How to serialize (JSON) a type reference?

(I know it's not a cattrs issue.)
I'm looking for a way to JSON-serialize a type-ref inside a attrs-class.
But type-refs aren't supported by json-lib.
See a litte example:

import attrs, cattrs, json

@attrs.mutable
class A:
    my_type: type

@attrs.mutable
class B:
    foo: int

a1=A(my_type=B)

a1u=cattrs.unstructure(a1)

a1uj=json.dumps(a1u)
Traceback (most recent call last):
  File "C:\Python310\lib\code.py", line 90, in runcode
    exec(code, self.locals)
  File "<input>", line 1, in <module>
  File "C:\Python310\lib\json\__init__.py", line 231, in dumps
    return _default_encoder.encode(obj)
  File "C:\Python310\lib\json\encoder.py", line 199, in encode
    chunks = self.iterencode(o, _one_shot=True)
  File "C:\Python310\lib\json\encoder.py", line 257, in iterencode
    return _iterencode(o, 0)
  File "C:\Python310\lib\json\encoder.py", line 179, in default
    raise TypeError(f'Object of type {o.__class__.__name__} '
TypeError: Object of type type is not JSON serializable

Are there any ideas on how to achieve my goal?

Start with the example's cattrs.unstructure(a1) output and Python's json.dumps handling of type objects. Determine whether the requested behavior belongs in cattrs or in caller-side conversion; done should be a clear, documented and tested path for the desired serialization.