ScalingStacks

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 $k$, we may consider enhancing its structure in three different ways, promoting it to

1. $\bullet$

   a differential graded (dg) category,

2. $\bullet$

   an $A_{\infty}$-category, or

3. $\bullet$

   a stable $\infty$-category.

Among the many excellent references for dg and $A_{\infty}$-categories, we recommend the survey [Ke]. Relative a ring of characteristic zero $k$, all three formalisms become equivalent: $k$-linear stable $\infty$-categories are equivalent to $k$-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 $k$-linear dg category” for “stable $\infty$-category” throughout the present paper. The distinction between $k$-linear stable $\infty$-categories, dg- and $A_{\infty}$-categories becomes important when considering commutative algebra away from characteristic zero, for which purpose we will only consider the former.