In Homotopy Field Theory in dimension 2 and group algebras, section 4.6, page 24, Turaev considers an annulus $C = S^1 \times [0,1]$ (thought of as a cobordism from $C_0 = S^1 \times \{ 0\}$ to $C_1 = S^1 \times \{ 1 \})$. The boundaries have basepoints $s_0 = s \times \{0\}$ and $s_1 = s \times \{ 1\}$, where $s$ is some generic basepoint on $S^1$.
The boundaries and the cylinder come equipped with principal $\pi$-bundles for some finite group $\pi$, given as maps from from $C_0$ and $C_1$ to $B\pi$. The bundle $\psi: C \to B\pi$ on the cylinder restricts to the bundles at the boundaries.
Here's my confusion: Turaev states that "the homotopy class of $\psi$ is determined by the homotopy classes $\alpha,\beta \in \pi$ represented by the loops $\psi\mid_{C_0}$ and $\psi|_{s \times [0,1]})$, where the interval $[0,1]$ is oriented from $0$ to $1$"
But $\psi|_{s \times [0,1]}$ is not a loop - it's just an arc! What does Turaev mean here? I couldn't find any specific convention earlier in the paper, and he keeps referring to arcs as loops later in the paper. What is the intended meaning here?
