# Filtered colimit and directed colimit, Example

This is a theorem in Locally presentable and accessible categories, by Adamek, Rosicky.

For every (small) filtered category $$D$$ there exists a (small) directed poset $$D_0$$ and a cofinal functor $$H:D_0 \rightarrow D$$.

It says as an example to illustrate the difficulty, consider

$$D$$ with one object $$d$$ two morphisms $$id$$ and $$f=f\cdot f$$. Then $$D_0$$ has to be an infintie directed category $$d \xrightarrow{f} d \xrightarrow{f} d \cdots$$

There must be some serious misunderstanding in my knowledge of a cofinal functor. According to the book, we require

Given $$f:d \rightarrow Hd_0, f':d \rightarrow Hd_1$$, there exists $$g:d_0 \rightarrow d , g': d_1 \rightarrow d'$$, such that $$Hg \cdot f = Hg' f'$$ where $$d \in D, d_0, d_1, d' \in D_0$$.

How could this be attaiened by the example?

Simply consider $$d \xrightarrow{id} Hd=d, d \xrightarrow{f} Hd=d$$ Then there is no way to find such $$g,g'$$.

What am I missing?

Take any $$d'$$ such that $$d_0,d_1 < d'$$ in the poset $$D_0$$. Then the arrows $$g,g'$$ are not identities, so that $$H(g)=f=H(g')$$, and thus $$H(g)\circ id=f=f\circ f=H(g')\circ f$$.
• Ah, I thought $f^2=id$... Thanks a lot. – Bryan Shih Aug 8 '19 at 8:03