For any countable1 diagram category $\mathcal{I}$ and any functor $F : \mathcal{I} \to \mathit{Sch}_\alpha $, the limit $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} F$ exists in $\mathit{Sch}_\alpha $ if and only if it exists in $\mathit{Sch}$ and moreover, in this case, the natural morphism between them is an isomorphism.
[1] Both the set of objects and the morphism sets are countable. In fact you can prove the lemma with $\aleph _0$ replaced by any cardinal whatsoever in (3) and (4).
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)