nLab
path infinity-groupoid in a lined topos

In every lined topos $(𝒯,R)$ the line $R$ may canonically be regarded as an interval object

$$*\stackrel{0}{\to }R\stackrel{1}{\leftarrow }*.$${*} \stackrel{0}{\to} R \stackrel{1}{\leftarrow} {*} \,.

As discussed there, this inuces a cosimplicial object

$${\Delta }_{𝒯}:\Delta \to 𝒯$$\Delta_{\mathcal{T}} : \Delta \to \mathcal{T}

and then in turn a functor

$$\Pi :𝒯\to [{\Delta }^{\mathrm{op}},𝒯]$$\Pi : \mathcal{T} \to [\Delta^{op},\mathcal{T}]

from $𝒯$ to simplicial objects in $𝒯$ given by

$$\Pi (X):=X^{{\Delta }_{𝒯}^n}.$$\Pi(X) := X^{\Delta^n_{\mathcal{T}}} \,.

Using the standard model structure on simplicial objects in a topos? this presents an ∞-groupoid internal to $𝒯$, the path $\mathrm{\infty }$-groupoid of $X$.

If $(𝒯,R)$ is even a smooth topos, then there is also the construction of the infinitesimal path infinity-groupoid in a smooth topos ${\Pi }^{\inf }(X)$ for each $X$ and a canonical inclusion

$${\Pi }^{\inf }(X)\hookrightarrow \Pi (X).$$\Pi^{inf}(X) \hookrightarrow \Pi(X) \,.