From 751a131ea8d855d6ff74a6555cf9cc974c7a26cf Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Tue, 31 Oct 2023 16:25:07 +0000
Subject: [PATCH 01/26] note what may raise

---
 spec/design_topics/python_builtin_types.md | 34 ++++++++++++++++++++--
 1 file changed, 32 insertions(+), 2 deletions(-)

diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index c85812eb..5144589c 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -30,8 +30,8 @@ builtin types to CPU. In the above example, the `.mean()` call returns a
 `float`. It is likely beneficial though to implement this as a library-specific
 scalar object which duck types with `float`. This means that it should (a) have
 the same semantics as a builtin `float` when used within a library, and (b)
-support usage as a `float` outside of the library (i.e., implement
-`__float__`). Duck typing is usually not perfect, for example `isinstance`
+support usage as a `float` outside of the library (see below).
+Duck typing is usually not perfect, for example `isinstance`
 usage on the float-like duck type will behave differently. Such explicit "type
 of object" checks don't have to be supported.
 
@@ -39,3 +39,33 @@ The following design rule applies everywhere builtin Python types are used
 within this API standard: _where a Python builtin type is specified, an
 implementation may always replace it by an equivalent library-specific type
 that duck types with the Python builtin type._
+
+## Required methods
+
+A ducktyped float scalar is required to implement all the methods which `float` implements,
+but note that those which require returning a Python scalar may raise
+(depending on the implementation).
+
+For example, if a library implements `FancyFloat` and `FancyBool` scalars,
+then the following should all be supported:
+```python
+df: DataFrame
+column_1: Column = df.col('a')
+column_2: Column = df.col('b')
+
+scalar: FancyFloat = column_1.std()
+result_1: Column = column_2 - column_1.std()
+result_2: FancyBool = column_2.std() > column_1.std()
+```
+The following, however, may raise, dependening on the
+implementation:
+```python
+df: DataFrame
+column = df.col('a')
+
+if column.std() > 0:  # this line may raise!
+    print('std is positive')
+```
+This is because `if column.std() > 0` will call `(column.std() > 0).__bool__()`,
+which must produce a Python scalar. Therefore, a purely lazy dataframe library
+may choose to raise here.

From fc65648b0e9f41ffdd1f0b4f47b6ff632f6c1f4f Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 8 Nov 2023 12:27:14 +0000
Subject: [PATCH 02/26] list required methods

---
 spec/design_topics/python_builtin_types.md | 36 +++++++++++++++++++---
 1 file changed, 32 insertions(+), 4 deletions(-)

diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 5144589c..430d3696 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -42,9 +42,36 @@ that duck types with the Python builtin type._
 
 ## Required methods
 
-A ducktyped float scalar is required to implement all the methods which `float` implements,
-but note that those which require returning a Python scalar may raise
-(depending on the implementation).
+If a library doesn't use the Python built-in scalars, then its scalars must implement
+at least the following operations which return scalars:
+- `__lt__`
+- `__le__`
+- `__eq__`
+- `__ne__`
+- `__gt__`
+- `__ge__`
+- `__add__`
+- `__radd__`
+- `__sub__`
+- `__rsub__`
+- `__mul__`
+- `__rmul__`
+- `__mod__`
+- `__rmod__`
+- `__pow__`
+- `__rpow__`
+- `__floordiv__`
+- `__rfloordiv__`
+- `__truediv__`
+- `__rtruediv__`
+- `__neg__`
+- `__abs__`
+
+Furthermore, unless the library exclusively allows for lazy execution,
+it must also implement the following unary operations which return Python scalars:
+- `__int__`
+- `__float__`
+- `__bool__`
 
