$\DeclareMathOperator\sAn{\mathit{sAn}}\DeclareMathOperator\An{\mathit{An}}$For the category of simpicial animas (simplicial $\infty$-groupoids if you like) $\sAn$, we have the evaluation functor $\mathrm{ev}_n:\sAn\rightarrow \An$ with a left adjoint $\operatorname{const}_n$ and the realization functor $\lvert\ \rvert:\sAn\rightarrow \An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega \lvert\operatorname{const}_1 X\rvert$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\operatorname{const}_1X$. Is this something concerning the theory of Segal spaces…?