# Fixed points as a full subcategory of prefixed points?

I'm working through these notes, which cover some fixed point theory for categories.

Here, we encounter the following definitions (I've rephrased them):

Let $C$ be a category and $F$ an endofunctor on $C$.

• A prefixed point of $F$ is pair $(A, \alpha)$ where $A$ is an object of $C$ and $\alpha$ a morphism from $F(A)$ to $A$;

• a fixed point of $F$ is a prefixed point $(A, \alpha)$ where $\alpha$ is an isomorphism.

Then. the category PFP of prefixed points of $F$ is defined in the following way:

• its objects are the prefixed points of $F$;

• for each $(A,\alpha)$, $(B,\beta)$ in PFP, the morphism set from $A$ to $B$ contains exactly the arrows $f \in \hom(A,B)$ such that the diagram \begin{aligned} &F(A) &\xrightarrow{\alpha} &A \\ F(f) &\downarrow & &\downarrow f\\ &F(B) &\xrightarrow{\beta} &B \end{aligned} commutes.

Can I then define the category FP of fixed points of $F$ as the full subcategory of PFP whose objects are the fixed points of $F$?

• Why couldn't you? – gniourf_gniourf Dec 22 '16 at 14:13
• To me, that's perfectly sound. However, in the linked notes another definition is given, so I thought there could be something wrong with the definition I've come up with. – Filippo De Bortoli Dec 22 '16 at 14:16
• Isn't it what they actually do? are you talking about the second point of Definition 2.20 on top of page 18 ? – gniourf_gniourf Dec 22 '16 at 14:20
• And they wrote it later in that same page! Not paying attention can lead to this. Thank you! I'll delete the question in a few minutes. :-) – Filippo De Bortoli Dec 22 '16 at 14:22
• Please don't do it! I'm sure we can make it interesting by asking some more. There surely is a relation between prefixed points and coalgebras for a functor. This has been extensively addressed in Kelly's "...and so on" paper. Give a look at that! It can answer questions like: is $\bf PF$ co/reflective in $\bf PFP$? Is this co/reflection exact? Is there a limit process yielding (pre)fixed points? Grazie! – Fosco Dec 22 '16 at 16:34