Find generators for the ideal of relations between the generating arrows of $\mathbb{Z}[\Delta]$

Question

I previously asked a question here: Equivalence of categories $\Delta$ and $\Delta_{\text{big}}$, and the generators of the algebra $\mathbb{Z}[\Delta]$. Using the same notations and definitions, I have a question which asks me to find generators for the ideal of relations between these generating arrows. I'm having a bit of trouble understanding what the question is asking, any help?

Answer 1

In the other question question you have seen that certain morphisms are the generator of the category-algebra $\mathbb Z[\Delta]$, that is every element of this algebra can written as a sum of product these generators-morphisms.

The point is that these generators satisfy some relations in the category $Delta$ hence these relations pass to the algebra (for instance, in the notations of the other question, $s^{(i)}_n \circ \delta^{(i)}_n = \text{id}$).

This implies that you can present the algebra $\mathbb Z[\Delta]$ as a quotient of the free algebra spanned by the morphisms of $\Delta$ quotiented by (the ideal) of the relations between the morphisms that holds in the category $\Delta$.

Hope this helps.