ENH: ``convert_dtypes`` should get an option to convert Categoricals as well

[phofl]

Feature Type

• Adding new functionality to pandas

• Changing existing functionality in pandas

• Removing existing functionality in pandas

Problem Description

Right now this is a no-op, but we can support EA dtypes in Indexes, which also makes them available for categoricals. What should we do for pyarrow dtype_backend? Cast to arrow dictionary type or to a Categorical backed by Arrow Indexes?

cc @mroeschke

Feature Description

Should probably add a new keyword

Alternative Solutions

Writing this yourself

Additional Context

No response

[jbrockmendel]

is the idea to convert Categorical to a dict-encoded arrow? or to convert cat.categories to an arrow dtype?

[phofl]

Not sure, ArrowDtype uses a dict-encoded arrow array, should make a decision what we want to do here.

[jbrockmendel]

did you have a particular use case in mind? or is the motivation just "the user asked for arrow, we should give them arrow"?