An example is in the category of the quantities, every non-empty quantity, especially the one element.

You can see that this is not something like a GIMS, which is clear except for isomorphism,

because otherwise all non-empty quantities would suddenly be isomorphic.

There can be different generators, but this is not a universal property, as we have seen before.

There is also no exact number of quantities.

Nevertheless, there is something that can be said to be more abstract, as we will see in a moment.

So, every non-empty quantity.

For example, G equals 1.

G equals 1.

Let's have a look at something more complicated.

What is mathematical? Vector spaces over a body. Let's take real numbers, because it is a bit more visual.

What is that?

Basically, we want to say that there are two vector spaces, two linear representations.

They are unequal if they differ on a vector.

Basically, I want to select this vector here with the help of a linear function.

Of course, one element vector space is not enough, because then one element would be zero.

Unfortunately, I can only select zero in A.

I need free space over a generator, and that is R itself.

If I map the basis of R as a vector space somewhere, I can select one vector and the rest is a linear combination.

So, this is, so to speak, then again every non-trivial vector space.

For example, the smallest possible would be G equals R itself.

Is that enlightening, or do I have to remember my own line of R algebra?

Well, how can I put it briefly?

The basis of this is a vector, for example 1.

A linear representation from here somewhere, for example to A, is determined by what happens to this 1.

I select a vector V in A and map the 1 there, and everything else has to be linear.

All the others are linear combinations of 1, so practically a scaling of 1.

Summing can be done, but that doesn't matter.

I choose a degree in A, that is the image of R.

But it is enough to look at the 1.

And I pick this one vector, which is different here, and the H is what it does.

Ok, a bit more complicated algebraically, if I look at groups, then it is also analogous.

I take the free on a generator.

I don't even have to know what it looks like, only that it is free and determines what I do on the generator.

So, every free group would be equal to a generator.

Unfortunately, this is a overload of the term generator, because in algebra there are elements of an algebra,

which generate the whole algebra by closing under operations.

And here we have the term generator for such an object.

But that is the universal algebraic term.

Now we do an object, which is a bit easier to understand, or a category.

And that is set cross set, and it doesn't seem to differ much from set, but it doesn't have a generator.

It doesn't have a generator.

So, what happens there?

The objects here are pairs of quantities, and the morphisms are pairs of representations.

So, I have A and A' which is one object, B and B' is another object,

and my two parallel morphisms are F, F' for example, and G, G'.

Composition and so on, all work in a component-based and independent manner.

And now they differ when a component makes a difference.

That means, you try to say, well, then I have to choose either one or the other element,

where one or the other differs.