In manifold theory, tangent vectors can be seen as equivalence classes of curves. Using the definition in Weintraub's Differential Forms: Theory and Practice, let $M$ be an $n$-dimensional manifold and $p \in U \subset M$. Let $k: O \rightarrow U$ be a coordinate chart where $O \subset \mathbb{R}^n$. Let $r_1$ and $r_2$ be the parametrizations of two curves on $M$ with $r_1(0) = r_2(0) = p$. We say that these two curves are equivalent if $(k^{−1} \circ r_1)'(0) = (k^{−1} \circ r_2)'(0) $. Then the tangent vectors to $M$ at $p$ are defined as the equivalence classes of curves on $M$ passing through $p$.
I wonder if cotangent vectors to $M$ can be seen as equivalence classes of any "co-curves" on $M$.