorified Fourier! $\endgroup$
    – fosco
    Dec 2, 2017 at 14:45

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