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.

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  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.)

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  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.

As for the technical content of this note, the skeptical reader is completely justified in asking: Why does this matter? We offer three related and complementary responses.

1. (a)

   The first and most obvious justification for model-independence is that it allows one to work model-independently. Being a vastly general and broadly applicable home for derived mathematics, the theory of $\infty$-categories has recently found widespread and exciting use in a plethora of different areas – for instance in the geometric Langlands program and in mathematical physics, to name but two (closely related) examples. In such applications, “$\infty$-categories” are generally manipulated in a purely formal fashion, a rule of thumb being that anything the typical user of $\infty$-categories would like to study should be accessible without reference to model-dependent notions such as quasicategories, various sorts of fibrations between them, their individual simplices, etc.

   For the most part, the theory of quasicategories – being itself soundly founded in the theory of model categories – allows for direct and straightforward manipulation of the corresponding model-independent notions: by and large, the definitions visibly descend to the underlying $\infty$-category of the model category $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$. However, the theory of co/cartesian fibrations is a glaring exception. It is therefore not a priori meaningful to work with these notions in a model-independent fashion. In order to allow the myriad users of $\infty$-category theory throughout mathematics to employ the theory of co/cartesian fibrations – and in particular, to allow them to appeal to the various results which have been proved about their incarnations in quasicategories –, it seems like nothing more than good homotopical manners to prove that these quasicategorical definitions do indeed descend to the underlying $\infty$-category of $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ as well.

   In particular, our results provide the crucial input to a model-independent reading of [Lur14], which work is premised heavily on the notion of co/cartesian morphisms in quasicategories.

2. (b)

   The next most obvious justification for model-independence is that it provides conceptual clarity: a model-independent definition is by definition unfettered by point-set or model-dependent notions which obfuscate its true meaning and significance.

   For a rather grotesque example, recall that one can define the “$n^{\textup{th}}$ homotopy group” of a based simplicial set $(Y,y)\in s{\mathcal{S}\textup{et}}_{*}$ as certain subquotient

   $$\pi_{n}(Y,y)=\left.\left\{\sigma\in(\mathbb{R}Y)_{n}:\delta^{n}_{i}(\sigma)=\sigma^{\circ(n-1)}(y)\textup{ for all }0\leq i\leq n\right\}\right/\sim$$

   of the set of $n$-simplices of a fibrant replacement $Y\stackrel{{\scriptstyle\approx}}{{\rightarrow}}\mathbb{R}Y\twoheadrightarrow\Delta^{0}$ (relative to the standard Kan–Quillen model category structure $s{\mathcal{S}\textup{et}}_{\textup{KQ}}$, and with respect to the induced basepoint $\Delta^{0}\xrightarrow{y}Y\rightarrow\mathbb{R}Y$). This definition completely obscures a number of important features of homotopy groups:

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     that they are independent of the choice of fibrant replacement;

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     that they actually form a group at all (for $n\geq 1$, let alone an abelian group for $n\geq 2$);

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     that a path connecting two basepoints (which itself may only be representable by a zigzag of edges in $Y$ itself) induces a conjugation isomorphism (for $n\geq 1$);

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     that they are corepresentable in the homotopy category $\ho(\mathcal{S}_{*})$ of pointed spaces.

   Though the quasicategorical definitions of co/cartesian morphisms and co/cartesian fibrations are not nearly so abstruse, they are nevertheless model-dependent, and hence the assertion that they have homotopical meaning requires further proof.

3. (c)

   Lastly, proving that quasicategorical definitions are model-independent also allows mathematicians employing specific but alternate models for $\infty$-categories to employ these notions and their attending results while continuing to work within their native context. As is particularly well-known to homotopy theorists, different model categories presenting the same $\infty$-category (e.g.​ and i.e.​ that of spectra) can be advantageous for different purposes. For a few examples of where such alternate models for $\infty$-categories arise:

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     ${\mathcal{T}\textup{op}}$-enriched categories appear in geometric topology;

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     $\mathcal{A}_{\infty}$-categories appear in mirror symmetry;

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     dg-categories appear in derived algebraic geometry and homological algebra;

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     Segal spaces (and indeed, Segal $\Theta_{n}$-spaces) appear in bordism theory;

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     $s{\mathcal{S}\textup{et}}$-enriched categories appear in homotopy theory (both as hammock localizations of relative categories (e.g.​ model categories) and as (the full subcategories of bifibrant objects of) simplicial model categories).

   In some sense, this is mainly a preservation-of-sanity issue. All of the various the model categories which present “the $\infty$-category of $\infty$-categories” are connected by an intricate web of Quillen equivalences (see [BSP]). It is therefore already possible to wrangle a given question asked in any of these model categories into a corresponding question asked in the model category $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$, so long as one is willing to take the appropriate co/fibrant replacements at every step to ensure that one is computing the derived values of these various Quillen equivalences. However, this procedure is a hassle at best, and at worst can effectively destroy all hope of answering the given question (since taking co/fibrant replacements generally drastically alters the object’s point-set features).

   This motivation can therefore be seen as something of a “down-then-up” maneuver: in proving that these notions actually descend to the underlying $\infty$-category which is common to all of these distinct model categories of $\infty$-categories, we prove that all of the results which have been proved in quasicategories can be brought to bear while working in any of these models – no wrangling required.

We refer to Lurie’s twin tomes [Lur09] and [Lur14] for general references on quasicategories. For further motivation regarding the Grothendieck construction (including a variety of applications), we refer the interested reader to [MGa]. Lastly, we refer to [MGb, §A] for details and specifics on working “within the $\infty$-category of $\infty$-categories” (as well as for further clarification regarding our conventions).

It is a pleasure to acknowledge our debt of influence to David Ayala, both for the generalities as well as for various specific observations that appear throughout this note. We are also deeply grateful to Piotr Pstragowski for his suggested outline of the proof of Theorem 3.3, and to Zhen Lin Low for countless conversations regarding the ideas that appear here. Finally, we would like to thank Montana State University for its hospitality as well as UC Berkeley’s Geometry and Topology RTG grant (which is part of NSF grant DMS-0838703) for its generous funding of our stay there (during which this note was written).