Relationship between Ehresmann connections and path-groupoid connections I've seen two seemingly very different approaches to defining connections on a smooth $G$-torsor $\alpha:A\to B$, with seemingly different intuitions.


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*A complement of the "vertical bundle" satisfying an equivariance condition.

*Smooth functors $\mathcal P_1(B)\to G$-$\mathsf{Tor}$ from the path groupoid to the category of $G$-torsors, intuitively sending a point to its fiber and a thin-homotopy class to the isomorphism of fibers it induces. (This functor needs to be locally trivial and smooth in a suitable sense.)
The second approach seems closer to general fiber bundles, but more troublesome technically. The first approach is very elegant but seems to "care" about smoothness more explicitly.
How are these approaches related? In general, how does one move between lifts of paths and complements of the vertical bundle? Why do Hurewicz connections deserve to be called connections?
 A: Let $\pi \colon P \rightarrow M$ be a principal $G$-bundle. 


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*Assume that we start with a notion of parallel transport. That is, given a smooth curve $\alpha \colon [0,1] \rightarrow M$ with $\alpha(0) = m$, we have the parallel transport maps $P_{\alpha, 0, t} \colon P_m \rightarrow P_{\alpha(t)}$ which allows us to identify the fiber $P_m = P_{\alpha(0)}$ over the starting point $m = \alpha(0)$ with the fibers $P_{\alpha(t)}$ along $\alpha$. Given a specific element $\sigma \in P_m$ in the fiber over the starting point, we can define a path $\tilde{\alpha}_{\sigma} \colon [0,1] \rightarrow P$ by $\tilde{\alpha}_{\sigma}(t) := P_{\alpha,0,t}(\sigma)$ (the parallel transport of $\sigma$ along $\alpha$). The path $\tilde{\alpha}_{\sigma}$ provides a lift of the path $\alpha$ with $\tilde{\alpha}_{\sigma}(0) = \sigma$. Given $m \in M$ and $\sigma \in P_m$, we can now define a subset $H_{(m,\sigma)}P \subseteq T_{(m,\sigma)}P$ called the horizontal space by
$$ H_{(m,\sigma)}P := \left \{ \frac{d}{dt} \tilde{\alpha}_{\sigma}|_{t = 0} \, : \, \alpha \in C^{\infty}([0,1],M), \alpha(0) = m \right \}.$$
The parallel transport maps are imposed to satisfy conditions which turns $HP$ into a smooth constant rank $G$-equivariant subbundle of $TP$ which complements the vertical bundle $VP = \ker(d\pi)$. For example, we want the lifts $\tilde{\alpha}_{\sigma}$ to be smooth so we require that the parallel transport maps are diffeomorphisms. We want the bundle to be $G$-equivariant so we require that the parallel transport maps are $G$-equivariant. We want $H_{(m,\sigma)}P$ to "depend smoothly" on $(m,\sigma)$ so we require that the parallel transport maps "depend smoothly" on $\alpha,\sigma$. We also want the parallel transport maps to be behave in the obvious way with respect to reparametrizations of $\alpha$, etc. Using the reparametrization invariance, it is easy to see that our parallel lifts $\tilde{\alpha}_{\sigma}$ which by definition satisfy $\frac{d}{dt} \tilde{\alpha}_{\sigma}|_{t = 0} \in H_{(m,\sigma)}P = H_{\tilde{\alpha}_{\sigma}(0)}P$ in fact satisfy $\frac{d}{dt} \tilde{\alpha}_{\sigma}(t) \in H_{\tilde{\alpha}_{\sigma}(t)}P$ for all $t \in [0,1]$. That is, the tangent vectors of horizontal lifts always belong to the horizontal bundle.

*Going the other way, assume we have a $G$-equivariant subbundle $HP$ of $TP$ such that $VP \oplus HP = TP$ and let $\alpha \colon [0,1] \rightarrow M$ be a smooth curve with $\alpha(0) = m$. Given $\sigma \in P_m$, we want to construct a unique horizontal lift $\tilde{\alpha}_{\sigma} \colon [0,1] \rightarrow P$ of $\alpha$ with $\tilde{\alpha}_{\sigma}(0) = \sigma$ which we will call the parallel transport of $\sigma$ along $\alpha$. More explicitly, we want $\pi \circ \tilde{\alpha}_{\sigma} = \alpha$, $\tilde{\alpha}_{\sigma}(0) = \sigma$ and $\frac{d}{dt} \tilde{\alpha}_{\sigma}(t) \in H_{\tilde{\alpha}_{\sigma}(t)}P$ for all $t \in [0,1]$. Note that for each $t \in [0,1]$ and $\mu \in P_{\alpha(t)}$, there exists a unique horizontal tangent vector in $H_{\alpha(t),\mu}P$ which projects under $d\pi$ to $\dot{\alpha}(t)$. This follows from the direct sum decomposition as $d\pi|_{H_{(\alpha(t),\mu)}P}$ is an isomorphism onto $T_{\alpha(t)}M$ so we just take the the horizontal tangent vector to be $d\pi^{-1}|_{H_{(\alpha(t),\mu)}P}(\dot{\alpha}(t))$. This horizontal vectors glue together to give a well-defined smooth vector field on the total space of $\alpha^{*}(E)$. An integral curve of this vector field starting at $(0,\sigma)$ will give us the required horizontal lift. This integral curve will exist for all $t \in [0,1]$ (and not just for a short time) because of the $G$-equivariance. Finally, using the horizontal lifts we can construct the parallel transport maps and verify that they satisfy the properties used in the previous item.



A Hurewicz connection abstracts and generalizes the first approach to the topological setting. Namely, if $s$ is a Hurewicz connection which provides us with uniquely defined lifts, we can define "parallel transport" by $P_{\alpha,0,t}(\sigma) := s(\sigma, \alpha)(t)$. This parallel transport depends continuously on $\alpha$ and $\sigma$ and that's about it. It doesn't have to provide us with a homeomorphism between the fibers along $\alpha$, it is not necessarily invariant under reparametrizations, etc.