 For example, if a library implements `FancyFloat` and `FancyBool` scalars,
 then the following should all be supported:
@@ -68,4 +95,5 @@ if column.std() > 0:  # this line may raise!
 ```
 This is because `if column.std() > 0` will call `(column.std() > 0).__bool__()`,
 which must produce a Python scalar. Therefore, a purely lazy dataframe library
-may choose to raise here.
+may choose to raise here, whereas as one which allows for eager execution may return
+a Python bool.

From 7c24afdbaf542520c19b8bcb61878ec9562c5725 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 8 Nov 2023 16:38:18 +0000
Subject: [PATCH 03/26] add scalar class

---
 .../dataframe_api/__init__.py                 |  4 +-
 .../dataframe_api/column_object.py            |  6 +-
 .../dataframe_api/dataframe_object.py         |  9 +-
 .../dataframe_api/scalar_object.py            | 88 +++++++++++++++++++
 .../API_specification/dataframe_api/typing.py | 37 +++-----
 5 files changed, 113 insertions(+), 31 deletions(-)
 create mode 100644 spec/API_specification/dataframe_api/scalar_object.py

diff --git a/spec/API_specification/dataframe_api/__init__.py b/spec/API_specification/dataframe_api/__init__.py
index 3a20f54b..ebad74c6 100644
--- a/spec/API_specification/dataframe_api/__init__.py
+++ b/spec/API_specification/dataframe_api/__init__.py
@@ -4,12 +4,12 @@
 """
 from __future__ import annotations
 
-from typing import Dict, Sequence, Any, TYPE_CHECKING
+from typing import TYPE_CHECKING, Any, Dict, Sequence
 
 from .column_object import *
 from .dataframe_object import DataFrame
-from .groupby_object import *
 from .dtypes import *
+from .groupby_object import *
 
 if TYPE_CHECKING:
     from .typing import DType, Scalar
diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 915f016c..1a2376aa 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -1,11 +1,13 @@
 from __future__ import annotations
 
-from typing import Any,NoReturn, TYPE_CHECKING, Literal, Protocol
+from typing import TYPE_CHECKING, Any, Literal, NoReturn, Protocol
 
 if TYPE_CHECKING:
-    from .typing import NullType, Scalar, DType, Namespace
     from typing_extensions import Self
 
+    from .scalar_object import Scalar
+    from .typing import DType, Namespace, NullType
+
 
 __all__ = ['Column']
 
diff --git a/spec/API_specification/dataframe_api/dataframe_object.py b/spec/API_specification/dataframe_api/dataframe_object.py
index f5c17d80..3691745b 100644
--- a/spec/API_specification/dataframe_api/dataframe_object.py
+++ b/spec/API_specification/dataframe_api/dataframe_object.py
@@ -1,13 +1,14 @@
 from __future__ import annotations
 
-from typing import Any, Literal, Mapping, Sequence, TYPE_CHECKING, NoReturn, Protocol
-
+from typing import TYPE_CHECKING, Any, Literal, Mapping, NoReturn, Protocol, Sequence
 
 if TYPE_CHECKING:
+    from typing_extensions import Self
+
     from .column_object import Column
     from .groupby_object import GroupBy
-    from .typing import NullType, Scalar, Namespace, DType, SupportsDataFrameAPI
-    from typing_extensions import Self
+    from .scalar_object import Scalar
+    from .typing import DType, Namespace, NullType, SupportsDataFrameAPI
 
 
 __all__ = ["DataFrame"]
diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
new file mode 100644
index 00000000..53ff5d00
--- /dev/null
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -0,0 +1,88 @@
+from __future__ import annotations
+
+from typing import Any, Protocol
+
+__all__ = ['Scalar']
+
+
+class Scalar(Protocol):
+    """
+    Scalar object
+
+    Not meant to be instantiated directly, but rather created via
+    `:meth:Column.get_value` or one of the column reductions such
+    as `:meth:`Column.sum`.
+    """
+    def __lt__(self, other: Any) -> Scalar:
+        ...
+
+    def __le__(self, other: Any) -> Scalar:
+        ...
+
+    def __eq__(self, other: Any) -> Scalar:  # type: ignore[override]
+        ...
+
+    def __ne__(self, other: Any) -> Scalar:  # type: ignore[override]
+        ...
+
+    def __gt__(self, other: Any) -> Scalar:
+        ...
+
+    def __ge__(self, other: Any) -> Scalar:
+        ...
+
+    def __add__(self, other: Any) -> Scalar:
+        ...
+
+    def __radd__(self, other: Any) -> Scalar:
+        ...
+
+    def __sub__(self, other: Any) -> Scalar:
+        ...
+
+    def __rsub__(self, other: Any) -> Scalar:
+        ...
+
+    def __mul__(self, other: Any) -> Scalar:
+        ...
+
+    def __rmul__(self, other: Any) -> Scalar:
+        ...
+
+    def __mod__(self, other: Any) -> Scalar:
+        ...
+
+    def __rmod__(self, other: Any) -> Scalar:
+        ...
+
+    def __pow__(self, other: Any) -> Scalar:
+        ...
+
+    def __rpow__(self, other: Any) -> Scalar:
+        ...
+
+    def __floordiv__(self, other: Any) -> Scalar:
+        ...
+
+    def __rfloordiv__(self, other: Any) -> Scalar:
+        ...
+
+    def __truediv__(self, other: Any) -> Scalar:
+        ...
+
+    def __rtruediv__(self, other: Any) -> Scalar:
+        ...
+
+    def __neg__(self) -> Scalar:
+        ...
+
+    def __abs__(self) -> Scalar:
+        ...
+
+    def __bool__(self) -> bool:
+        """
+        Note that this return a Python scalar.
+
+        Depending on the implementation, this may raise or trigger computation.
