ScalingStacks

2.1. ∞\infty-categories

We are interested in studying algebraic operations on categories of a homotopy-theoretic nature such as derived categories of sheaves or spectra. It is well established that the theory of triangulated categories, through which derived categories are usually viewed, is inadequate to handle many basic algebraic and geometric operations. Examples include the absence of a good theory of gluing or of descent, of functor categories, or of generators and relations. The essential problem is that passing to homotopy categories discards essential information (in particular, homotopy coherent structures, homotopy limits and homotopy colimits).

This information can be captured in many alternative ways, the most common of which is the theory of model categories. Model structures keep weakly equivalent objects distinct but retain the extra structure of resolutions which enables the formulation of homotopy coherence. This extra structure can be very useful for calculations but makes some functorial operations difficult. In particular, it can be hard to construct certain derived functors because the given resolutions are inadequate. There are also fundamental difficulties with the consideration of functor categories between model categories. However, much of the essential information encoded in model categories can be captured by the Dwyer-Kan simplicial localization. This construction uses the weak equivalences to construct simplicial sets (or alternatively, topological spaces) of maps between objects, refining the sets of morphisms in the underlying homotopy category (which are recovered by passing to π0\pi_{0}).

This intermediate regime between model categories and homotopy categories is encoded by the theory of (∞,1)(\infty,1)-categories, or simply ∞\infty-categories. The notion of ∞\infty-category captures (roughly speaking) the notion of a category whose morphisms form topological spaces and whose compositions and associativity properties are defined up to coherent homotopies. Thus an important distinction between ∞\infty-categories and model categories or homotopy categories is that coherent homotopies are naturally built in to all the definitions. Thus for example all functors are naturally derived and the natural notions of limits and colimits in the ∞\infty-categorical context correspond to homotopy limits and colimits in more traditional formulations.

The theory of ∞\infty-categories has many alternative formulations (as topological categories, Segal categories, quasi-categories, etc; see [Ber] for a comparison between the different versions). We will follow the conventions of [L2], which is based on Joyal’s quasi-categories [Jo]. Namely, an ∞\infty-category is a simplicial set, satisfying a weak version of the Kan condition guaranteeing the fillability of certain horns. The underlying simplicial set plays the role of the set of objects while the fillable horns correspond to sequences of composable morphisms. The book [L2] presents a detailed study of ∞\infty-categories, developing analogues of many of the common notions of category theory (an overview of the ∞\infty-categorical language, including limits and colimits, appears in [L2, Chapter 1.2]).

Among the structures we will depend on are the ∞\infty-category of ∞\infty-categories [L2, 3], adjoint functors [L2, 5.2], and ind-categories and compact objects [L2, 5.3] (see also Section 3.1). Most of the objects we encounter form presentable ∞\infty-categories [L2, 5.5]. Presentable ∞\infty-categories are ∞\infty-categories which are closed under all (small) colimits (as well as limits, [L2, Proposition 5.5.2.4]), and moreover are generated in a weak sense by a small category. In particular, by a result of Simpson [L2, Theorem 5.5.1.1], they are given by suitable localizations of ∞\infty-categories of presheaves on a small ∞\infty-category. Presentable ∞\infty-categories form an ∞\infty-category 𝒫​rL{\mathcal{P}r}^{\rm L} whose morphisms are continuous functors, that is, functors that preserve all colimits [L2, 5.5.3]. Note that since presentable categories are closed under all (co)products, categories with a finiteness condition (like compact spaces, coherent sheaves, etc.) do not fall under this rubric. A typical example is the ∞\infty-category of spaces.

Algebra (and algebraic geometry, see below) in the ∞\infty-categorical (or derived) setting has been developed in recent years by Toën-Vezzosi [ToVe1, ToVe2] and Lurie [L3, L4, L5]. This has resulted in a very powerful and readily applicable formalism, complete with ∞\infty-analogues of many of the common tools of ordinary category theory. We single out two powerful tools that are crucial for this paper and available in the ∞\infty-context thanks to [L4] (but are not available in a suitable form in the triangulated or model contexts):

  1. ∙\bullet

    The general theory of descent, as embodied by the ∞\infty-categorical version of the Barr-Beck theorem ([L4, Theorem 3.4.5]).

  2. ∙\bullet

    The theory of tensor products of and functor categories between presentable ∞\infty-categories (reviewed in Section 2.2).

2.1.1. Enhancing triangulated categories

The ∞\infty-categorical analogue of the additive setting of homological algebra is the setting of stable ∞\infty-categories [L3]. A stable ∞\infty-category can be defined as an ∞\infty-category with a zero-object, closed under finite limits and colimits, and in which pushouts and pullbacks coincide [L3, 2,4]. The result of [L3, 3] is that stable categories are enhanced versions of triangulated categories, in the sense that the homotopy category of a stable ∞\infty-category has the canonical structure of a triangulated category. We will mostly be concerned with ∞\infty-categories that are both presentable and stable, as studied in [L3, 17]. Typical examples are the ∞\infty-categorical enhancements of the derived categories of modules over a ring, quasi-coherent sheaves on a scheme, and the ∞\infty-category of spectra.

Given a triangulated category which is linear over a ring kk, we may consider enhancing its structure in three different ways, promoting it to

  1. ∙\bullet

    a differential graded (dg) category,

  2. ∙\bullet

    an A∞A_{\infty}-category, or

  3. ∙\bullet

    a stable ∞\infty-category.

Among the many excellent references for dg and A∞A_{\infty}-categories, we recommend the survey [Ke]. Relative a ring of characteristic zero kk, all three formalisms become equivalent: kk-linear stable ∞\infty-categories are equivalent to kk-linear pre-triangulated dg categories (that is, those whose homotopy category is triangulated). Thus we recommend the reader interested in characteristic zero applications substitute the term “pre-triangulated kk-linear dg category” for “stable ∞\infty-category” throughout the present paper. The distinction between kk-linear stable ∞\infty-categories, dg- and A∞A_{\infty}-categories becomes important when considering commutative algebra away from characteristic zero, for which purpose we will only consider the former.

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5