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Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$?

You may assume that you know $G$ and $H$ are already deloopable, i.e., are $A_\infty$-spaces such that the induced monoid on $\pi_0$ is a group (this has been discussed before, see for example this questionthis question.)

Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$?

You may assume that you know $G$ and $H$ are already deloopable, i.e., are $A_\infty$-spaces such that the induced monoid on $\pi_0$ is a group (this has been discussed before, see for example this question.)

Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$?

You may assume that you know $G$ and $H$ are already deloopable, i.e., are $A_\infty$-spaces such that the induced monoid on $\pi_0$ is a group (this has been discussed before, see for example this question.)

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user2529
user2529

Criterion for deloopable based map

Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$?

You may assume that you know $G$ and $H$ are already deloopable, i.e., are $A_\infty$-spaces such that the induced monoid on $\pi_0$ is a group (this has been discussed before, see for example this question.)