+        """
+        ...
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index 901c0f50..0c1390d0 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -3,38 +3,32 @@
 """
 from __future__ import annotations
 
-from typing import (
-    TYPE_CHECKING,
-    Any,
-    Literal,
-    Dict,
-    Protocol,
-    Sequence,
-    Union,
-)
+from typing import TYPE_CHECKING, Any, Dict, Literal, Protocol, Sequence, Union
 
 from dataframe_api.column_object import Column
 from dataframe_api.dataframe_object import DataFrame
-from dataframe_api.groupby_object import GroupBy, Aggregation as AggregationT
+from dataframe_api.groupby_object import Aggregation as AggregationT
+from dataframe_api.groupby_object import GroupBy
 
 if TYPE_CHECKING:
     from .dtypes import (
         Bool,
-        Float64,
-        Float32,
-        Int64,
-        Int32,
-        Int16,
-        Int8,
-        UInt64,
-        UInt32,
-        UInt16,
-        UInt8,
         Date,
         Datetime,
         Duration,
+        Float32,
+        Float64,
+        Int8,
+        Int16,
+        Int32,
+        Int64,
         String,
+        UInt8,
+        UInt16,
+        UInt32,
+        UInt64,
     )
+    from .scalar_object import Scalar
 
     DType = Union[
         Bool,
@@ -54,9 +48,6 @@
         Duration,
     ]
 
-# Type alias: Mypy needs Any, but for readability we need to make clear this
-# is a Python scalar (i.e., an instance of `bool`, `int`, `float`, `str`, etc.)
-Scalar = Any
 # null is a special object which represents a missing value.
 # It is not valid as a type.
 NullType = Any

From 2714c13f51220c6a185356cdd2fdd44e61726f27 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 8 Nov 2023 16:52:16 +0000
Subject: [PATCH 04/26] reword

---
 spec/design_topics/python_builtin_types.md | 60 +++++-----------------
 1 file changed, 12 insertions(+), 48 deletions(-)

diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 430d3696..0146e144 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -18,7 +18,7 @@ class DataFrame:
         ...
 
 class Column:
-    def mean(self, skip_nulls: bool = True) -> float | NullType:
+    def mean(self, skip_nulls: bool = True) -> Scalar | NullType:
         ...
 
 larger = df2 > df1.col('foo').mean()
@@ -27,51 +27,11 @@ larger = df2 > df1.col('foo').mean()
 For a GPU dataframe library, it is desirable for all data to reside on the GPU,
 and not incur a performance penalty from synchronizing instances of Python
 builtin types to CPU. In the above example, the `.mean()` call returns a
-`float`. It is likely beneficial though to implement this as a library-specific
-scalar object which duck types with `float`. This means that it should (a) have
-the same semantics as a builtin `float` when used within a library, and (b)
-support usage as a `float` outside of the library (see below).
-Duck typing is usually not perfect, for example `isinstance`
-usage on the float-like duck type will behave differently. Such explicit "type
-of object" checks don't have to be supported.
+`Scalar`. It is likely beneficial though to implement this as a library-specific
+scalar object which (partially) duck types with `float`. The required methods it
+must implement are listed in `:class:Scalar`.
 
-The following design rule applies everywhere builtin Python types are used
-within this API standard: _where a Python builtin type is specified, an
-implementation may always replace it by an equivalent library-specific type
-that duck types with the Python builtin type._
-
-## Required methods
-
-If a library doesn't use the Python built-in scalars, then its scalars must implement
-at least the following operations which return scalars:
-- `__lt__`
-- `__le__`
-- `__eq__`
-- `__ne__`
-- `__gt__`
-- `__ge__`
-- `__add__`
-- `__radd__`
-- `__sub__`
-- `__rsub__`
-- `__mul__`
-- `__rmul__`
-- `__mod__`
-- `__rmod__`
-- `__pow__`
-- `__rpow__`
-- `__floordiv__`
-- `__rfloordiv__`
-- `__truediv__`
-- `__rtruediv__`
-- `__neg__`
-- `__abs__`
-
-Furthermore, unless the library exclusively allows for lazy execution,
-it must also implement the following unary operations which return Python scalars:
-- `__int__`
-- `__float__`
-- `__bool__`
+### Example
 
 For example, if a library implements `FancyFloat` and `FancyBool` scalars,
 then the following should all be supported:
@@ -84,6 +44,10 @@ scalar: FancyFloat = column_1.std()
 result_1: Column = column_2 - column_1.std()
 result_2: FancyBool = column_2.std() > column_1.std()
 ```
+
+Note that the scalars above are library-specific ones - they may be used to keep
+data on GPU, or to keep data lazy.
+
 The following, however, may raise, dependening on the
 implementation:
 ```python
@@ -94,6 +58,6 @@ if column.std() > 0:  # this line may raise!
     print('std is positive')
 ```
 This is because `if column.std() > 0` will call `(column.std() > 0).__bool__()`,
-which must produce a Python scalar. Therefore, a purely lazy dataframe library
-may choose to raise here, whereas as one which allows for eager execution may return
-a Python bool.
+which is required by Python to produce a Python scalar.
+Therefore, a purely lazy dataframe library may choose to raise here, whereas as
+one which allows for eager execution may return a Python bool.

