Example 1.5 (0MWB) · ScalingStacks

0MWB

Example 1.5. Consider the category ${\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}$ of vector bundles: its objects are the pairs $(M,V)$ of a manifold $M$ and a vector bundle $V\downarrow M$ , and its morphisms are commutative squares

(of a morphism of manifolds and a compatible morphism of (total spaces of) vector bundles). Then, the forgetful functor

to the category of manifolds is a cartesian fibration. Given a morphism $M\xrightarrow{\varphi}N$ in ${\mathcal{M}\textup{fld}}$ and a vector bundle $W\downarrow N$ , a cartesian lift is provided by the pullback vector bundle, which is defined by a pullback square

of underlying topological spaces. The fiber/cartesian factorization system in this cartesian fibration translates into the assertion that for any vector bundle $V\downarrow M$ , the induced diagram

is a pullback square in ${\mathcal{S}\textup{et}}$ , where ${\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}(M)$ denotes the fiber $\textup{U}_{\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}^{-1}(M)$ of the forgetful functor over the object $M\in{\mathcal{M}\textup{fld}}$ (or equivalently, the value at $M$ of the functor

classifying our cartesian fibration).

There is a canonical section

of the forgetful functor, called the tangent bundle: this takes a manifold $M\in{\mathcal{M}\textup{fld}}$ to its tangent bundle $TM\downarrow M$ . Note that this is not a cartesian section: it does not take morphisms in ${\mathcal{M}\textup{fld}}$ to cartesian morphisms of the cartesian fibration. (Correspondingly, this does not arise from a natural transformation

between ${\mathcal{C}\textup{at}}$ -valued functors on ${\mathcal{M}\textup{fld}}^{op}$ .) In other words, given an arbitrary morphism $M\xrightarrow{\varphi}N$ of manifolds, we obtain a canonical map $TM\rightarrow\varphi^{*}(TN)$ in ${\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}(M)$ , but this map is not generally an isomorphism. However, it is an isomorphism (and the morphism $TM\rightarrow\varphi^{*}(TN)$ is a cartesian lift of $\varphi$ ) whenever the morphism $\varphi$ is an open embedding.