Criteria for being an $\infty$-category?

Question

Let $\mathcal{C}$ be a simplicial category, such that for any two objects $X, Y\in\mathcal{C}$, $\text{Hom}_{\mathcal{C}}(X,Y)$ is a simplicial commutative monoid. Is the simplicial nerve $\text{N}(\mathcal{C})$ an $(\infty, 1)$-category?

If for any two objects $X,Y\in \mathcal{C}$, $\text{Hom}_{\mathcal{C}}(X,Y)$ was a simplicial abelian group, this would be true as a consequence of Prop. 1.1.5.10 in J. Lurie's Higher Topos Theory.

Thanks

Answer 1

Since this example is kinda fun, let me spell it out. (The intuition should be clear though: the simplicial category I defined is really the result of taking a not-so-exciting (2,2)-category and applying the nerve. This 2-category has non-invertible 2-morphisms, so its nerve shouldn't magically have invertible 2-morphisms. And indeed that's what we'll see: the horn filling condition for 3-horns forces the 2-morphisms to be invertible (up to yet higher morphisms, which won't play a role here))

First of all, for any poset $P$ with pairwise maxima, the assignment $(a,b) \mapsto \mathrm{max}(a,b)$ gives $P$ the structure of a commutative monoid in the category of posets. Taking the nerve now yields a simplicial commutative monoid $NP$, and we can then build a simplicial category with one object, which I'll call $\mathcal{C}_P$ since $BNP$ looks strange.

Here is what the coherent nerve of $\mathcal{C}_P$ looks like in low dimensions:

• 0-simplices: there's only one.
• 1-simplices: elements of $P$
• 2-simplices: A pair of elements $(a,b)$ together with a third element $c$ such that $c \le \mathrm{max}(a,b)$.
• 3-simplices: A triple of elements $(a_1, a_2, a_3)$, a pair of elements $(b_{12}, b_{23})$ such that $b_{i, i+1} \le \mathrm{max}(a_i, a_{i+1})$, and an element $c$ such that $c \le \mathrm{max}(b_{12}, a_3), \mathrm{max}(a_1, b_{23})$

(It's helpful to draw pictures here. One should view $\mathrm{max}(a,b)$ as the 'actual' composite, and a 2-simplex as a 2-morphism from $c$ to the composite. Hopefully I've gotten my arrows in the right direction...)

Anyway, a map from $\Lambda^3_1$ is described by the same data of a 3-simplex except we don't necessarily require $c \le \mathrm{max}(b_{12}, a_{3})$. It can fill to a 3-simplex if and only if this condition happens to be satsfied, in which case it fills uniquely. Of course, said this way, it's clear that we should be able to find un-fillable horns.

As an explicit example, take $P= [1]$, and consider the horn specified by $a_2 = a_3 = b_{12} = b_{23} = 0$, $a_1 = c= 1$.

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Some bonus info which you may or may not care about (mostly in response to Porter's comment):

• Porter is of course correct that the proper birthplace for the modern version of these ideas is Cordier (and Porter himself!) and Boardmann-Vogt/Vogt, but I might add Segal and Leitch to that list, who were also thinking about homotopy coherent diagrams in the 70s but using topological categories instead of simplicial categories. I guess Dwyer and Kan's work is also related at the very least in spirit. Mainly I wanted to mention this because Segal's name for $\mathfrak{C}(K)$ is the 'explosion of $K$', and we should all totally use this name because it's great.
• Given a category enriched in marked simplicial sets (a model for $(\infty,1)$-categories) its coherent nerve can be given the structure of a "scaled simplicial set", which is a riff on Verity's general notion of stratified simplicial sets/ complicial sets and model $(\infty,2)$-categories. This is developed in Lurie's paper on $(\infty,2)$-categories and Goodwillie calculus.
• If instead you have a category enriched in quasi-categories, you can associate to each quasi-category $X$ the marked simplicial set $(X, E_X)$ where $E_X$ is the collection of 1-simplices which are equivalences. This will produce a fibrant marked simplicial set (I think) and then you can do the procedure in the previous bullet to get a (fibrant) scaled simplicial set.