From 99b91a55ca3f1893a30533ed8944e83b46ace379 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 8 Nov 2023 16:56:10 +0000
Subject: [PATCH 05/26] fixup

---
 spec/design_topics/python_builtin_types.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 0146e144..2339451a 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -31,7 +31,7 @@ builtin types to CPU. In the above example, the `.mean()` call returns a
 scalar object which (partially) duck types with `float`. The required methods it
 must implement are listed in `:class:Scalar`.
 
-### Example
+## Example
 
 For example, if a library implements `FancyFloat` and `FancyBool` scalars,
 then the following should all be supported:

From a867f00f7eaaaadcf2424db9f7682d997142ec4d Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 8 Nov 2023 16:58:19 +0000
Subject: [PATCH 06/26] fixup

---
 spec/design_topics/python_builtin_types.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 2339451a..0b2ed617 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -29,7 +29,7 @@ and not incur a performance penalty from synchronizing instances of Python
 builtin types to CPU. In the above example, the `.mean()` call returns a
 `Scalar`. It is likely beneficial though to implement this as a library-specific
 scalar object which (partially) duck types with `float`. The required methods it
-must implement are listed in `:class:Scalar`.
+must implement are listed in the spec for class `Scalar`.
 
 ## Example
 

From f197672db4abb9c1a55976f4fd5f670e18b2eb7a Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 8 Nov 2023 18:17:19 +0000
Subject: [PATCH 07/26] fixup

---
 spec/API_specification/dataframe_api/column_object.py | 2 --
 spec/API_specification/dataframe_api/scalar_object.py | 2 +-
 spec/API_specification/dataframe_api/typing.py        | 4 ++--
 3 files changed, 3 insertions(+), 5 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index ca00dadc..e02ccc02 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -8,8 +8,6 @@
     from .scalar_object import Scalar
     from .typing import DType, Namespace, NullType
 
-    from .typing import DType, Namespace, NullType, Scalar
-
 
 __all__ = ["Column"]
 
diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
index 0f5f9fb5..a15db639 100644
--- a/spec/API_specification/dataframe_api/scalar_object.py
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -6,7 +6,7 @@
 
 
 class Scalar(Protocol):
-    """Scalar object
+    """Scalar object.
 
