Canonical colimit and cartesian product of simplicial sets

Question

Let $K$ be a simplicial set and let $\Delta K$ be the category of simplices, i.e the category where the objects are simplicial maps $$ \Delta[n]\to K $$ and the maps $\phi\: : \: (\Delta[n]\to K)\to (\Delta[m]\to K)$ are simplicial maps $\phi\: : \: \Delta[n]\to \Delta[m]$ such that the triangle commute. Now let $L$ be another simplicial set, here my question: is it possible build $\Delta K\times L$ from $\Delta K$, $\Delta L$ in a canonical way?

Here a concrete example: Let $x\: : \:\Delta[n]\to K$, $y\: : \:\Delta[l]\to L$ be two objects of the two categories of simplices. The cartesian products $$ x\times y\: : \:\Delta[n]\times\Delta[l] \to K\times L $$ is not an element of $\Delta (K\times L)$ (for $l,n\neq 0$), well is it possible to find a canonical representant of $x\times y$ inside $\Delta (K\times L)$?

Equivalently: there is a canonical map $\Delta[n]\times\Delta[l]\to \Delta[something]$?

Answer 1

You can do this for any category of presheaves. Let $\mathcal{C}$ be a small category. For a presheaf $X$ on $\mathcal{C}$, we write $\mathbf{El} (X)$ for the category of elements of $X$, i.e. the comma category $(h \downarrow X)$ where $h : \mathcal{C} \to [\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ is the Yoneda embedding. (So the objects of $\mathbf{El} (X)$ are pairs $(c, x)$ where $c$ is an object in $\mathcal{C}$ and $x$ is an element of $X (c)$.) I claim:

For any presheaves $X$ and $Y$ on $\mathcal{C}$, the following diagram is a pullback square: $$\require{AMScd} \begin{CD} \mathbf{El} (X \times Y) @>>> \mathbf{El} (Y)\\ @VVV @VVV \\ \mathbf{El} (X) @>>> \mathbf{El} (1) \end{CD}$$

This can be verified straightforwardly. The point is that $\mathbf{El} (X \times Y)$ can be identified with the subcategory of $\mathbf{El} (X) \times \mathbf{El} (Y)$ consisting of those pairs of the form $((c, x), (c, y))$.

More generally, you might like to verify that $\mathbf{El}(-)$ preserves pullbacks.