In this talk we will consider different flavours of homotopy theory for different types of geometric contexts. First we will recall the usual definitions of the homotopy category, and equivariant homotopy categories, and we will show how we can think of these in terms of (equivariant) smooth manifolds. This will give us a suitable framework with which to adapt the notion of homotopy theory to different geometries.

We first use this framework to give a brief overview of the story in algebraic geometry, which produces motivic homotopy theory. Next, we will use our framework to discuss notions of homotopy theories parametrized by a geometric object, and discuss some consequences.

Depending on time constraints and interest, we will end either by discussing work relating this homotopical framework to 6-functor formalisms, by going over a version of this story in the complex analytic setting, or by introducing C^∞-schemes and discussing possibilities for an extension of the story for manifolds to this setting.