     Not meant to be instantiated directly, but rather created via
     `:meth:Column.get_value` or one of the column reductions such
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index 13ac5181..e7ee8e56 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -14,6 +14,8 @@
 from dataframe_api.groupby_object import Aggregation as AggregationT
 from dataframe_api.groupby_object import GroupBy
 
+from .scalar_object import Scalar
+
 if TYPE_CHECKING:
     from collections.abc import Sequence
 
@@ -34,7 +36,6 @@
         UInt32,
         UInt64,
     )
-    from .scalar_object import Scalar
 
     DType = Union[
         Bool,
@@ -181,5 +182,4 @@ def __column_consortium_standard__(
     "Scalar",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
-    "Scalar",
 ]

From 409d8f3cf401479a28234bc0abfbdcb1a1e0245b Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Tue, 14 Nov 2023 19:09:43 +0000
Subject: [PATCH 08/26] replace Scalar|NullType with Scalar

---
 .../dataframe_api/column_object.py             | 18 +++++++++---------
 spec/design_topics/python_builtin_types.md     |  2 +-
 2 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index e02ccc02..33f6a560 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -518,7 +518,7 @@ def all(self, *, skip_nulls: bool = True) -> bool | NullType:
         """
         ...
 
-    def min(self, *, skip_nulls: bool = True) -> Scalar | NullType:
+    def min(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a scalar.
 
         Any data type that supports comparisons
@@ -526,7 +526,7 @@ def min(self, *, skip_nulls: bool = True) -> Scalar | NullType:
         """
         ...
 
-    def max(self, *, skip_nulls: bool = True) -> Scalar | NullType:
+    def max(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a scalar.
 
         Any data type that supports comparisons
@@ -534,7 +534,7 @@ def max(self, *, skip_nulls: bool = True) -> Scalar | NullType:
         """
         ...
 
-    def sum(self, *, skip_nulls: bool = True) -> Scalar | NullType:
+    def sum(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -543,7 +543,7 @@ def sum(self, *, skip_nulls: bool = True) -> Scalar | NullType:
         """
         ...
 
-    def prod(self, *, skip_nulls: bool = True) -> Scalar | NullType:
+    def prod(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical data types.
@@ -551,7 +551,7 @@ def prod(self, *, skip_nulls: bool = True) -> Scalar | NullType:
         """
         ...
 
-    def median(self, *, skip_nulls: bool = True) -> Scalar | NullType:
+    def median(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -561,7 +561,7 @@ def median(self, *, skip_nulls: bool = True) -> Scalar | NullType:
         """
         ...
 
-    def mean(self, *, skip_nulls: bool = True) -> Scalar | NullType:
+    def mean(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -576,7 +576,7 @@ def std(
         *,
         correction: int | float = 1,
         skip_nulls: bool = True,
-    ) -> Scalar | NullType:
+    ) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -609,7 +609,7 @@ def var(
         *,
         correction: int | float = 1,
         skip_nulls: bool = True,
-    ) -> Scalar | NullType:
+    ) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -738,7 +738,7 @@ def unique_indices(self, *, skip_nulls: bool = True) -> Self:
         """
         ...
 
-    def fill_nan(self, value: float | NullType, /) -> Self:
+    def fill_nan(self, value: Scalar, /) -> Self:
         """Fill floating point ``nan`` values with the given fill value.
 
         Parameters
diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 0b2ed617..11411267 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -18,7 +18,7 @@ class DataFrame:
         ...
 
 class Column:
-    def mean(self, skip_nulls: bool = True) -> Scalar | NullType:
+    def mean(self, skip_nulls: bool = True) -> Scalar:
         ...
 
 larger = df2 > df1.col('foo').mean()

From 0db987154f9aa52287b886b4ff693e6a2d784bd7 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Tue, 14 Nov 2023 19:14:24 +0000
Subject: [PATCH 09/26] type null as Scalar

---
 spec/API_specification/dataframe_api/column_object.py    | 6 +++---
 spec/API_specification/dataframe_api/dataframe_object.py | 4 ++--
 spec/API_specification/dataframe_api/typing.py           | 7 +------
 spec/conf.py                                             | 1 -
 4 files changed, 6 insertions(+), 12 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 33f6a560..ec921b2d 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -6,7 +6,7 @@
     from typing_extensions import Self
 
     from .scalar_object import Scalar
-    from .typing import DType, Namespace, NullType
+    from .typing import DType, Namespace
 
 
 __all__ = ["Column"]
@@ -498,7 +498,7 @@ def __invert__(self) -> Self:
         """
         ...
 
-    def any(self, *, skip_nulls: bool = True) -> bool | NullType:
+    def any(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a bool.
 
