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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,

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2 Answers 2

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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 $|K|$.

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Have you considered the sequence Omega X --> PX --> X in the two categories where Omega denotes based loop space and P based path space

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