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BUG: categorical dtype equality for level in different type #55468

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@kwhkim

Description

@kwhkim

Pandas version checks

  • I have checked that this issue has not already been reported.

  • I have confirmed this bug exists on the latest version of pandas.

  • I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd

pd.CategoricalDtype([1,2,'a', 'b'], ordered=False) == pd.CategoricalDtype(['a', 'b', 1,2], ordered=False)

Issue Description

According to Equality semantics of CategoricalDtype,
the code above should result in True but it came out False

Expected Behavior

CategoricalDtype([1,2,'a', 'b'], ordered=False) == \
  CategoricalDtype(['a', 'b', 1,2], ordered=False)

True

Installed Versions

/usr/local/lib/python3.10/dist-packages/_distutils_hack/init.py:33: UserWarning: Setuptools is replacing distutils.
warnings.warn("Setuptools is replacing distutils.")

INSTALLED VERSIONS

commit : e86ed37
python : 3.10.12.final.0
python-bits : 64
OS : Linux
OS-release : 5.15.120+
Version : #1 SMP Wed Aug 30 11:19:59 UTC 2023
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : en_US.UTF-8
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 2.1.1
numpy : 1.23.5
pytz : 2023.3.post1
dateutil : 2.8.2
setuptools : 67.7.2
pip : 23.1.2
Cython : 3.0.3
pytest : 7.4.2
hypothesis : None
sphinx : 5.0.2
blosc : None
feather : None
xlsxwriter : None
lxml.etree : 4.9.3
html5lib : 1.1
pymysql : None
psycopg2 : 2.9.9
jinja2 : 3.1.2
IPython : 7.34.0
pandas_datareader : 0.10.0
bs4 : 4.11.2
bottleneck : None
dataframe-api-compat: None
fastparquet : None
fsspec : 2023.6.0
gcsfs : 2023.6.0
matplotlib : 3.7.1
numba : 0.56.4
numexpr : 2.8.7
odfpy : None
openpyxl : 3.1.2
pandas_gbq : 0.17.9
pyarrow : 9.0.0
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : 1.11.3
sqlalchemy : 2.0.21
tables : 3.8.0
tabulate : 0.9.0
xarray : 2023.7.0
xlrd : 2.0.1
zstandard : None
tzdata : 2023.3
qtpy : None
pyqt5 : None

PS) I'd like to be assigned for this task

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