EDIT The question was based on an error, as it turns out. In fact my example is a category (and therefore a groupoid), by Eric Wofsey's argument. I can't remember why I thought it wasn't, and I feel a little silly, but I am very glad that it is.

In the nerve of a small category an $n$-simplex is determined by $n+1$ objects and $n$ arrows: $$x_0\to x_1\to x_2\to\dots \to x_n $$

What do you call a simplicial set in which an $n$-simplex is determined by a different part its $1$-skeleton, as follows: $$ (x_0\to x_1\ ,\ x_0\to x_2\ ,\ \dots\ ,\ x_0\to x_n)\ ?$$ You could say that instead of a composition law for arrows you have a certain kind of decomposition law.

Have people run into this before? What is it called? Do you know any good ways of thinking about it?

I have run into something formally analogous to this (with cosimplicial commutative rings rather than simplicial sets), and I'm trying to get to know it better.

EDIT I see that I did not say all of what I meant. I should have said that given $$ (x_0\to x_1\ ,\ x_0\to x_2\ ,\ \dots\ ,\ x_0\to x_n)\ $$ (i.e. a map into $X$ from the $1$-dimensional object indicated by the above) there is a unique map $\Delta^n\to X$ extending it. By the way, the dual thing is also true in my examples: given $$ (x_0\to x_n\ ,\ x_1\to x_2\ ,\ \dots\ ,\ x_{n-1}\to x_n)\ $$ (i.e. a map into $X$ from the $1$-dimensional object indicated by the above) there is a unique map $\Delta^n\to X$ extending it. (In fact, there is also an arrow-reversing involution $X_n\to X_n$ for all $n$, if you know what I mean.)