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Is there a definition of $(\infty, n)$-category using just simplicial sets?

This is the case for $n \leq 2$.

Is the forgetful functor from saturated $n$-trivial complicial sets to simplicial sets an embedding of 1-categories?

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    $\begingroup$ Wait, Is it true for $n=2$? $\endgroup$
    – Simon Henry
    Commented Sep 16, 2023 at 19:11
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    $\begingroup$ @SimonHenry First question: yes (see kerodon.net/tag/01W9) Second question: I don't know. $\endgroup$
    – Daniel Bruegmann
    Commented Sep 16, 2023 at 19:29
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    $\begingroup$ Interesting! In any case, it is known that Street Nerve (from strict infinity category to simplicial sets) is not fuly faitfull (but it become fuly faithfull when viewed as valued in complicial sets with the appropriate notion of thiness. I don't quite remember what is the counter-exemple, but I'd expect it would provide also a negative answer to your question... So the answer might be in old paper of Street and/or Roberts... but I haven't been able to find it. $\endgroup$
    – Simon Henry
    Commented Sep 16, 2023 at 22:29
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    $\begingroup$ I don't know what's the current status of the theory, but didn't Riehl's complicial sets project precisely model (oo,n)-categories as simplicial sets with structure? $\endgroup$
    – Daniel Teixeira
    Commented Sep 20, 2023 at 15:04
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    $\begingroup$ @DanielTeixeira Yes, complicial sets are simplicial sets with extra structure. I am asking if the forgetful functor is faithful when restricted to a certain subcategory, which I am open to changing. $\endgroup$
    – Daniel Bruegmann
    Commented Oct 21 at 18:38

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