ScalingStacks

A.6 Set-theoretic considerations[00IM]

In order to deal with set-theoretic issues, we systematically use the device of Grothendieck universes (see e.g. [Lur09, § 1.2.15]). Specifically, we fix a triple of strongly inaccessible cardinals \(\kappa_0 < \kappa_1 < \kappa_2\). The sets of cardinality \(<\kappa_i\) for \(0\leq i \leq 2\) will be called \(\kappa_i\)-small and they form Grothendieck universes \(U_0\in U_1\in U_2\). Likewise, a category is called \(\kappa_i\)-small if the sets of isomorphism classes of objects and the homotopy groups of morphisms spaces are of cardinality \(<\kappa_i\). We refer to \(\kappa_0\)-small objects as small, to \(\kappa_1\)-small objects as large, and to \(\kappa_2\)-small objects as huge (and the latter play almost no role in in our work). So for instance, the \(\infty\)-category \(\mathrm{Cat}_\infty\) of small \(\infty\)-categories is large, as is the \(\infty\)-category \(\mathcal S\) of (small) spaces.

We occasionally write e.g. \(\widehat{\mathrm{Cat}}_\infty\) to refer to the huge \(\infty\)-category of large \(\infty\)-categories. Its main use is that it contains the \(\infty\)-category of presentable \(\infty\)-categories (see Subsection A.7). We often prove results for \(\mathrm{Cat}_\infty\) and then apply them to \(\widehat{\mathrm{Cat}}_\infty\) (which is easily justified by a change of Grothendieck universe) in order to discuss specializations to presentable \(\infty\)-categories.

We may sometimes emphasize smallness (e.g. of a set or of an \(\infty\)-category). On the other hand, we may also omit the word “small” for brevity; for instance, when we say that an \(\infty\)-category admits all colimits we certainly mean that it admits all small colimits.

We generally refer to a large set as a “class” (and to a class that is not small as a “proper class”). However, in related contexts we will have occasion to contemplate large spaces, and rather than belabor the distinction we simply also refer to these as “classes”.

Relatedly, in the most invariant terms, given an \(\infty\)-category \(\mathcal C\), “a set of objects of \(\mathcal C\)” refers to a set \(S\) equipped with a functor \(S \xrightarrow{F} \mathcal C\). We say that an object of \(\mathcal C\) lies in the set if it is in the image of \(F\) (up to equivalence). Said differently, when we refer to a set of objects of \(\mathcal C\), we generally intend to implicitly refer to its image (a subgroupoid of \(\mathcal C\)). Note that if \(S\) is small and \(\mathcal C\) is locally small, then the image of \(S\) in \(\mathcal C\) is also small; hence, in such cases this implicit passage to images does not change size.