TITLE: This is not a category. What is it?
QUESTION [10 upvotes]: 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.)

REPLY [2 votes]: One of the intuitions of (simplicial) T-complexes is that you can make the simplicial singular complex $SX$ of a space into a natural Kan complex, by making choices of the appropriate retractions in the  models. But of course these choices are not unique, and what is unclear is what are the relations among these natural fillers! The idea has something to do with "composition" since in a Kan complex the filler of a horn makes the remaining face in some sense the composition of the other faces; the filler is a "program" for determining this composition. 
Keith Dakin (1977) then axiomatised the idea that these natural fillers are in some sense "thin". So the axioms are: 

Degenerate elements are thin. 
Every horn has a unique thin filler. 
If every face but one of a thin element is thin, then so also is the remaining face. 

A T-complex is of rank $\leq n$ if every element of dimension $>n$ is thin. 
T-complexes of rank $1$ are equivalent to groupoids, and (Dakin) those of rank $2$ are equivalent to crossed modules over groupoids. Nick Ashley (1978) completed the job by showing, among other things, that  $T$-complexes are equivalent to crossed complexes, and giving the topological example, the fundamental $T$-complex of a filtered space. 
These theses are available from here.