Equivalence of categories $\Delta$ and $\Delta_{\text{big}}$, and the generators of the algebra $\mathbb{Z}[\Delta]$

Question

I have been given that $\Delta_{\text{big}}$ is the category of all finite ordered sets with order preserving maps as the morphisms and $\Delta \subset \Delta_{\text{big}}$ be its full small subcategory formed by sets $[n]:=\{0,1,\cdots,n\}, \ n \geq 0$, ordered usually.

I need to show that $\Delta$ and $\Delta_{\text{big}}$ are equivalent, and that the algebra $\mathbb{Z}[\Delta]$ is generated by the identity arrows $e_n=\text{Id}_{[n]}$, the inclusions $\partial_n^{(i)}:[n-1]\hookrightarrow[n], \ 0 \leq i \leq n, \ i \notin \partial_n^{(i)}([n-1])$, and surjections $s_n^{(i)}:[n]\twoheadrightarrow[n-1], \ 0 \leq i \leq n-1, \ (i+1) \mapsto i$

I'm having trouble translating the definition I know of equivalent categories and use it solve the problem, and for the second part I have no worldly clue what $\mathbb{Z}[\Delta]$ even means. Any kind of help will be appreciated!

Answer 1

You can use the folowing characterization of equivalence of categories: a functor $F:C\rightarrow D$ is an equivalence of categories if and only if it is fully faithful and essentially surjective.

The canonical embedding $\Delta\rightarrow \Delta_{big}$ is fully faithful and essentially surjective. This implies that $\Delta$ and $\Delta_{big}$ are equivalent categories:

Fully faithful: $Hom_{\Delta}([n],[m])=Hom_{\Delta_{big}}([n],[m])$

Essentially surjective: every finite set whose cardinal is $n$ is isomorphic to $[n]$.

For the second part, use the fact that any map $f\in Hom_{\Delta}([n],[m])$ is a composition of a surjection with an injection. A surjection which respects the order is a composition of $s^i_n$ and an injection which respects the order is a composition of $\partial^i_n$.