# For which topological spaces does pullback along $\operatorname{ev}_0:B^I\to B$ have a right adjoint?

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?

• What are you pulling back? Maps? Sheaves? – Qiaochu Yuan Nov 7 '18 at 18:19
• Continuous maps. – Arrow Nov 7 '18 at 18:22
• 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? – Tim Campion Nov 9 '18 at 17:08
• @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. – Arrow Nov 9 '18 at 21:32