The affine Grassmannian associated to a reductive group G is a key object in the Geometric Langlands Program. We will recall its definition, as well as that of the \infty-category of constructible sheaves over it, which is a model for the so-called spherical Hecke category. We will explain how both algebro-geometric and homotopy-theoretic properties of the affine Grassmannian induce key features of the spherical Hecke category, including its E_3-monoidal structure. We will make substantial use of the theory of stratified spaces and exodromy, whose basic concepts we will briefly recall.