# Degeneracies for semi-simplicial Kan complexes

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.)

• I think this could help you ( I haven't read it careful): arxiv.org/abs/1210.5650 – Buschi Sergio Dec 3 '12 at 12:20
• @Buschi: Yep, I did see that when it came out recently. – Mike Shulman Dec 4 '12 at 2:55
• Thakns for a nice question, Mike. It is going to save me lots of work. – Kolya Ivankov May 4 '14 at 12:54

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."

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).

• There have been a few people who have explicitly adopted the term "semi-simplicial set" for this type of object in recent years; McClure is one. – Tyler Lawson Mar 7 '11 at 13:26
• I like the term "simplicial set w/o degeneracies," even though its long-winded. – John Klein Mar 7 '11 at 13:31
• Also, as far as I can tell "semi-simplicial complex" was originally used by Eilenberg and Zilber for the version without degeneracies; they added the adjective "complete" when there the degeneracies were given. Later people seem to have dropped the "complete" in referring to the version with degeneracies, and then later dropped the "semi-". So calling the degeneracy-less ones "semi-simplicial" is actually the more faithful to the original! (-: – Mike Shulman Mar 7 '11 at 18:05
• Kan refers to what we now call "simplicial sets" as c.s.s. complexes. I guess "c.s.s." refers to the Eilenberg-Zilber "complete semi-simplicial." – John Klein Mar 7 '11 at 18:11
• There is a nice quick description of the left adjoint to the forgetful functor from simplicial sets to semi-simplicial sets in Fritsch and Piccinini Cellular structures in algebraic topology'' together some not so usual material. See Section 4.4. They use the name presimplicial sets'' for these animals. They are related to categories without identity morphisms, which appear occasionally in the combinatorial literature. – Peter May Dec 3 '12 at 3:32

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 .

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).

• Thanks! This doesn't seem to me to be any different from Rourke and Sanderson's proof, though; it just has some bits omitted. In addition to their 2.1, he seems to use that a semisimplicial set which satisfies the "extension condition" has the RLP versus inclusions that induce inclusions of strong deformation retracts upon geometric realization, which is what R+S need their simplicial approximation theorem to prove (if "extension condition" means the usual RLP against horns). – Mike Shulman Dec 4 '12 at 2:54
• You are right. I interpreted your question 1. as a reference request and didn't see Kan's paper mentioned, so I thought it might be useful to provide that reference. I didn't have the necessary privileges to comment, so I decided to post a slightly expanded comment as an answer. Did you see that Peter May posted a comment to John Klein's answer suggesting section 4.4 of Fritsch and Piccinini? In the bibliography of that book there are some papers of Fritsch (although most of them in German) that seem relevant, especially (1972), but I don't have access to those papers right now. – Martin Dec 4 '12 at 7:21