Krdytkyu

I've heard the words "internal" and "external" generalization of concepts in category theory.

Specifically, i heard the idea that the concept of 'power set' has an internal and an external generalization in category theory.

What is the difference between these two?

First of all one need to understand the concept of internalization.

Generally many classical constructions which can be given inside some specific category (usually $mathbf{Set}$) can be expressed in the language of category theory in terms of objects, arrows and more generally diagrams.

Once one has a such diagrammatic definition of the construction it is possible to use the same definition to other categories, providing a new version of the construction internal to the new category.

So internalization is about defining concepts in terms of diagrams in a (possibly structured) category, in such a way that once one interprets these concepts in some specific categories (usually $mathbf{Set}$) they get the classical notions that have been internalized.

As an example you can consider an internal monoid in a monidal category, which is a diagram made of morphisms of the form $X otimes X to X$ and $I to X$ that make commute certain diagrams.

Externalization is about turing the internalized data in $mathbf{Set}$-theoretic data.
More technically externalization is the process of mapping the internal data via the yoneda embedding.

So the externalization of an internal data (which amounts to a diagram satisfying certain properties) in a category $mathbf C$ is basically the corresponding diagram internal to $[mathbf C^text{op},mathbf{Set}]$.

Continuing with the example of a monoidal category $mathbf C$, the externalization turns the data of an internal monoid $(X,X otimes X to X,I to X)$ are in a monoid object $$(hom(-,X),hom(-,X)timeshom(-,X) to hom(-,X),hom(-,I) to hom(-,X))$$ in $[mathbf C^text{op},mathbf{Set}]$.

So far it should be clear why internalization is basically a generalization of classical notions: because classical notion are special version (i.e. usually internal to $mathbf{Set}$) of the internal concept.

Externalization provides a different way to generalize, or if you like internalize, concepts.
Neverless this would be difficult to explain in the general case, so I prefer to stop here.

Anyway if you feel the need for additional details feel free to ask.

I hope this helps.

Categorial internalisation is about taking a statement
that involves “points”, which is the usual Set theoretic rendition,
and turning it “point free” so that it is soley rendered in the
language of category theory.

For example, an adjoint between preorders is a pair $f, g$ such that
$$∀ x, y • quad f,x ≤ y ;;≡;; x ≤′ g, y$$
Notice the “points” $x$ and $y$ from each preorder being utilised.
However, if we move from the category Set to the category Rel, for example,
to consider relations. Then a preorder is reflexive and transitive relation;
let us use $E$ in-place of $≤$. Then the above can be rephrased with no points
$$ f˘;E ;=; E′;g˘$$
Where $-;-$ is relational composition and $-˘$ is relational converse.

This is another form of internalisation; it is about rephrasing statements that use,
e.g., logical connectives $forall, Rightarrow$, into forms that do not use them.
For example, see this presentation of Cartesian Closed Categories in the case of preorders
where properties are shown using, e.g., ∀, then later obtained without it; e.g., having
internal homs $[X, Y]$ the fact
$$text{there is a unique map from $X$ to the initial object 𝑰}$$
Can be internalised, i.e., rendered without using the logical notion of existence as
$$ [X, 𝑰] ;≅; 𝑰$$