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Stream: learning: questions

Topic: ✔ Sheaf condition as limit condition

Sergei Burkin (Sep 24 2023 at 03:27):

Let $\mathcal{C}$ be a category, $X$ be an object in $\mathcal{C}$, and $\mathcal{U}=\{U_i\to X\}$ be a sieve on $X$. Is it true that for any presheaf $F$ on $\mathcal{C}$ the sheaf condition for $F(X)$ with respect to $\mathcal{U}$ is the same as the limit condition with respect to the diagram $\mathcal{B}^{op}\to Sets$ induced by $F$, where $\mathcal{B}$ is full subcategory of the category $\mathcal{C}/X$, with objects of $\mathcal{B}$ being precisely the elements of $\mathcal{U}$?

This seems to be trivially true, but for some reason I cannot find the sheaf condition stated as the limit condition, so maybe I've missed something.

Jonas Frey (Sep 24 2023 at 10:31):

Let $S\subseteq \mathcal C/X$ be the sieve generated by $\mathcal U$, viewed as a full subcategory of the slice category $\mathcal C/X$. The diagram $S\rightarrow\mathcal C/X\to\mathcal C$ admits a canonical cocone with vertex $X$, and the sheaf condition for $\mathcal U$ is equivalent to $F$ mapping this cocone to a limit cone.

Notification Bot (Sep 24 2023 at 14:23):

Sergei Burkin has marked this topic as resolved.

Sergei Burkin (Sep 24 2023 at 14:25):

Thank you

Notification Bot (Sep 26 2023 at 03:04):

Ned Summers has marked this topic as unresolved.

Notification Bot (Sep 26 2023 at 03:04):

Ned Summers has marked this topic as resolved.