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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.

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    $\begingroup$ See eg Ara's Higher quasi-categories vs higher Rezk spaces arxiv.org/abs/1206.4354 or Barwick's (∞, n)-Cat as a closed model category proquest.com/docview/305445747 $\endgroup$
    – David Roberts ♦
    Commented Oct 17 at 6:22
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    $\begingroup$ Segal spaces are another model for $(\infty,1)$-category, this time using simplicial spaces instead of simplicial sets ( so bi-simplicial sets). Not a model of $(\infty,n)$ or $(\infty,\infty)$. They are however easier to generalise to $(\infty,n)$ or $(\infty,\infty)$ than quasicategories (using either $\Theta$-spaces, or iterated Segal spaces). $\endgroup$
    – Simon Henry
    Commented Oct 17 at 13:09
  • $\begingroup$ @SimonHenry Could you briefly elaborate a bit on your comment, especially \theta spaces? Apologies for lack of background. $\endgroup$
    – Pinak Banerjee
    Commented Oct 18 at 2:22
  • $\begingroup$ You'll find a general overview on theta spaces and lots of references here : ncatlab.org/nlab/show/Theta-space in short they are a generalisation of Segal spaces that provide a model for $(\infty,n)$-categories. $\endgroup$
    – Simon Henry
    Commented Oct 18 at 14:10
  • $\begingroup$ Every model of $(\infty,n)$-category that I'm aware of arises as the fibrant objects of a model structure on some complete and cocomplete 1-category. That's usually the way the model is defined. This includes the case $n=\infty$. Many of these models are discussed in Barwick and Schommer-Pries, the most notable exception being Verity's complicial model. $\endgroup$
    – Tim Campion
    Commented Oct 20 at 17:19

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