4 added 108 characters in body

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).

3 added 300 characters in body

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."

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?). Kan? (I don't have the paper at hand, so this is speculation on my partpart.) 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.

If

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

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."

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.

If this is true, then it also provides an answer to (3).

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