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

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?

• I don't really see how it changes anything if two $X_n$ are equal as sets. Commented Feb 3, 2020 at 19:59
• I'm not even convinced there's a discrepancy. If I say the 1-simplices are $S_1$ and the 0-simplices are $S_0$, what is guaranteeing these are disjoint sets? Commented Feb 3, 2020 at 23:55
• @KevinCarlson The original definition of Eilenberg and Zilber says that a semi-simplicial complex is a set of elements, called simplexes, together with two functions: The first associates with each simplex an integer $\ge 0$ called its dimension, the second function describes the face operators. Thus we get pairwise disjoint sets of $i$-simplexes. Commented Feb 6, 2020 at 14:50
• @PaulFrost Ah, got it. Commented Feb 6, 2020 at 14:54

## 2 Answers

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.

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.

• Although the simplicial identity $d_i d_j = d_{j-1}d_i$ for $i<j$ is not formulated as such in the original EZ paper, the formula (1.1) contains it implicitly. Granted, the degeneracy operators are missing in that paper - but we do not need them for semi-simplicial (not simplicial!) objects, do we?
– p_k
Commented Feb 10, 2020 at 8:27
• @p_k Concerning the $d_i$ you are right. I edited my answer. You are also right that many authors make a distinction between semi-simplicial and simplicial objects. See Chapter 3 of arxiv.org/pdf/0809.4221.pdf for a discussion of notation. You see that notation is a bit "fuzzy" in the literature, but I admit that my interpretation may be old-fashioned. Concerning the purpose of degeneracies have a look at the discussion after Definition 2.4 in mathi.uni-heidelberg.de/~rueschoff/ss17sset/sset.pdf. Commented Feb 10, 2020 at 9:53