We assume ZFC+U. A category is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying some conditions.

Let $\mathbb{U}$ be a Grothendieck universe. An element of $\mathbb{U}$ is called a $\mathbb{U}$-set. A set is called $\mathbb{U}$-small if it is isomorphic to a $\mathbb{U}$-set. In the following, we suppose that $\mathbb{N} \in \mathbb{U}$.

In SGA4, a category $\mathcal{C}$ is called $\mathbb{U}$-small if $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ is $\mathbb{U}$-small as a set (if my understanding is correct). However, I don't see this definition working well. For any set $a$ and $b$, an ordered pair $(a,b)$ is always $\mathbb{U}$-small since $(a,b)=\{\{a\},\{a,b\} \}$ is a set consisting of exactly two elements, which is isomophic to $2:=\{\emptyset,\{\emptyset\}\} \in \mathbb{U}$. Thus, $\mathbb{U}$-smallness imposes nothing on categories. In particular, it is not equivalent to $\operatorname{Ob} \mathcal{C}$ and $\operatorname{Mor} \mathcal{C}$ are $\mathbb{U}$-small.

I think I am mistaken somewhere, where is it?

  • $\begingroup$ Could you refer specifically to where this definition occurs? $\endgroup$
    – LSpice
    Jan 25 at 4:33
  • 4
    $\begingroup$ In the new edition of SGA 4, there is an editor's footnote on p. 2: "A category is viewed as a set of arrows". $\endgroup$
    – abx
    Jan 25 at 5:06
  • $\begingroup$ It has been a long time since I looked at SGA 4 but do they actually define smallness as you say...? My impression was that they define a notion that is not bijection-invariant. $\endgroup$
    – Zhen Lin
    Jan 25 at 10:39
  • $\begingroup$ @Zhen Lin In page 2 of SGA 4 (see abx's link above), he says 'On dit qu’un ensemble est U-petit (ou, quand aucune confusion n’en résulte, petit) s’il est isomorphe à un élément de U.' $\endgroup$
    – LOCOAS
    Jan 25 at 12:08
  • 2
    $\begingroup$ In my view this is a completely artificial problem. We all know very well what is a small category, don't we? $\endgroup$
    – abx
    Jan 25 at 14:36


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