ScalingStacks

This note concerns the notions of cocartesian fibrations and of cartesian fibrations, as well as their connection to the Grothendieck construction, in the $\infty$-categorical context.

• •

  The ur-example of a cartesian fibration is the forgetful functor ${\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}\rightarrow{\mathcal{M}\textup{fld}}$ from the category of vector bundles – that is, of pairs $(M,V)$ of a manifold $M$ and a vector bundle $V\downarrow M$ – to the category of manifolds; the fact that this is a cartesian fibration encodes the observation that vector bundles can be pulled back in a category-theoretically meaningful way. (See Example 1.5.)

• •

  Meanwhile, the ur-example of the Grothendieck construction accounts for the equivalence between the two competing definitions of a stack in algebraic geometry, either as a functor to the category of groupoids or as a “category fibered in groupoids”.

The purpose of this note is twofold.

1. (1)

   On the one hand, we offer a relaxed and informal discussion of co/cartesian fibrations and their motivation coming from the Grothendieck construction, which assumes no prerequisites beyond a vague sense of the meaning of the signifier “$\infty$-category”.

2. (2)

   On the other hand, we carefully prove that our relaxed and informal discussion was in fact actually completely rigorous all along. More precisely, we show that our definitions, which are formulated within the $\infty$-category of $\infty$-categories, are suitably compatible with the corresponding notions in quasicategories (see Theorem 3.3 and Corollary 3.4).¹¹ 1 As the quasicategorical definitions are not manifestly model-independent, the proofs of these results are nontrivial. Indeed, they are fairly involved, and rely on rather subtle model-categorical manipulations.

The bulk of this note consists in §​​ 1 (where we offer our relaxed and informal discussion) and in §​​ 3 (where we present our proofs). Separating these, in §​​ 2 we distill the discussion of §​​ 1 into precise model-independent definitions of co/cartesian morphisms and of co/cartesian fibrations.