For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms $d$-dimensional cobordisms. We identify homotopy type its classifying space (i.e. geometric realization simplicial nerve) as fixed points infinite loop spectrum.

In this short expository note, we discuss, with plenty of examples, the bestiary of fibrations in quasicategory theory. We underscore the simplicity and clarity of the constructions these fibrations make available to end-users of higher category theory.