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Suppose we are given a category enriched over semi-simplicial sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this category all inner Kan conditions are satisfied, i.e. we have $\mathrm{Kan}(n,k)$ for all $n$ and $1\le k \le n-1$, but not necessarily $\mathrm{Kan}(n,0)$ and $\mathrm{Kan}(n,n)$.

Is there any "canonical" way to make this category become enriched over simplicial sets, i.e. to find "canonical up to equivalence" degeneracy maps, so that the category will become an $(\infty,1)$-category?

There are several papers on making a Kan-simplicial set out of Kan-semi-simplicial one, including the classical paper by Rouke and Sanderson and a more recent by McClure, but I haven't found anything concerning quasi-categories.

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The short answer is no. If you take the semi-simplicial set $X$ with one $0$-simplex and no higher simplices, then it has fillers for all inner horns (because it has no inner horns) but clearly does not admit degeneracies.

The question then becomes: what in addition to inner horn fillers could we assume to make this true (without restricting attention to Kan complexes)? One possible guess would be: add lifts along the inclusion of $X$ into $E[1]$, i.e. (the underlying semi-simplicial set of) the nerve of the groupoid freely generated by one isomorphism. I don't know whether this is sufficient, but this is a notable lifting property that follows from fillers for inner horns in the simplicial setting, but doesn't in the semi-simplicial one.

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Kolya! Thanks an $\infty$ many times for your question, and for pointing (me) to McClure's paper! :)

You might be interested to take a look at this paper (of mine), where a definition of the notion of a "cat" is presented: the globular (see, e.g., John Baez's An Introduction to n-Categories) multi-simplicial sets satisfying the (multi-simplicial version of the) Kan condition presented by McClure in Definition 5.2 of his paper ( - I think "higher Kan complexes" are a good name for them - ) are special kinds of cats, just like Kan complexes are special kinds of $\infty$-categories (in the sense of Boardman-Vogt, Joyal, Lurie, ...).

I did not read McClure's proof yet, but I'm "pretty confident" that what you're asking for should be possible. Also, an analogue of McClure's Theorem 5.3 should be true: it should be possible to canonically build a cat from any given "semi-cat", even if the "horizontal composition" in the semi-cat isn't strict. But as I said, I did not prove this, it's just a guess.

Another question that one could come up with is whether the analogue of Moore's proposition ( - whose formulation should be obvious - ) in this "higher" situation holds.

For several months now I tried to find out whether this definition of a "cat" has already been studied, and if so, why it can't be found in the literature ( - at least I did not find it). If anybody knows of further papers in this direction, for example answering my previous question, please drop a reference in the comments section below ( - if that is allowed).

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  • $\begingroup$ @Kolya Are you in Bonn, too? $\endgroup$
    – Daniel Gerigk
    Commented May 10, 2014 at 2:03
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    $\begingroup$ I wrote (somewhat sloppy) that "what you're asking for should be possible". Perhaps this is misleading. As Karol pointed out, it is not possible to obtain a simplicial set (which then is an $\infty$-category) from a semi-simplicial set satisfying the inner horn conditions just by adding degeneracy maps. So the immediate analogue of McClure's Theorem 5.3 does not hold, of course. What I had in mind was "building" a cat, i.e. not only adjoining degeneracy maps, but also degenerate simplices. $\endgroup$
    – Daniel Gerigk
    Commented May 10, 2014 at 16:13
  • $\begingroup$ Thanks for the answer and the comment. No, I wrote my PhD in Bonn, presently I'm in Goettingen, just haven't updated my profile. $\endgroup$
    – Kolya Ivankov
    Commented May 11, 2014 at 8:05

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