nLab diffeological singular simplicial set -- definition

Definition

(diffeological singular simplicial set)

Consider the simplicial diffeological space

$$\array{ \Delta & \overset{ \Delta^\bullet_{diff} }{ \longrightarrow } & DiffeologicalSpaces \\ [n] &\mapsto& \Delta^n_{diff} \mathrlap{ \coloneqq \big\{ \vec x \in \mathbb{R}^{n+1} \;\vert\; \underset{i}{\sum} x^i = 1 \big\} } }$$

which in degree $n$ is the standard extended n-simplex inside Cartesian space $\mathbb{R}^{n+1}$, equipped with its sub-diffeology.

This induces a nerve and realization adjunction between diffeological spaces and simplicial sets:

(1)$$DiffeologicalSpaces \underoverset { \underset{Sing_{\mathrlap{diff}}}{\longrightarrow} } { \overset{ \left\vert - \right\vert_{\mathrlap{diff}} }{\longleftarrow} } { \phantom{AA}\bot\phantom{AA} } SimplicialSets \,,$$

where the right adjoint is the diffeological singular simplicial set functor $Sing_{diff}$.

(e.g. Christensen-Wu 13, Def. 4.3)

Remark

(diffeological singular simplicial set as path ∞-groupoid)

Regarding simplicial sets as presenting ∞-groupoids, we may think of $Sing_{diff}(X)$ (Def. ) as the path ∞-groupoid of the diffeological space $X$.

In fact, by the discussion at shape via cohesive path ∞-groupoid we have that $Sing_{diff}$ is equvialent to the shape of diffeological spaces regarded as objects of the cohesive (∞,1)-topos of smooth ∞-groupoids:

$$Sing_{diff} \;\simeq\; Shp \circ i \;\;\colon\;\; DiffeologicalSpaces \overset{i}{\hookrightarrow} SmoothGroupoids_{\infty} \overset{Shape}{\longrightarrow} Groupoids_\infty$$