         Raises
@@ -508,7 +508,7 @@ def any(self, *, skip_nulls: bool = True) -> bool | NullType:
         """
         ...
 
-    def all(self, *, skip_nulls: bool = True) -> bool | NullType:
+    def all(self, *, skip_nulls: bool = True) -> Scalar:
         """Reduction returns a bool.
 
         Raises
diff --git a/spec/API_specification/dataframe_api/dataframe_object.py b/spec/API_specification/dataframe_api/dataframe_object.py
index ddba939a..5454893e 100644
--- a/spec/API_specification/dataframe_api/dataframe_object.py
+++ b/spec/API_specification/dataframe_api/dataframe_object.py
@@ -9,7 +9,7 @@
 
     from .column_object import Column
     from .groupby_object import GroupBy
-    from .typing import DType, Namespace, NullType, Scalar, SupportsDataFrameAPI
+    from .typing import DType, Namespace, Scalar, SupportsDataFrameAPI
 
 
 __all__ = ["DataFrame"]
@@ -803,7 +803,7 @@ def unique_indices(self, *keys: str, skip_nulls: bool = True) -> Column:
         """
         ...
 
-    def fill_nan(self, value: float | NullType, /) -> Self:
+    def fill_nan(self, value: Scalar, /) -> Self:
         """Fill ``nan`` values with the given fill value.
 
         The fill operation will apply to all columns with a floating-point
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index 6cccf39d..ca5f12b6 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -63,10 +63,7 @@ class Bool:
     class Date:
         ...
 
-    class NullType:
-        ...
-
-    null: NullType
+    null: Scalar
 
     class Datetime:
         def __init__(  # noqa: ANN204
@@ -130,7 +127,6 @@ def date(self, year: int, month: int, day: int) -> Scalar:
         ...
 
 
-NullType = Namespace.NullType
 DType = Union[
     Namespace.Bool,
     Namespace.Float64,
@@ -174,7 +170,6 @@ def __column_consortium_standard__(
     "DType",
     "GroupBy",
     "Namespace",
-    "NullType",
     "Scalar",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
diff --git a/spec/conf.py b/spec/conf.py
index cc6e3270..054c012d 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -85,7 +85,6 @@
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),
-    ('py:class', 'NullType'),
     ('py:class', 'Namespace'),
     ('py:class', 'SupportsDataFrameAPI'),
     ('py:class', 'Self'),

From 6520ac438f222408d24f312fbc83705a5eb9cca5 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Tue, 14 Nov 2023 19:21:19 +0000
Subject: [PATCH 10/26] add example of working with scalars

---
 .../examples/05_scalars_example.py            | 19 +++++++++++++++++++
 1 file changed, 19 insertions(+)
 create mode 100644 spec/API_specification/examples/05_scalars_example.py

diff --git a/spec/API_specification/examples/05_scalars_example.py b/spec/API_specification/examples/05_scalars_example.py
new file mode 100644
index 00000000..b6b5f41b
--- /dev/null
+++ b/spec/API_specification/examples/05_scalars_example.py
@@ -0,0 +1,19 @@
+from __future__ import annotations
+
+from typing import TYPE_CHECKING
+
+if TYPE_CHECKING:
+    from dataframe_api.typing import SupportsDataFrameAPI
+
+
+def main(df_raw: SupportsDataFrameAPI) -> SupportsDataFrameAPI:
+    df = df_raw.__dataframe_consortium_standard__(api_version="2023-11.beta")
+
+    # We can fill nulls using a Scalar object.
+    df = df.fill_null(df.col("a").mean())
+
+    # Python Scalars also implement the Scalar Protocol (indeed, the Scalar
+    # Protocol is designed to be a subset of the Python Scalar types), so we
+    # can pass Python scalars too.
+    df = df.fill_null(3)
+    return df.dataframe

