pullback of this diagram exists and given by the subring of the product ring defined by along with the morphisms given by and for all . We then have Groups and modules In complete analogy to the example of commutative rings above, one can show that all pullbacks exist in the category of groups and in the category of modules over some fixed ring. Sets In the category of sets, the pullback of functions and always exists and is given by the set together with the restrictions of the projection maps and to . Alternatively one may view the pullback in asymmetrically: where is the disjoint union of sets (the involved sets are not disjoint on their own unless resp. is injective). In the first case, the projection extracts the index while forgets the index, leaving elements of . This example motivates another way of characterizing the pullback: as the equalizer of the morphisms where is the binary product of and and and are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers. In fact, by the existence theorem for limits, all finite limits exist in a category with binary products and equalizers; equivalently, all finite limits exist in a category with terminal object and pullbacks (by the fact that binary product = pullback on the terminal object, and that an equalizer is a pullback involving binary product). Fiber bundles Another example of a pullback comes from the theory of fiber bundles: given a bundle map and a continuous map , the pullback (formed in the category of topological spaces with continuous maps) is a fiber bundle over called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles. This is also the case in the category of differentiable manifolds. A special case is the pullback of two fiber bundles . In this case is a fiber bundle over , and pulling back along the diagonal map gives a space homeomorphic (diffeomorphic) to , which is a fiber bundle over . The pullback of two smooth transverse maps into the same differentiable manifold is also a differentiable manifold, and the tangent space of the pullback is the pullback of the tangent spaces along the differential maps. Preimages and intersections Preimages of sets under functions can be described as pullbacks as follows: Suppose , . Let be the inclusion map .