The Stacks project

Remark 21.31.5 (Set theoretic issues). The category $\textit{LC}$ is a “big” category as its objects form a proper class. Similarly, the coverings form a proper class. Let us define the size of a topological space $X$ to be the cardinality of the set of points of $X$. Choose a function $Bound$ on cardinals, for example as in Sets, Equation (3.9.1.1). Finally, let $S_0$ be an initial set of objects of $\textit{LC}$, for example $S_0 = \{ (\mathbf{R}, \text{euclidean topology})\} $. Exactly as in Sets, Lemma 3.9.2 we can choose a limit ordinal $\alpha $ such that $\textit{LC}_\alpha = \textit{LC} \cap V_\alpha $ contains $S_0$ and is preserved under all countable limits and colimits which exist in $\textit{LC}$. Moreover, if $X \in \textit{LC}_\alpha $ and if $Y \in \textit{LC}$ and $\text{size}(Y) \leq Bound(\text{size}(X))$, then $Y$ is isomorphic to an object of $\textit{LC}_\alpha $. Next, we apply Sets, Lemma 3.11.1 to choose set $\text{Cov}$ of qc covering on $\textit{LC}_\alpha $ such that every qc covering in $\textit{LC}_\alpha $ is combinatorially equivalent to a covering this set. In this way we obtain a site $(\textit{LC}_\alpha , \text{Cov})$ which we will denote $\textit{LC}_{qc}$.