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LOCOAS
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LOCOAS Asks: 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?
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?
^2(dn^2 - dx^2)$ where $dn$ is the conformal time differential form. The Klein Gordon equation in curved spacetime is $(\frac{1}{g^{1/2}}\partial_{\mu}(g^{1/2}g^{\mu\nu}\partial_{\nu}) + m^2)\phi = 0$
