Bug in concat casting all na levels to float

doc/source/whatsnew/v1.4.0.rst

@@ -806,6 +806,7 @@ Reshaping
 - Bug in :func:`crosstab` would fail when inputs are lists or tuples (:issue:`44076`)
 - Bug in :meth:`DataFrame.append` failing to retain ``index.name`` when appending a list of :class:`Series` objects (:issue:`44109`)
 - Fixed metadata propagation in :meth:`Dataframe.apply` method, consequently fixing the same issue for :meth:`Dataframe.transform`, :meth:`Dataframe.nunique` and :meth:`Dataframe.mode` (:issue:`28283`)
+- Bug in :func:`concat` casting levels of :class:`MultiIndex` to float if the only consist of missing values (:issue:`44900`)
 - Bug in :meth:`DataFrame.stack` with ``ExtensionDtype`` columns incorrectly raising (:issue:`43561`)
 - Bug in :meth:`Series.unstack` with object doing unwanted type inference on resulting columns (:issue:`44595`)
 - Bug in :class:`MultiIndex` failing join operations with overlapping ``IntervalIndex`` levels (:issue:`44096`)

pandas/core/arrays/categorical.py

@@ -422,7 +422,14 @@ def __init__(
                     # We remove null values here, then below will re-insert
                     #  them, grep "full_codes"
                     arr_list = [values[idx] for idx in np.where(~null_mask)[0]]
-                    arr = sanitize_array(arr_list, None)
+
+                    # GH#44900 Do not cast to float if we have only missing values
+                    if arr_list or arr.dtype == "object":
+                        sanitize_dtype = None
+                    else:
+                        sanitize_dtype = arr.dtype
+
+                    arr = sanitize_array(arr_list, None, dtype=sanitize_dtype)
                 values = arr
 
         if dtype.categories is None:

pandas/tests/arrays/categorical/test_missing.py

@@ -197,3 +197,17 @@ def test_compare_categorical_with_missing(self, a1, a2, categories):
         result = Series(a1, dtype=cat_type) == Series(a2, dtype=cat_type)
         expected = Series(a1) == Series(a2)
         tm.assert_series_equal(result, expected)
+
+    @pytest.mark.parametrize(
+        "na_value, dtype",
+        [
+            (pd.NaT, "datetime64[ns]"),
+            (None, "float64"),
+            (np.nan, "float64"),
+            (pd.NA, "float64"),
+        ],
+    )
+    def test_categorical_only_missing_values_no_cast(self, na_value, dtype):
+        # GH#44900
+        result = Categorical([na_value, na_value])
+        tm.assert_index_equal(result.categories, Index([], dtype=dtype))

pandas/tests/reshape/concat/test_datetimes.py

@@ -528,3 +528,16 @@ def test_concat_timedelta64_block():
     result = concat([df, df])
     tm.assert_frame_equal(result.iloc[:10], df)
     tm.assert_frame_equal(result.iloc[10:], df)
+
+
+def test_concat_multiindex_datetime_nat():
+    # GH#44900
+    left = DataFrame({"a": 1}, index=MultiIndex.from_tuples([(1, pd.NaT)]))
+    right = DataFrame(
+        {"b": 2}, index=MultiIndex.from_tuples([(1, pd.NaT), (2, pd.NaT)])
+    )
+    result = concat([left, right], axis="columns")
+    expected = DataFrame(
+        {"a": [1.0, np.nan], "b": 2}, MultiIndex.from_tuples([(1, pd.NaT), (2, pd.NaT)])
+    )
+    tm.assert_frame_equal(result, expected)