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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.)

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Simplices are labeled stars in the graph underlying the category, and faces/degenerations are adding identities and dropping edges in the star? –  Mariano Suárez-Alvarez Dec 14 '12 at 16:51
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Do you, like in a category, have an extension property that tells you how much you need to check for a cell to exist? –  Will Sawin Dec 14 '12 at 17:25
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It's either a painting of a category or a pipe. –  Will Jagy Dec 14 '12 at 22:25
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(Will Jagy, I meant.) –  Todd Trimble Dec 15 '12 at 14:56
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I believe that any distinction between a category and a painting of a category would be considered "evil". –  Tom Goodwillie Dec 16 '12 at 0:38

3 Answers 3

up vote 17 down vote accepted

It's called a groupoid. Given an object $A$, call the degenerate edge from $A$ to itself the identity map at $A$. Given an edge $f:A\to B$, let $f^{-1}:B\to A$ denote the unique edge that fills in a 2-simplex whose other two edges are $f$ and the identity. Given $f:A\to B$ and $g:B\to C$, define $gf$ to be the unique edge that fills in a 2-simplex whose other two edges are $f^{-1}$ and $g$. The condition for $n=3$ using the maps $f$, $1$, and $f$ gives that $(f^{-1})^{-1}=f$. It is now easy to see that the identity maps are units for our composition operation, and $f^{-1}f=ff^{-1}=1$.

The condition for $n=3$ now says that $(hf^{-1})(gf^{-1})^{-1}=hg^{-1}$. Setting $g=1$ tells us that $(hf^{-1})f=h$, and so we have cancellation on the right. Setting $h=1$, $f=x$, and $g=yx$ gives $x^{-1}y^{-1}=(yx)^{-1}$. Setting $f=y$, $g=z^{-1}$, and $h=xy$ now says that $x(z^{-1}y^{-1})^{-1}=(xy)z$. Since $(z^{-1}y^{-1})^{-1}=yz$, this says our composition operation is associative. It follows that composition defines a groupoid, and it is now easy to see the simplicial set is the nerve of this groupoid.

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A groupoid gives an example if this, but I don't think every example arises from a groupoid. –  David Roberts Dec 14 '12 at 22:31
    
Is there a mistake in my argument? –  Eric Wofsey Dec 15 '12 at 0:06
    
Ah, I didn't understand Tom fully from his first description. –  David Roberts Dec 15 '12 at 0:30
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I have to go back to my example and think. I was convinced it was not a category at all (of course if it were a category it would be a groupoid), but your argument seems right. –  Tom Goodwillie Dec 15 '12 at 12:50

For related ideas, (rather than being strictly an answer), have a look at 'simplicial T-complex' as explored by some of Ronnie Brown's students is related. The nerve of a groupoid is one. I would also suggest that a strict Segal space is going to give you another related notion. The idea of $T$-complex is that it is a Kan complex with a subset of 'thin' simplices in each dimension such that any horn has a unique thin filler. These are to satisfy some axioms that I will not give here but which are at 'simplicial T-complex' in the n-Lab. Another idea to check out is that of complicial set, due to Dom Verity.

I think that Eric may be correct, but that he does not give all the detail, (I just checked that $f= (f^{-1})^{-1}$, which seems to be true), so the point of my `answer' is that the generalisations of this are also important.

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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:

  1. Degenerate elements are thin.
  2. Every horn has a unique thin filler.
  3. 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.

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