# On the definition of small categories in SGA4

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?

• Could you refer specifically to where this definition occurs? Jan 25 at 4:33
• 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".
– abx
Jan 25 at 5:06
• 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. Jan 25 at 10:39
• @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.' Jan 25 at 12:08
• In my view this is a completely artificial problem. We all know very well what is a small category, don't we?
– abx
Jan 25 at 14:36