# Is there a citeable source for generators and relations of simplicial sets?

Simplicial sets can be specified using generators and relations, in complete analogy with groups, rings, etc.

More precisely, a system of generators of relations for a simplicial set consists of a set G of generating simplices equipped with a dimension function dim: G→N and a set R of relations between simplices, equipped with a function F that sends a relation r∈R to the tuple F(r)=(g_1(r),f_1(r),g_2(r),f_2(r)), where g_i(r)∈G and f_i(r):[m(r)]→[dim g_i(r)] is a map of simplices of indicated dimension.

Intuitively, each element g∈G yields a simplex of dimension dim(g), and for any r∈R the generalized face of the simplex g_1(r) corresponding to the map of simplices f_1(r) must be equal to the generalized face of g_2(r) corresponding to f_2(r).

The simplicial set generated by (G,R) is the initial object in the category of pairs (X,u), where X is a simplicial set and u is a function from G to the set of simplices of X (of any dimension) such that u(g) is a simplex of dimension dim(g) and for any relation r∈R with F(r)=(g_1(r),f_1(r),g_2(r),f_2(r)) we have X_{f_1(r)}(g_1(r)) = X_{f_2(r)}(g_2(r)), which formalizes the above intuitive description.

This allows one to specify various elementary simplicial sets very efficiently. For instance, the real projective plane can be specified as a single 2-simplex α that satisfies two relations d_0(α)=d_2(α) and d_1(α)=s_0(d_1(d_1(α))).

Are there any citeable sources that discuss this in more detail than 1-2 lines of text? For the purposes of this question, “citeable” means indexed by MathSciNet/zbMATH or published on arXiv.

I prefer expository sources with examples akin to the one given above, but so far I am not aware of any sources that treat this in any nontrivial detail, e.g., more than a simple formula with a pushout or coequalizer.

I want to specifically emphasize that I am not interested in sources that use simplicial complexes instead of simplicial sets. (The above example makes no sense for simplicial complexes.)

• Shouldn't $m_1(r) = m_2(r)$, since they are the dimensions of$X_{f_1(r)}(g_1(r)) = X_{f_2(r)}(g_2(r))$? – Omar Antolín-Camarena Sep 25 '18 at 1:58
• I wouldn't be too suprised if you don't find a reference that discusses this in those terms. I think more commonly the simplicial set presented that way would simply be called the coequalizer of two morphisms $f_1, f_2 \colon \coprod_{r \in R} \Delta^{m(r)} \to \coprod_{g \in G} \Delta^{\dim(g)}$. – Omar Antolín-Camarena Sep 25 '18 at 2:01
• @OmarAntolín-Camarena: Yes, m_1(r)=m_2(r)=m(r). Your formula with coequalizer is "1-2 lines of text" that I referred to in the main post. The question is about sources that do a bit more than these 1-2 lines of text, e.g., consider some examples etc. – Dmitri Pavlov Sep 25 '18 at 2:41