This question came up when I was reading over some information about sheaves; specifically, that if $\mathscr{F}$ is a sheaf on the topological space $X$, $x\in X$, and $Z\subseteq X$, then $(\mathscr{F}|\_Z )\_x =\mathscr{F}\_x$. I don't know if this is supposed to be trivial, and while it definitely seems to be a desirable property, I didn't see it as being obvious that it is true. After thinking for a while, I came to this conclusion:


Let $\mathfrak{I}$ be a directed category, and $F:\mathfrak{I}\rightarrow\mathfrak{C}$ a covariant functor; for $i\in \mathfrak{I}$, denote $F(i)$ by $A\_i$. Fix a particular $I\in\mathfrak{I}$, and let $\mathfrak{J}$ be the full subcategory of $\mathfrak{I}$ satisfying $\mathrm{obj}(\mathfrak{J})=\lbrace j\in \mathfrak{I}\mid \mathrm{Mor}\_\mathfrak{I}(I,j)\neq \emptyset\rbrace$. Then $\underrightarrow{\mathrm{lim}}\_{i\in \mathfrak{I}}A\_i =\underrightarrow{\mathrm{lim}}\_{j\in \mathfrak{J}}A\_j$.


I believe that this is true (well, I wrote a proof that convinced myself, anyway). If this is correct, it would explain why the statement about the stalks of the sheaves earlier is true. I was wondering if something more general can be said about when a colimit of a subcollection is equal to the colimit of the entire collection? Is there an even more general principle at work here, other than just a property of colimit?



When is a colimit of a subcollection the same as the overall colimit?
Dylan's comment is right. More generally, if $D \colon J \to C$ is a diagram and $L \colon J' \to J$ a functor, then the colimit of D is isomorphic to the colimit of DL if and only if L is (co)final. See Mac Lane, *Cats Work*, section IX.3.


