While I have only a shallow understanding, I like category theory. I find definitions and proofs in terms of category theoretic concepts to be very clean and deep, often cutting to the core of a concept and what its "about", and why we care about it, especially definitions in terms of universal properties.
I wanted to know if there is a definition of the exponential map from differential geometry in category theoretic terms. The definition I know of is for a manifold $M$, and a point $p \in M$,
$\exp: (v \in T_pM) \mapsto \gamma_v(1)$
Where $\gamma$ is the locally unique geodesic though $p$ of velocity $v$ such that $\gamma_v(0) = p$. I may have slightly messed up the definiton. I would be especially interested in definitions in geometrical/topological terms with as few constructions/parametrisations as possible.