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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 6th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted a subsection ([here](https://ncatlab.org/nlab/show/real+vector+bundle#InKRTheory)) on Atiyah Real vector bundles <a href="https://ncatlab.org/nlab/revision/diff/real+vector+bundle/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/real+vector+bundle/5">v5</a>, <a href="https://ncatlab.org/nlab/show/real+vector+bundle">current</a>

   started a subsection (here) on Atiyah Real vector bundles

   diff, v5, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 9th 2023
   • (edited Nov 9th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded to the example "real vector bundles as Real vector bundles" ([here](https://ncatlab.org/nlab/show/real+vector+bundle#RealVectorBundlesAsRealVectorBundles)) a line saying that the equivalence continues to hold over varying base spaces. Then below that I added a Remark ([here](https://ncatlab.org/nlab/show/real+vector+bundle#ComplexVectorBundlesAsRealBundles)) that, in contrast, complex vector bundles over varying bases do $\,$ not $\,$ form a full subcategory of Real vector bundles over varing $\mathbb{Z}/2$-spaces --- but they do form a full subcategory of "Real vector bundles equipped with complex structure", where "complex structure" means endomorphisms (of Real vector bundles) squaring to $-1$. In this sense, the term "Real vector bundle" is quite apt. <a href="https://ncatlab.org/nlab/revision/diff/real+vector+bundle/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/real+vector+bundle/9">v9</a>, <a href="https://ncatlab.org/nlab/show/real+vector+bundle">current</a>

   Added to the example “real vector bundles as Real vector bundles” (here) a line saying that the equivalence continues to hold over varying base spaces.

   Then below that I added a Remark (here) that, in contrast, complex vector bundles over varying bases do $\,$ not $\,$ form a full subcategory of Real vector bundles over varing $\mathbb{Z}/2$-spaces — but they do form a full subcategory of “Real vector bundles equipped with complex structure”, where “complex structure” means endomorphisms (of Real vector bundles) squaring to $-1$.

   In this sense, the term “Real vector bundle” is quite apt.

   diff, v9, current