Now we come to a bit more category theory,

while we did everything quite quantitatively last time,

because it was a result of quantitative functions,

that these finite functions get these omega-chain co-limits.

And now we want to say,

now we have everything together,

to actually look at the generalization of the clinch fix point set.

So we have the omega-chains,

these are the omega-chains,

so now categorically considered,

the supremes are the co-limits of them, categorically considered,

monotone imaging are functions,

and the stability of the imaging is now generalized

to the preservation of these co-limits.

And now we only have to generalize the statement of the set,

of the clinch fix point set,

and then the proof, and then we have it.

So that means,

we start now.

So we look at a co-compatible category,

and then an endo-function,

which is omega-co-stable,

so it gets co-limits of omega-chains.

So, and the clinch fix point set,

it says, so, assuming I have a co-compatible and a constant function,

then the smallest fix point is a supremum of such a rising chain.

So then F has the initial algebra,

so first of all it has one, so that's an existing result,

but it is also described in detail as,

well, notation is mi of F,

and the description is that it is the co-limit of the chain of the functor iterations.

So that is now very strongly shortened,

because there is actually no chain,

so where is the chain of morphisms,

that means, we need that now first,

in the first step,

that you make it clear,

what I have simply written here as objects,

so here that is about n omega,

or n element n,

so natural numbers indicated,

what is actually the chain?

So that means,

you look at the omega chain,

and now it comes,

which is a diagram of natural numbers with a small equal to c,

with the object i, the natural number i,

that is F high i of 0,

so 0 is the initial object of the category,

which is co-compatible,