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Let $A,B$ be sufficiently connected spaces so as to have the category of coverings equivalent to the category of representations of the fundamental groupoid.

Given a covering map of $A$ and a continuous map $A\overset{f}{\to} B$, we may pass to the associated representation of $\pi_1A$ and take its left Kan extension along $\pi_1A\overset{\pi_1f}{\to} \pi_1B$ to obtain a representation of $\pi_1B$. Finally, we may integrate it into a covering map of $B$. This defines the pushforward of a covering map along a continuous map.

Question. How to calculate - or at least anticipate - this pushforward?

For instance, what is the pushforward of the $n$-fold cover of $\mathbb S^1$ along the $k$-fold cover of $\mathbb S^1$?

I'm trying to use the formula for pointwise Kan extensions, but I'm struggling. At least for $f:\bf 1\to \mathbb S^1$, the fibers of the pushforward of the identity on the point seem to be $\mathbb Z$, but even in this case I'm struggling to calculate the parallel transport.

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  • $\begingroup$ just induce up: $V \mapsto \uparrow_{\pi_1A}^{\pi_1B}V$. To check it's correct, notice you're describing a left adjoint to pulling back {coverings of B}--->{coverings of A}. Under the identification of coverings with representations, it's easy to see this is restriction, so the left adjoint has no choice but to be induction. $\endgroup$ Sep 30, 2019 at 17:22
  • $\begingroup$ @DylanWilson I agree with everything you've written, but I don't know how to calculate that tensor product either. I don't even know the covering given by the pushforward of an identity map. $\endgroup$
    – Arrow
    Sep 30, 2019 at 17:36
  • $\begingroup$ the pushforward along the identity will be the identity functor... pushing forward along inclusion of a point will send a cover to copies of the universal cover... etc. in general, write your pi_1A-set as a coproduct of stuff like pi_1A/H and send each of those to \pi_B/f(H) where f:\pi_1A-->pi_1B. $\endgroup$ Sep 30, 2019 at 17:55
  • $\begingroup$ @DylanWilson I asked about the pushforward of the identity cover, not along the identity map. Why does pushing along a point take covers to copies of the universal cover? (Thanks for your patience!) $\endgroup$
    – Arrow
    Sep 30, 2019 at 19:47
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    $\begingroup$ @DylanWilson "just"! Strong words... $\endgroup$
    – David Roberts
    Oct 1, 2019 at 6:18

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