We know that Kan complexes are fibrant objects in the classical model structure on the category of simplicial sets. Similarly, if we take the Joyal model structure, the fibrant objects are quasi categories, which are models for ($\infty$,1) categories. I was wondering if ($\infty$,n)categories can be modelled as fibrant objects in some model "space". I have just learned that there do exist some spaces called Segal spaces where model structure does exist. Could someone kindly tell me briefly the idea of Segal spaces and the relevant model structure here? Also, what's the case for ($\infty,\infty$) categories? I am a physics first year grad student, starting with higher category theory. Thanks.

We know that Kan complexes are fibrant objects in the classical model structure on the category of simplicial sets. Similarly, if we take the Joyal model structure, the fibrant objects are quasi categories, which are models for $(\infty,1)$ categories. I was wondering if $(\infty,n)$ categories can be modelled as fibrant objects in some model "space". I have just learned that there do exist some spaces called Segal spaces where model structure does exist. Could someone kindly tell me briefly the idea of Segal spaces and the relevant model structure here? Also, what's the case for $(\infty,\infty)$ categories?

I am a physics first year grad student, starting with higher category theory. Thanks.