Pcks

IDEA

Then it's the recording in German.

Then it's the recording in German, of course.

Dirty.

Speaking in front of a purely German audience in English always feels so stupid.

They have to get a mix of it or let it translate or something.

This chalk is not only bad, I think it's also unhealthy, but whatever.

So, we'll look at, as I said, special objects and morphisms.

And we'll start with a very simple term.

Maybe the simplest definition that can be written in the category theory at all.

And at the same time a very central one, which also goes through much more complicated terms.

And which also has much more complex examples than our very simple initial examples suggest.

So, let's call this child.

So, it's an property of a single object, B.

And such a thing is called terminal or final.

If the following property applies.

For every other object A there exists exactly one morphism from A to B.

That was the definition.

Definition can be a lot.

Two things are part of it.

To check or convince ourselves that a definition made once is meaningful.

What could that be?

These things are part of a good definition.

Yes?

So, there should be something that has this property.

Well, it shouldn't have all this property.

So, the definition itself and its negation should be achievable.

That was purely formalistic.

Conditions for a good definition.

A little less formalistic would be to demand that there are no trivial examples for the definition.

There are always these famous cases where a system of actions is calculated.

Until it is determined that the only model is the empty quantity.

Well, the empty quantity would be an example.

But you would also demand that there are no trivial examples.

But typically there should be as many interesting examples as possible.

A definition that is only valid for one object.

Or for a very narrow restriction on what it is.

Depending on that, it can be justified to make very strong restrictions on things.

Where there are only a few examples.

Ideally, there are many examples for a definition.

That is what most people think about when they write scientific papers.

As many of you have already done or will do.

There is another requirement that is often forgotten when you look at scientific literature.

Which is this?

No, that would be included in the first part.

You have to be able to prove something with a definition.

You have to be able to write a term and list 500 examples.

But if you read through the literature, many papers are like this.

You can see that I did not refer to them.