Keith wrote:

> Can't we reduce \\(\mathcal{V}\\)-functors to changes-of-base, mainly a change-of-base between a \\(\mathcal{V}\\)-category and itself?

No, you're making a classic level slip here. There's _no such thing_ as change-of-base between a \\(\mathcal{V}\\)-category and itself! Change-of-base is not a kind of map between \\(\mathcal{V}\\)-categories.

It's great that you made this mistake, because everybody does it. Please think about this! It's easy to get confused... but it's very important to get unconfused.

It's like "a function from an element of a set to itself."

There's _no such thing_ as a function from an element of a set to itself. A function \\(f: X \to Y\\) goes between _sets_ \\(X\\) and \\(Y\\). If \\(X = Y\\) this function may _map_ an element \\(x \in X\\) to itself: \\(f(x) = x\\). But it's still a function from \\(X\\) to \\(X\\), not a function from \\(x\\) to \\(x\\).

Similarly, there's _no such thing_ as a change-of-base from a \\(\mathcal{V}\\)-category to itself. Change-of-base turns \\(\mathcal{V}\\)-categories into \\(\mathcal{W}\\)-categories, so it's a functor \\(\Phi: \mathcal{V}\mathrm{Cat}\to \mathcal{W}\mathrm{Cat} \\). If \\(\mathcal{V} = \mathcal{W}\\) this functor may _map_ a \\(\mathcal{V}\\)-category \\(\mathcal{C}\\) to itself: \\(\Phi(\mathcal{C}) = \mathcal{C}\\). But it's still a functor from \\(\mathcal{V}\mathrm{Cat}\\) to \\(\mathcal{V}\mathrm{Cat}\\), not an enriched functor from \\(\mathcal{C}\\) to \\(\mathcal{C}\\).

This is why change-of-base and enriched functors are both crucial, indispensable concepts.

I've been through this confusion myself a couple of times! It only takes 8 hours of agony to figure out what's going on.

Let me review the whole story:

First I told you about enriched functors between \\(\mathcal{V}\\)-categories. They're called \\(\mathcal{V}\\)-functors.

Then I told you about enriched functors from a \\(\mathcal{V}\\)-category to a \\(\mathcal{W}\\)-category. But these only makes sense when you first have a trick for turning \\(\mathcal{V}\\)-categories into \\(\mathcal{W}\\)-categories. This trick is "change of base".

Suppose \\(\mathcal{V}\\) and \\(\mathcal{W}\\) are monoidal posets and \\(f: \mathcal{V} \to \mathcal{W}\\) is a monoidal monotone. Suppose \\(\mathcal{X}\\) is a \\(\mathcal{V}\\)-category. Then there's a \\(\mathcal{W}\\)-category \\(\Phi(\mathcal{X})\\) with the same objects as \\(\mathcal{X}\\) and

\[ \Phi(\mathcal{X})(x,x') = f(\mathcal{X}(x,x')). \]

This trick, \\(\Phi\\), is called **change of base**.

Once you've got change of base, you can get an enriched functor from a \\(\mathcal{V}\\)-category \\(\mathcal{X}\\) to a \\(\mathcal{W}\\)-category \\(\mathcal{Y}\\). First you use change of base to turn \\(\mathcal{X}\\) into a \\(\mathcal{W}\\)-category \\(\Phi(\mathcal{X})\\). Then you pick a \\(\mathcal{W}\\)-functor

\[ F : \Phi(\mathcal{X}) \to \mathcal{Y} .\]

In short: _first you move \\(\mathcal{X}\\) to the world where \\(\mathcal{Y}\\) lives, and then it can talk to \\(\mathcal{Y}\\)_.

> Can't we reduce \\(\mathcal{V}\\)-functors to changes-of-base, mainly a change-of-base between a \\(\mathcal{V}\\)-category and itself?

No, you're making a classic level slip here. There's _no such thing_ as change-of-base between a \\(\mathcal{V}\\)-category and itself! Change-of-base is not a kind of map between \\(\mathcal{V}\\)-categories.

It's great that you made this mistake, because everybody does it. Please think about this! It's easy to get confused... but it's very important to get unconfused.

It's like "a function from an element of a set to itself."

There's _no such thing_ as a function from an element of a set to itself. A function \\(f: X \to Y\\) goes between _sets_ \\(X\\) and \\(Y\\). If \\(X = Y\\) this function may _map_ an element \\(x \in X\\) to itself: \\(f(x) = x\\). But it's still a function from \\(X\\) to \\(X\\), not a function from \\(x\\) to \\(x\\).

Similarly, there's _no such thing_ as a change-of-base from a \\(\mathcal{V}\\)-category to itself. Change-of-base turns \\(\mathcal{V}\\)-categories into \\(\mathcal{W}\\)-categories, so it's a functor \\(\Phi: \mathcal{V}\mathrm{Cat}\to \mathcal{W}\mathrm{Cat} \\). If \\(\mathcal{V} = \mathcal{W}\\) this functor may _map_ a \\(\mathcal{V}\\)-category \\(\mathcal{C}\\) to itself: \\(\Phi(\mathcal{C}) = \mathcal{C}\\). But it's still a functor from \\(\mathcal{V}\mathrm{Cat}\\) to \\(\mathcal{V}\mathrm{Cat}\\), not an enriched functor from \\(\mathcal{C}\\) to \\(\mathcal{C}\\).

This is why change-of-base and enriched functors are both crucial, indispensable concepts.

I've been through this confusion myself a couple of times! It only takes 8 hours of agony to figure out what's going on.

Let me review the whole story:

First I told you about enriched functors between \\(\mathcal{V}\\)-categories. They're called \\(\mathcal{V}\\)-functors.

Then I told you about enriched functors from a \\(\mathcal{V}\\)-category to a \\(\mathcal{W}\\)-category. But these only makes sense when you first have a trick for turning \\(\mathcal{V}\\)-categories into \\(\mathcal{W}\\)-categories. This trick is "change of base".

Suppose \\(\mathcal{V}\\) and \\(\mathcal{W}\\) are monoidal posets and \\(f: \mathcal{V} \to \mathcal{W}\\) is a monoidal monotone. Suppose \\(\mathcal{X}\\) is a \\(\mathcal{V}\\)-category. Then there's a \\(\mathcal{W}\\)-category \\(\Phi(\mathcal{X})\\) with the same objects as \\(\mathcal{X}\\) and

\[ \Phi(\mathcal{X})(x,x') = f(\mathcal{X}(x,x')). \]

This trick, \\(\Phi\\), is called **change of base**.

Once you've got change of base, you can get an enriched functor from a \\(\mathcal{V}\\)-category \\(\mathcal{X}\\) to a \\(\mathcal{W}\\)-category \\(\mathcal{Y}\\). First you use change of base to turn \\(\mathcal{X}\\) into a \\(\mathcal{W}\\)-category \\(\Phi(\mathcal{X})\\). Then you pick a \\(\mathcal{W}\\)-functor

\[ F : \Phi(\mathcal{X}) \to \mathcal{Y} .\]

In short: _first you move \\(\mathcal{X}\\) to the world where \\(\mathcal{Y}\\) lives, and then it can talk to \\(\mathcal{Y}\\)_.