From 2dd480145d61bd1ab583a5cc8c76636ada866a0f Mon Sep 17 00:00:00 2001
From: Patrick Hoefler <61934744+phofl@users.noreply.github.com>
Date: Tue, 3 Oct 2023 22:01:34 +0200
Subject: [PATCH] Backport PR #55364: BUG: eq not implemented for categorical
 and arrow backed strings

---
 doc/source/whatsnew/v2.1.2.rst                  | 1 +
 pandas/core/arrays/arrow/array.py               | 5 ++++-
 pandas/tests/indexes/categorical/test_equals.py | 6 ++++++
 3 files changed, 11 insertions(+), 1 deletion(-)

diff --git a/doc/source/whatsnew/v2.1.2.rst b/doc/source/whatsnew/v2.1.2.rst
index 090df1489e493..b6cacecfdc5f8 100644
--- a/doc/source/whatsnew/v2.1.2.rst
+++ b/doc/source/whatsnew/v2.1.2.rst
@@ -21,6 +21,7 @@ Fixed regressions
 
 Bug fixes
 ~~~~~~~~~
+- Fixed bug in :meth:`Categorical.equals` if other has arrow backed string dtype (:issue:`55364`)
 - Fixed bug in :meth:`DataFrame.idxmin` and :meth:`DataFrame.idxmax` raising for arrow dtypes (:issue:`55368`)
 -
 
diff --git a/pandas/core/arrays/arrow/array.py b/pandas/core/arrays/arrow/array.py
index 190fd8fd54e02..2e3ad8bc13091 100644
--- a/pandas/core/arrays/arrow/array.py
+++ b/pandas/core/arrays/arrow/array.py
@@ -32,6 +32,7 @@
 
 from pandas.core.dtypes.cast import infer_dtype_from_scalar
 from pandas.core.dtypes.common import (
+    CategoricalDtype,
     is_array_like,
     is_bool_dtype,
     is_integer,
@@ -628,7 +629,9 @@ def __setstate__(self, state) -> None:
 
     def _cmp_method(self, other, op):
         pc_func = ARROW_CMP_FUNCS[op.__name__]
-        if isinstance(other, (ArrowExtensionArray, np.ndarray, list, BaseMaskedArray)):
+        if isinstance(
+            other, (ArrowExtensionArray, np.ndarray, list, BaseMaskedArray)
+        ) or isinstance(getattr(other, "dtype", None), CategoricalDtype):
             result = pc_func(self._pa_array, self._box_pa(other))
         elif is_scalar(other):
             try:
diff --git a/pandas/tests/indexes/categorical/test_equals.py b/pandas/tests/indexes/categorical/test_equals.py
index 1ed8f3a903439..a8353f301a3c3 100644
--- a/pandas/tests/indexes/categorical/test_equals.py
+++ b/pandas/tests/indexes/categorical/test_equals.py
@@ -88,3 +88,9 @@ def test_equals_multiindex(self):
         ci = mi.to_flat_index().astype("category")
 
         assert not ci.equals(mi)
+
+    def test_equals_string_dtype(self, any_string_dtype):
+        # GH#55364
+        idx = CategoricalIndex(list("abc"), name="B")
+        other = Index(["a", "b", "c"], name="B", dtype=any_string_dtype)
+        assert idx.equals(other)