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Let $X$ be a topological space. Is there a path object functor $\mathrm{P}:\mathbf{Top}\rightarrow \mathbf{Top}$, with fibration $ev_{0},ev_{1}: \mathrm{P}(X)\rightarrow X\times X$ such that:

(1) $\mathrm{P}$ is a cartesian functor i.e., $\mathrm{P}(X\times Y)$ is isomorphic to $\mathrm{P}(X)\times \mathrm{P}(Y).$

(2) and the concatenation map $-\ast-:\mathrm{P}(X)\times_{X}\mathrm{P}(X)\rightarrow \mathrm{P}(X)$ is strictly associative.

The condition (1) (but not (2)) is verified by the standard path object $\mathrm{P}(-)= \mathrm{Map}([0,1],-)$ and the condition (2) (but not (1)) is verified by the moore path object $\mathrm{M}(-)$ which is not a cartesian functor.

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  • $\begingroup$ Sure: let $P = \text{Hom}(\{0, 1 \}, -)$. $\endgroup$ Commented Mar 28, 2012 at 16:20
  • $\begingroup$ That is not a path object! Just to be clear a path object $\mathrm{P}(X)$ factors the diagonal $\Delta: X\rightarrow X\times X$ to $X\rightarrow \mathrm{P}(X)\rightarrow X\times X$ where the first map is a weak equivalence and the second is a fibration. $\endgroup$
    – Ilias A.
    Commented Mar 28, 2012 at 16:51
  • $\begingroup$ I have undeleted this question which you have deleted six years ago. -- Please do not self-delete your useful questions! $\endgroup$
    – Stefan Kohl
    Commented Apr 30, 2021 at 21:13

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