In category theory, a **section** of a morphism is a chosen right-inverse to that morphism. More precisely, let $\mathcal{C}$ be a category and consider a morphism $f: A \to B$. A **section** of $f$ is a morphism $s: B \to A$ such that the composite $f \circ s = \mathrm{id}_B$, where $\mathrm{id}_B$ is the identity morphism on the object $B$. A section provides a canonical “retraction” along $f$, selecting a distinguished element of each fiber of $f$. (From the internal perspective of a topos or an elementary fibration, a section corresponds to a global point of the associated “bundle” (or object of families): it picks out one element in each fiber-object in a way compatible with the structure encoded by the morphism $f$.) <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsMixbMCwwLCJBIl0sWzEsMCwiQiJdLFswLDEsImYiXSxbMSwwLCJzIiwwLHsiY3VydmUiOi0xfV1d&embed" width="304" height="176" style="border-radius: 8px; border: none;"></iframe> In this diagram, the curved arrow $s$ is a right-inverse to $f$, meaning the composition $f \circ s$ returns to the identity on $B$. <!-- https://q.uiver.app/#q=WzAsMixbMCwwLCJBIl0sWzIsMCwiQiJdLFswLDEsInNlY3Rpb24iLDAseyJjdXJ2ZSI6Mywic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibW9ubyJ9fX1dLFsxLDAsInJldHJhY3QiLDAseyJjdXJ2ZSI6Mywic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV1d --> <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsMixbMCwwLCJBIl0sWzIsMCwiQiJdLFswLDEsInNlY3Rpb24iLDAseyJjdXJ2ZSI6Mywic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibW9ubyJ9fX1dLFsxLDAsInJldHJhY3QiLDAseyJjdXJ2ZSI6Mywic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV1d&embed" width="432" height="176" style="border-radius: 8px; border: none;"></iframe> If you conceive of the function $f$ as projecting from a total space $E$ to a base space $B$ (a subspace), a section maps back to the target space in a way that respects $f$.