ScalingStacks

In this work, we use Joyal’s quasi-category model of $\oo$-category theory [Jo]. Boardman and Vogt first introduced these simplicial sets in [BV], as weak Kan complexes, and their and Joyal’s theory has been developed in great depth by Lurie in [Lu1] and [Lu2], our primary references; see the first chapter of [Lu1] for an introduction. We use this model, rather than model categories or simplicial categories, because of the great technical advantages for constructions involving categories of functors, which are ubiquitous in this work.

More specifically, we work inside of the quasi-category associated to this model category of Joyal’s. In particular, each map between quasi-categories is understood to be an iso- and inner-fibration; (co)limits among quasi-categories are equivalent to homotopy (co)limits with respect to Joyal’s model structure. As we work in this way, we refer the reader to these sources for $\infty$-categorical versions of numerous familiar results and constructions among ordinary categories. To point, we will make repeated use of the $\infty$-categorical adjoint functor theorem (Corollary 5.5.2.9 of [Lu1]); the straightening-unstraightening equivalence between Cartesian fibrations over an $\infty$-category $\mathcal{C}$ and $\Cat$-valued contravariant functors from $\mathcal{C}$ (Theorem 3.2.0.1 of [Lu1]), and likewise between right fibrations over $\mathcal{C}$ and space-valued presheaves on $\mathcal{C}$ (Theorem 2.2.1.2 of [Lu1]); the $\infty$-categorical version of the Yoneda functor $\mathcal{C}\to\Psh(\mathcal{C})\simeq{\sf RFib}_{\mathcal{C}}$ as it evaluates on objects as $c\mapsto\mathcal{C}_{/c}$ (see §5.1 of [Lu1]).

We will also make use of topological categories, such as $\mfld_{n}$ of $n$-manifolds and embeddings among them. By a functor $\mathcal{S}\rightarrow\mathcal{C}$ from a topological category to an $\oo$-category $\mathcal{C}$ we will always mean a functor $\mathsf{N}\Sing\mathcal{S}\rightarrow\mathcal{C}$ from the simplicial nerve of the ${\sf Kan}$-enriched category obtained by applying the product preserving functor $\Sing$ to the morphism topological spaces.

The reader uncomfortable with this language can substitute the words “topological category” for “$\oo$-category” wherever they occur in this paper to obtain the correct sense of the results, but they should then bear in mind the proviso that technical difficulties may then abound in making the statements literally true. The reader only concerned with algebras in chain complexes, rather than spectra, can likewise substitute “pre-triangulated differential graded category” for “stable $\oo$-category” wherever those words appear, with the same proviso.