Why are simplicial sets contravariant functors and not covariant? The definition of a simplicial set is given as a contravariant functor $\textbf{F}:\Delta\to\textbf{Set}$ where $\Delta$ is the category of finite ordinals with order preserving functions as morphisms. The way I understand this definition and how it is a good model for topological simplicial complexes is that the image of each $[n]$ represents the n-dimensional simplex and the images of the coface and codegeneracy maps tell you how the simplexes fit together. 
This all makes sense to me (though please tell me if there's some conceptual error here) but what I don't understand is why we use a contravariant functor rather than a covariant one. It seems to me that if we just swapped the names of coface and codegeneracy maps and changed it to a covariant functor we would get the same result. I'd love to either get confirmation that this is the case and the choice is purely historical or get some intuition on how these two definitions differ and why contravariant is a better choice.
Thanks for taking the time to read my question
 A: First, let's look at the Simplex Category $\Delta$. We can see that $\Delta$, rather than $\Delta^{op}$, is a natural category to look at, both because it has a nice combinatorial description (the objects are finite total orders and the morphisms are order preserving maps) and, more importantly for us, because it has a topological visualisation which goes as follows:


*

*The objects are simplicies.

*The face maps $[n] \rightarrow [n+1]$ are given by the inclusion of an $n$-simplex as a face of an $n+1$ simplex.

*The degeneracy maps $[n+1] \rightarrow [n]$ are given by the projection onto a face.

*In particular, all the morphisms correspond to continuous maps.


Therefore, we expect that our functors will have source $\Delta$ and would prefer to make functors contravariant rather than switch  to having $\Delta^{op}$ as the source.
Now lets look at what's going on with a simplicial set $X$ and see how we can construct it as a functor. The set $X_n$ is the set of all instances of $n$-simplices in the simplicial set. Now, for any $n$-simplex we can pick out each of its faces: for each choice of face we get a function $X_n \rightarrow X_{n-1}$. But this function was constructed from the face map $[n-1] \rightarrow [n]$: it is a function sending each $n$-simplex to the $n-1$-simplex that was included into it by that. So, face maps should be sent to functions going the other way round.
Similarly, for each $n$-simplex we can construct a degenerate instance of an $n+1$-simplex occupying the same space. We can view this constructed $n+1$-simplex as the map that first projects and $n+1$-simplex onto a face using a degeneracy map, then includes the original $n$-simplex into the simplicial set. Thus the degeneracy maps $[n+1] \rightarrow [n]$ are turned into functions $X_n \rightarrow X_{n+1}$.
Combining this, we see that $X$ is a covariant functor $\Delta^{op} \rightarrow \underline{Set}$. Since $\Delta$ is a more natural category to work with and can be viewed as a category consisting precisely of the simplices as topological spaces and continuous maps, we think of this as being a contravariant functor $\Delta \rightarrow \underline{Set}$.
A: There is a fairly strong relationship between the simplex category $\mathbf{\Delta}$ and the presheaf category $[\mathbf{\Delta}^\mathrm{op}, \mathbf{Set}]$.
For example, $\mathbf{\Delta}$ is canonically a full subcategory of $[\mathbf{\Delta}^\mathrm{op}, \mathbf{Set}]$, via the Yoneda embedding.
The presheaf category is, in a suitable sense, the free completion of $\mathbf{\Delta}$ under the operation of taking (small) colimits.
This remains true if you replace $\mathbf{\Delta}$ with any small category.

The dual picture is even more twisted: a small category $\mathbf{C}$ is a full subcategory of $[\mathbf{C}^\mathrm{op}, \mathbf{Set}^\mathrm{op}]$ (or equivalently, $[\mathbf{C}, \mathbf{Set}]^\mathrm{op}$), and the latter is the free completion under taking small colimits.
I find this very awkward to work with. (admittedly, I haven't seen it come up very often)
A: Contrary to what you seem to be implying, the face and degeneracy maps are not "interchangeable" when you reverse the direction that they go.  In other words, the category $\Delta$ is not isomorphic to its opposite category by just swapping face maps with degeneracy maps.  Indeed, you can't even do this "swap" at all, since the numbers don't work out: there are $n+2$ face maps $[n]\to[n+1]$ in $\Delta$ but only $n+1$ degeneracy maps $[n+1]\to[n]$.
So contravariant and covariant functors out of $\Delta$ really are different; they involve different collections of maps with different combinatorics.  As Chessanator's answer explains, the combinatorics of contravariant functors exactly captures the geometric properties of spaces built out of simplices.  The combinatorics of covariant functors, being different, do not.  For instance, if you have a covariant functor $F:\Delta\to\mathbf{Set}$, then you have only one map $F([1])\to F([0])$, and so only one way to get a "vertex" from an "edge", but an edge should have two vertices!
