Search results
Results From The WOW.Com Content Network
The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the ...
In other words, the pushforward of the tangent vector to the curve at is the tangent vector to the curve at Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by
For example, if X, Y are manifolds, R the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms. The homotopy invariance of cohomology states that if two maps f, g: X → Y are homotopic to each other, then they determine the same pullback: f * = g *.
Pullback, a name given to two different mathematical processes; Pullback (cohomology), a term in topology; Pullback (differential geometry), a term in differential geometry; Pullback (category theory), a term in category theory; Pullback attractor, an aspect of a random dynamical system; Pullback bundle, the fiber bundle induced by a map of its ...
In other words, it is the k-th homotopy group of the suspension spectrum of X. homotopy pullback A homotopy pullback is a special case of a homotopy limit that is a homotopically-correct pullback. homotopy quotient
The limit L of F is called a pullback or a fiber product. It can nicely be visualized as a commutative square: Inverse limits. Let J be a directed set (considered as a small category by adding arrows i → j if and only if i ≥ j) and let F : J op → C be a diagram. The limit of F is called an inverse limit or projective limit.
If S is a sieve on X, and f: Y → X is a morphism, then left composition by f gives a sieve on Y called the pullback of S along f, denoted by f S. It is defined as the fibered product S × Hom(−, X ) Hom(−, Y ) together with its natural embedding in Hom(−, Y ).
The pullback diffeology of a diffeological space by a function : is the diffeology on whose plots are maps such that the composition is a plot of . In other words, the pullback diffeology is the smallest diffeology on X {\displaystyle X} making f {\displaystyle f} differentiable.