Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology.

For which spaces $B$ does the pullback functor $\mathsf{Top}_{/B}\overset{\operatorname{ev}_0^\ast}{\longrightarrow}\mathsf{Top}_{/B^I}$ have a right adjoint? Is local connectedness enough?

(If it's more convenient - same question with $\mathsf{Top}$ replaced by the full subcategory of locally connected spaces.)

Added. Perhaps my question is overly general. Here's what I am specifically looking for.

Let $s$ be a Hurewicz connection on a bundle $\begin{smallmatrix}A\\ \;\;\downarrow \alpha\\ B \end{smallmatrix}$. By definition $s$ is a bundle map $\operatorname{ev}_0^\ast \alpha \to \alpha^I$. I am wondering if there is some sort of bundle over $B$, worthy of being called $\Pi_{\operatorname{ev}_0}\alpha^I$, such that the Hurewicz connection uniquely corresponds to a bundle arrow $s_\flat:\alpha\to \Pi_{\operatorname{ev}_0}\alpha^I$. In other words, given a Hurewicz connection, can we (continuously!) separate its input into "path" and "initial point of lift" portions?

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    $\begingroup$ What are you pulling back? Maps? Sheaves? $\endgroup$ – Qiaochu Yuan Nov 7 '18 at 18:19
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    $\begingroup$ Continuous maps. $\endgroup$ – Arrow Nov 7 '18 at 18:22
  • $\begingroup$ I assume you want the exponential topology on $B^I$, right? If $B$ is Hausdorff, this is the compact-open topology, but not otherwise. Are your spaces Hausdorff? I don't see a reason that local connectedness would simplify things -- maybe I'm missing something? Have you looked at general characterizations of exponentiable maps such as here and here? $\endgroup$ – Tim Campion Nov 9 '18 at 17:08
  • $\begingroup$ @TimCampion thank you for your remarks. I forgot about Niefield's paper and did not know about the second. At any rate, I added motivation to my question. $\endgroup$ – Arrow Nov 9 '18 at 21:32

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