Are functors defining (semi-)simplicial sets injective on objetcs?

Question

The original definition of semi-simplicial sets by Eilenberg and Zilber deals with a collection of sets (or a graded set), one for each dimension. Thus, the sets of simplices of distinct dimensions are disjoint by design. On the other hand, the categorical definition of semi-simplicial sets as contravariant functors from $\mathbf{\Delta}_\mathrm{inj}$ to $\mathbf{Set}$ does not require this functor to be injective on objects. It is thus possible to have a simplicial set $X_\bullet$ with $X_1=X_0=\{*\}$ and $X_i=\varnothing$ for $i>1$, that is the situation when the set of 1-simplices and 0-simplices is the same singleton set. I have the impression that this discrepancy in definitions is not essential (in the sense that within the class of natural equivalence of functors $\mathbf{\Delta}_\mathrm{inj}^\mathrm{op} \to \mathbf{Set}$ it is always possible to find a functor injective on objects. Do I miss something?

Answer 1

Certainly any functor $F:\mathcal C\to \mathrm{Set}$ is naturally isomorphic to one injective on objects, at least if $\mathcal C$ is small. Then the objects of $\mathcal C$ themselves form a set, and we can replace the values $F(c)$ with the pairs $(c,F(c))$, which are distinct. This is related to the canonical model structure on the category of categories, in which the cofibrations are precisely the injective-on-objects functors.

Answer 2

There is a serious discrepancy in definitions, but it is not injectivity on objects. Eilenberg and Zilber define a semi-simplicial complex as a set $S$ of elements, called simplexes, together with two functions: The first associates with each simplex $s \in S$ an integer $\dim(s) \ge 0$ called its dimension, the second function assoaciates to each pair $(s,i)$ with $s \in S$ and $0 \le i \le \dim(s)$ its $i$-th face $d_is$ such that $\dim(d_is) = \dim(s) - 1$ and $d_i d_js = d_{j-1}d_i s$ for $i < j$ and $\dim s > 1$. What is missing are degeneracy operators.

However, if you add degeneracy operators to the Eilenberg and Zilber definition and require the adequate identities for faces and degeneracies, then the remaining discrepancy is injectivity on objects. But as you say: It is irrelevant. Each semi-simplicial set is isomorphic to one which is injective on objects.