---- > [!definition] Definition. ([[morphism of vector bundles]]) > Let $E \xrightarrow{\pi}B$ and $E' \xrightarrow{\pi'}B'$ be two [[vector bundle|vector bundles]]. Let $f:B \to B'$ be a map. > A [[continuous|map]] $F:E \to E'$ is called a **bundle morphism from $E$ to $E'$** if there is a [[continuous|map]] $f:B \to B'$ such that the [[diagram|square]] > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFEQBfU9TXfIRQBGclVqMWbdgHJuvEBmx4CRMsPH1mrRCABC8vssFFRG6lqm69cruJhQA5vCKgAZgCcIAWyRkQOBBIAEwWkjogADqRaFgg1Ax0AEYwDAAK-CpCIB5YjgAWOIYgnj5IogFBiADMYdps0bG2CqW+iP6B5XVWIABixa0h1J013RFu8SCJKemZJrq5BUV2XEA > \begin{tikzcd} > E \arrow[d, "\pi"'] \arrow[r, "F"] & E' \arrow[d, "\pi'"] \\ > B \arrow[r, "f"'] & B' > \end{tikzcd} > \end{document} > ``` > commutes, and such that the map on fibers $E_{x} \to E'_{f(x)}$ induced by $f$ is a [[linear map|linear map]] of [[vector space|vector spaces]]. In this case, we say **$F$ is a bundle morphism covering $f$**. > > ^definition > > > An **isomorphism of vector bundles** is a morphism covering the [[identity map|identity]] that is an [[isomorphism]] of $E,E'$ in the [[category]] to which they belong (e.g. a [[homeomorphism]] in $\mathsf{Top}$). > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFEQBfU9TXfIRQAmclVqMWbdgHJuvEBmx4CRAIyk14+s1aIO3cTCgBzeEVAAzAE4QAtklEgcER9QYQIaImUuM4MOIMdABGMAwACvwqQiDWWCYAFjgg1DpS+gA6mTgwAB44wFhQXPJWtg6IZM6uiBoSumzZAMYEJmUgNvZI1S5uDRkg2WhYqSDBYZHRgmzxSSk85d111H2ITul6Q5kjcu6h4VHKM-oMMJYLFFxAA > \begin{tikzcd} > E \arrow[rr, "\cong"] \arrow[rd, "\pi"'] & & E' \arrow[ld, "\pi'"] \\ > & B \arrow["\text{id}"', loop, distance=2em, in=305, out=235] & > \end{tikzcd} > \end{document} > ``` > ---- #### (Maybe a relationship to [[fibered two-object slice category]] or [[fibered two-object coslice category]]) ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```