0
$\begingroup$

Let $C$ be a cocomplete category and $S$ be a set such that $S \subset \operatorname{Ob} C$ and for any object $a$ in $C$ there is at least one arow from $a$ to some object in $S$. Is there terminal object in $C$.

I know that answer is yes, but I have some difficulties with understanding why.

Suppose a diagram which consists of all objects in $S$ and $Id$ arrows. Since this diagram is small there is a colimit $K$ with following property: for any object $a$ in $C$ there is an arrow from $a$ to $K$. $K$ is not a terminal object since there is nothing to guarantee that such arrows are unique.

I was given a hint to take a colimit of diagram with only object $K$ and all endomorphisms of $K$ as arrows. The claim is that described colimit $E$ is terminal object and I can't see why.

Thanks!

$\endgroup$
  • 3
    $\begingroup$ You can find the proof of the dual statement in Mac Lane, "Categories for the Working Mathematician", p.120. $\endgroup$ – Oskar Feb 11 at 23:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.