[[Fibre bundle]]
# Bundle map

Let $F\to E \stackrel{\pi}{\twoheadrightarrow} X$ and $F'\to E' \stackrel{\pi'}{\twoheadrightarrow} X$ be [[Fibre bundle|fibre bundles]] over a common base space $X$ in a category $\cat C$.
A **bundle map** $f : E \to E'$ is a map $f \in \cat C(E,E')$ such that

![[bundle-map.svg#invert|https://q.uiver.app/#q=WzAsMyxbMCwwLCJFIl0sWzIsMCwiRSciXSxbMSwyLCJYIl0sWzAsMSwiZiJdLFswLDIsIlxccGkiLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMSwyLCJcXHBpJyIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dXQ==]]

commutes in $\cat C$.
Bundle maps are the morphisms of [[Category of fibre bundles]].

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