Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise.

In my understanding, there are several models for $(\infty,1)$-categories: Joyal-Lurie's quasi-categories, simplicially enriched categories, Segal catgories, etc. . They are all, for the most part, models for the same mathematical objects and can be used interchangeably to develop higher category theory.

Here is something that sounds like it might be a model. Assume all categories below are as small as necessary.

Definition 1: A (special) simplicial set of categories $C_\bullet$ is a functor $C:\Delta^{\text{op}} \to \bf{Cat}$, denoted by $[k] \mapsto C_{[k]}$, satisfying the following pair of properties.

  • The map $\text{Mor}(C_\bullet) \to \text{Obj}(C_\bullet) \times \text{Obj}(C_\bullet)$ given by sending a morphism $f:x \to y$ to the pair $(x,y)$ is a map of simplicial sets. (Not sure if this is automatic)
  • The simplicial set $\text{Map}(x,y)$, defined as the union over all $k$ of the morphisms $f_{[k]}$ whose image under any degeneration map $C_{[k]} \to C_{[0]}$ is a morphism $x \to y$, is a Kan complex.

My question is just the following one.

Question: Does Definition 1 (or a corrected/similar definition) provide a known model for $(\infty,1)$-categories? If so, where can I find a reference?

It would be great if this reference related this model to one of the more popular ones, such as quasi-categories.

  • $\begingroup$ Your 'simplicial set of categories' is better known as 'a simplicial object in $\mathbf{Cat}$'. The first part of the definition is automatic, since $\mathrm{Mor},\mathrm{Obj}\colon \mathbf{Cat} \to \mathbf{Set}$ are functors and $(s,t)\colon \mathrm{Mor} \Rightarrow \mathrm{Obj}\times \mathrm{Obj}$ is a natural transformation. $\endgroup$ – David Roberts Jul 24 '19 at 0:45
  • $\begingroup$ Added [ct.category-theory] tag (and removed the AMScd command, not sure why that was there) $\endgroup$ – David Roberts Jul 24 '19 at 0:51
  • 2
    $\begingroup$ ...and a simplicial object in Cat is also equivalently an internal category in simplicial sets. That version of the question is what was answered by the one I linked in the duplicate mark. $\endgroup$ – Mike Shulman Jul 24 '19 at 0:56
  • $\begingroup$ Thanks, Mike. I was sure this was known. $\endgroup$ – David Roberts Jul 24 '19 at 3:21
  • $\begingroup$ Great, this is very helpful. $\endgroup$ – Julian Chaidez Jul 24 '19 at 7:42