= Adjoints to constant presheaves on open sets
{title2=$\operatorname{ev}_{\varnothing}\dashv\Delta\dashv\operatorname{ev}_S$}
For a <topological space> $S$, let $\Delta A$ be the <constant presheaf of sets> on its open-set category, with identity restriction maps, including the value at the empty open set. A <natural transformation> $P\to\Delta A$ is determined by its map $P(\varnothing)\to A$; its component at $U$ composes with $P(U)\to P(\varnothing)$. A transformation $\Delta A\to P$ is determined by $A\to P(S)$; its component at $U$ restricts sections from $S$. These two bijections prove the displayed <adjunctions>. This concerns presheaves, so no sheafification is involved.
Back to article page
