The simplex category $\mathbf{\Delta}$ is, for our purposes, the category whose objects are $[n]=\{0,1, \dots , n-1, n\}$ for each $n = 0,1,2 \dots$, and whose morphisms are (all) order-preserving maps, i.e. for any morphism $f$, $x \leq y \implies f(x)\leq f(y)$, when such symbols are defined.
My question (as in the title) comes after the simplex category was briefly defined in a lecture I observed, although I did not really understand what its role was. An answer at any level would do, but for context, I have an grad a.t. course under my belt and some basic category theory, although the latter is still a bit raw in my mind. My very weak understanding is that it helps us develop the notion of homotopy without passage to topological spaces (or at least point-set topology?)
Best Answer
Here is some motivation for the category $sSet := Fun({\bf\Delta}^{op},Set)$ of simplicial sets, which is one of the primary uses of the category ${\bf\Delta}$.
In short, simplicial sets are meant to be a combinatorial model for spaces (up to weak homotopy equivalence), not unlike simplicial complexes. The primary difference is that simplicial sets keep track of degenerate simplices. These are pretty confusing at first, so this is what I'll focus on.
Degenerate simplices exist because of our expectations for the behavior of the morphisms in our category (of combinatorial models for spaces). For instance, surely there ought to be a morphism from the 1-simplex $\Delta^1$ to the 0-simplex $\Delta^0$! This is accommodated by simply declaring that the object $\Delta^0$ "has" a 1-simplex, which happens to be degenerate.
More broadly, if you know what all of the morphisms between simplices should be, then you know what all of the morphisms should be (since your objects are all glued together from simplices). This is where ${\bf\Delta}$ comes in: the object $[n]$ is "the universal $n$-simplex" (equipped with a total ordering of its vertices, which simplifies the theory).$^*$
This implies that the appropriate category sits fully faithfully inside of $Fun({\bf\Delta}^{op},Set)$. At the level of objects, this is because to know an object is to know its $n$-simplices for all $n$ along with the information of how they fit together: namely, the functor that it represents on ${\bf\Delta}$. At the level of morphisms, this is because a morphism should be uniquely determined by how it acts on the simplices of the source, and moreover any such function should be allowable as long as it respects the structure maps of the source -- that is, as long as it is a natural transformation.
From here, the remaining observation is that all objects of $Fun({\bf\Delta}^{op},Set)$ can be obtained in this way. Namely, a simplicial set can be viewed as a recipe for gluing together its nondegenerate simplices (or all of its simplices -- you get the same answer either way).
In case it clarifies things, let me note that above I am distinguishing between the object $[n] \in {\bf\Delta}$ and the object $\Delta^n \in sSet$. (The relationship between them is that $\Delta^n := hom_{\bf\Delta}(-,[n])$.)
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$^*$ If you think of these objects as topological simplices, then the morphisms between them are all required to be linear functions, and so are determined by their values on the vertices -- and this is what is being recorded by the morphisms in the category ${\bf\Delta}$. Said differently, there is a functor $${\bf\Delta} \xrightarrow{\Delta^\bullet_{top}} Top$$ from the simplex category to the category of topological spaces that realizes this intuition. Namely, it carries $[n]$ to the topological $n$-simplex $\Delta^n_{top}$, and it carries a morphism $[m] \xrightarrow{f} [n]$ to the unique linear function $\Delta^m_{top} \xrightarrow{\Delta^f_{top}} \Delta^n_{top}$ that carries the $i^{th}$ vertex of $\Delta^m_{top}$ to the $f(i)^{th}$ vertex of $\Delta^n_{top}$.