Degeneracies for semi-simplicial Kan complexes

Question

By a semi-simplicial set I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. Unlike when degeneracies are present, we have to include 1-dimensional horns: having fillers for 1-dimensional horns means that every vertex is both the source of some 1-simplex and the target of some 1-simplex.

I have been told that it is possible to choose degeneracies for any semi-simplicial Kan complex to make it into an ordinary simplicial Kan complex. For instance, to obtain a degenerate 1-simplex on a vertex $x$, we first find (by filling a 1-dimensional 1-horn) a 1-simplex $f\colon x\to y$, then we fill a 2-dimensional 2-horn to get a 2-simplex $f g \sim f$, and we can choose $g\colon x\to x$ to be degenerate. But obviously there are many possible choices of such a $g$.

I have three questions:

1. Where can I find this construction written down?

2. Is the choice of degeneracies unique in some "up to homotopy" sense? Ideally, there would be a space of choices which is always contractible.

3. Does a morphism of semi-simplicial Kan complexes necessarily preserve degeneracies in some "up to homotopy" sense? (A sufficiently positive answer to this would imply a corresponding answer to the previous question, by considering the identity morphism.)

Answer 1

The answer to (1) is to be found in

Rourke, C. P.; Sanderson, B. J.$\Delta$-sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. (2) 22 (1971), 321–338.

It is shown there that a Kan "semi-simplicial" set admits a compatible system of degeneracies.

By the way, the term "semi-simplicial" set is not the usual name for this term; it is usually called a "$\Delta$-set."

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Added: Given Mike's comment below, I realize now that the following sketch doesn't do the job.

I haven't looked at this paper recently, but I would imagine the way it goes is as follows: Let $$ \Delta^{\text{inj}} = \text{category of finite ordered sets and order preserving injections} $$

$$ \Delta = \text{category of finite ordered sets and order preserving maps} $$ The we have an inclusion functor $j: \Delta^{\text{inj}} \to \Delta$. Given a "semi-simplicial" set $X$ (i.e., a functor $X: \Delta^{\text{inj}} \to \text{Sets}$) we can form the left Kan extension $j_*X$ and then the restriction $j^*j_*X$ to get a natural map $X \to j^*j_*X$. One can ask whether this is a weak equivalence of simplicial sets. Perhaps what Rourke and Sanderson are doing is showing this map to be a weak homotopy equivalence, or maybe just so when $X$ is Kan? (I don't have the paper at hand, so this is speculation on my part.) One might argue as follows: Step a): show that $j^\ast j_\ast$ preserves colimits, Step b) show that the map $X \to j^\ast j_\ast X$ is a weak equivalence when $X$ is a standard "semi-simplicial" $n$-simplex, Step c) infer the general case by induction on simplices.

At any rate, if this is how it goes, then the outcome also provides an answer to (3).

Answer 2

The paper by Rourke and Sanderson writes that "it is a standard exercise to show that $g$ induces an isomorphism of homology and of fundamental groups" and then "applies" Whitehead's theorem; unfortunately that is not Whitehead's theorem on homotopy equivalences. A detailed proof due M. Zisman of the map from the "thick" to the usual realisation being a homotopy equivalence is given on p. 573 of "Nonabelian algebraic topology" (EMS Tract vol 15, August 2011), pdf available from my web page http://groupoids.org.uk/nonab-a-t.html .

Answer 3

Kan wrote a very short note while the paper by Rourke and Sanderson (mentioned in the other answers) was in the publishing process:

Daniel M. Kan, Is an ss complex a css complex? Advances in Mathematics Volume 4, Issue 2, April 1970, Pages 170–171.

(as explained in the comments "ss" refers to semi-simplicial set -- without degeneracies -- and "css" to what is known as simplicial set nowadays).

The proposition is: "An ss complex $X$ which satisfies the extension condition can be completed (although, in general, in many diferent ways)." and he states afterwards that "It is clear that any two such completions will have the same homotopy type."

The proof proceeds by an inductive construction of a simplicial set $WX$ such that there is an isomorphism $FWX \to X$ where $F$ is the forgetful functor from simplicial sets to semi-simplicial sets. As a crucial ingredient he uses the geometric realization from Rourke and Sanderson (he gives a reference to [1,1.3], which should probably be Proposition 2.1. on p.325 in the published version of Rourke and Sanderson's On $\Delta$-sets, I).

Answer 4

The Rourke and Sanderson paper.