In differential geometry, the existence of pullbacks is a delicate matter, since the category of smooth manifolds does not admit all of them. This issue is reflected in tangent categories, which offer a minimal categorical framework for differential geometry.
Often, in differential geometry, one solves this issue by working with submersions, which form a well-behaved class of morphisms. In this talk, we introduce another class of maps in a tangent category, called tangent display maps, which are designed to work well with respect to pullbacks and that generalize submersions in differential geometry.
From this definition, we also show how to construct a notion of open sub-objects in a tangent category, reconstructing the topological information of the objects.