I'm following A primer for unstable motivic homotopy theory written by Antieau, Elmanto as my first intro to motivic homotopy theory. Their treatment of the singular construction on p. 30 confuses me. Let me introduce you to their setup:
We write $\mathbf{sPre}(\mathbf{Sm}_S) = \operatorname{Fun}(\mathbf{Sm}_S, \mathbf{sSet})$ where $\mathbf{Sm}_S$ is the category of nice enough smooth $S$-schemes. Moreover, we set $\Delta^n = \operatorname{Spec} k[x_0, \cdots, x_n]/(x_0+\cdots+x_n-1)$ which gets promoted to a cosimplicial scheme. Then, we define $$ \operatorname{Sing}^{\mathbb{A}^1} : \mathbf{sPre}(\mathbf{Sm}_S) \to \mathbf{sPre}(\mathbf{Sm}_S), \ X \mapsto |X(- \times \Delta^{\bullet})|. $$ We write $\operatorname{map}$ for the simplicial mapping spaces on the simplicial model categories that are floating around.
There should be a natural map $X \to \operatorname{Sing}^{\mathbb{A}^1} X$. On p. 30 they then write:
Observe that the functor $U \mapsto X(U \times_S \Delta^n)$ is the same as the functor $U \mapsto \operatorname{map}( \times_S \Delta^n, X)$. We have a natural map $X \simeq \operatorname{map}(\Delta^0, X) \to \operatorname{map}(\Delta^n, X)$ for each $n$, so we think of the map $X \to \operatorname{Sing}^{\mathbb{A}^1}X$ as the canonical map from the zero simplices.
I'm confused about the entire paragraph. Shouldn't it be $\operatorname{map}(\Delta^0, X) \simeq X(\Delta^0)$ and how does this discussion even yield the desired map?
Some remarks that you don't need to read to answer my question:
- Formally, the Yoneda embedding induces for $\Delta^{\bullet} : \Delta \to \mathbf{Sm}_S$ an adjunction $$|-|_{\Delta^{\bullet}} : \mathbf{sSet} \rightleftarrows \mathbf{Sm}_S : \operatorname{Sing}_{\Delta^{\bullet}}$$ but this is not our $\operatorname{Sing}^{\mathbb{A}^1}$.
- The way I understood this $\mathrm{Sing}^{\mathbb{A}^1}$ is as the composition \begin{align*} \mathbf{Sm}_S^{\mathrm{op}} &\to \operatorname{Fun}(\Delta, \mathbf{Sm}_S^{\mathrm{op}}) \to \operatorname{Fun}(\Delta, \mathbf{sSet}) \to \mathbf{sSet}, \\ Y &\mapsto Y \times \Delta^{\bullet} \mapsto X(Y \times \Delta^{\bullet}) \mapsto |X(Y \times \Delta^{\bullet})| = \operatorname{colim}_{n \in \Delta} X(Y \times \Delta^n).\end{align*} But perhaps I misunderstood.
Since the singular construction sounds like a very important construction in the theory, it would be helpful for me to understand it properly, so I'm grateful for any help or comments!