I am trying to understand the relationship between the simplicial path space and loop space with the path space of a topological space, and the loop space of a topological space.


I have understood that the simplicial path space of a simplicial object $A$ is homotopy equivalent to the constant simplicial object $A\_0$ but I feel I should be able to say more. 


Any help would be greatly appreciated, 



Simplicial path and loop spaces
I'll refer to my ancient book "Simplicial objects in algebraic topology". It is best to restrict to Kan complexes $K$ with a single vertex. In 23.3 and 23.4, it is shown that the path projection $PK \to K$ is a particularly nice kind of simplicial bundle provided that its
fiber $L(K)$ is a simplicial group, which usually fails. The Kan loop group (Section 26)
$G(K)$ substitutes for $L(K)$. It is the fiber of a different simplicial bundle over $K$ with a contractible total space. The geometric realization of this bundle is equivalent to the path space fibration of the realization $|K|$.


