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Bundle theory, originated by the 18th century Scottish philosopher David Hume, is the ontological theory about objecthood in which an object consists only of a collection (bundle) of properties, relations or tropes.
In philosophy of mind, Hume is well known for his development of the bundle theory of the self. It states that the self is to be understood as a bundle of mental states and not as a substance acting as the bearer of these states, as is the traditional conception. Many of these positions were initially motivated by Hume's empirical outlook. It ...
One example of a principal bundle is the frame bundle. If for each two points b 1 and b 2 in the base, the corresponding fibers p −1 (b 1) and p −1 (b 2) are vector spaces of the same dimension, then the bundle is a vector bundle if the appropriate conditions of local triviality are satisfied. The tangent bundle is an example of a vector ...
For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H ∗ ( B G ) {\displaystyle H^{*}(BG)} was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H ∗ ...
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
A G-bundle is a fiber bundle with an equivalence class of G-atlases. The group G is called the structure group of the bundle; the analogous term in physics is gauge group. In the smooth category, a G-bundle is a smooth fiber bundle where G is a Lie group and the corresponding action on F is smooth and the transition functions are all smooth maps.
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From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover. A special case of this is the orientable double cover of a differentiable manifold , where the corresponding line bundle is the determinant bundle of the tangent bundle (see below).