Remark 4.2.2. Big categories. In some texts a category is allowed to have a proper class of objects. We will allow this as well in these notes but only in the following list of cases (to be updated as we go along). In particular, when we say: “Let $\mathcal{C}$ be a category” then it is understood that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is a set.

The category $\textit{Sets}$ of sets.

The category $\textit{Ab}$ of abelian groups.

The category $\textit{Groups}$ of groups.

Given a group $G$ the category $G\textit{-Sets}$ of sets with a left $G$-action.

Given a ring $R$ the category $\text{Mod}_ R$ of $R$-modules.

Given a field $k$ the category of vector spaces over $k$.

The category of rings.

The category of schemes.

The category $\textit{Top}$ of topological spaces.

Given a topological space $X$ the category $\textit{PSh}(X)$ of presheaves of sets over $X$.

Given a topological space $X$ the category $\mathop{\mathit{Sh}}\nolimits (X)$ of sheaves of sets over $X$.

Given a topological space $X$ the category $\textit{PAb}(X)$ of presheaves of abelian groups over $X$.

Given a topological space $X$ the category $\textit{Ab}(X)$ of sheaves of abelian groups over $X$.

Given a small category $\mathcal{C}$ the category of functors from $\mathcal{C}$ to $\textit{Sets}$.

Given a category $\mathcal{C}$ the category of presheaves of sets over $\mathcal{C}$.

Given a site $\mathcal{C}$ the category of sheaves of sets over $\mathcal{C}$.

One of the reason to enumerate these here is to try and avoid working with something like the “collection” of “big” categories which would be like working with the collection of all classes which I think definitively is a meta-mathematical object.

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