From 46bc08c0d2427e69c9d2c1114822c939ec7505c8 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 11:44:56 +0000
Subject: [PATCH 11/26] use AnyScalar;

---
 .../dataframe_api/column_object.py            | 58 +++++++++----------
 .../dataframe_api/dataframe_object.py         | 52 ++++++++---------
 .../API_specification/dataframe_api/typing.py |  6 ++
 .../examples/05_scalars_example.py            |  8 +--
 spec/conf.py                                  |  1 +
 5 files changed, 66 insertions(+), 59 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index ec921b2d..777c7593 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -6,7 +6,7 @@
     from typing_extensions import Self
 
     from .scalar_object import Scalar
-    from .typing import DType, Namespace
+    from .typing import AnyScalar, DType, Namespace
 
 
 __all__ = ["Column"]
@@ -187,7 +187,7 @@ def sorted_indices(
         """
         ...
 
-    def __eq__(self, other: Self | Scalar) -> Self:  # type: ignore[override]
+    def __eq__(self, other: Self | AnyScalar) -> Self:  # type: ignore[override]
         """Compare for equality.
 
         Nulls should follow Kleene Logic.
@@ -205,7 +205,7 @@ def __eq__(self, other: Self | Scalar) -> Self:  # type: ignore[override]
         """
         ...
 
-    def __ne__(self, other: Self | Scalar) -> Self:  # type: ignore[override]
+    def __ne__(self, other: Self | AnyScalar) -> Self:  # type: ignore[override]
         """Compare for non-equality.
 
         Nulls should follow Kleene Logic.
@@ -223,7 +223,7 @@ def __ne__(self, other: Self | Scalar) -> Self:  # type: ignore[override]
         """
         ...
 
-    def __ge__(self, other: Self | Scalar) -> Self:
+    def __ge__(self, other: Self | AnyScalar) -> Self:
         """Compare for "greater than or equal to" `other`.
 
         Parameters
@@ -239,7 +239,7 @@ def __ge__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __gt__(self, other: Self | Scalar) -> Self:
+    def __gt__(self, other: Self | AnyScalar) -> Self:
         """Compare for "greater than" `other`.
 
         Parameters
@@ -255,7 +255,7 @@ def __gt__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __le__(self, other: Self | Scalar) -> Self:
+    def __le__(self, other: Self | AnyScalar) -> Self:
         """Compare for "less than or equal to" `other`.
 
         Parameters
@@ -271,7 +271,7 @@ def __le__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __lt__(self, other: Self | Scalar) -> Self:
+    def __lt__(self, other: Self | AnyScalar) -> Self:
         """Compare for "less than" `other`.
 
         Parameters
@@ -287,7 +287,7 @@ def __lt__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __and__(self, other: Self | bool) -> Self:
+    def __and__(self, other: Self | AnyScalar) -> Self:
         """Apply logical 'and' to `other` Column (or scalar) and this Column.
 
         Nulls should follow Kleene Logic.
@@ -308,7 +308,7 @@ def __and__(self, other: Self | bool) -> Self:
         """
         ...
 
-    def __or__(self, other: Self | bool) -> Self:
+    def __or__(self, other: Self | AnyScalar) -> Self:
         """Apply logical 'or' to `other` Column (or scalar) and this column.
 
         Nulls should follow Kleene Logic.
@@ -329,7 +329,7 @@ def __or__(self, other: Self | bool) -> Self:
         """
         ...
 
-    def __add__(self, other: Self | Scalar) -> Self:
+    def __add__(self, other: Self | AnyScalar) -> Self:
         """Add `other` column or scalar to this column.
 
         Parameters
@@ -345,7 +345,7 @@ def __add__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __sub__(self, other: Self | Scalar) -> Self:
+    def __sub__(self, other: Self | AnyScalar) -> Self:
         """Subtract `other` column or scalar from this column.
 
         Parameters
@@ -361,7 +361,7 @@ def __sub__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __mul__(self, other: Self | Scalar) -> Self:
+    def __mul__(self, other: Self | AnyScalar) -> Self:
         """Multiply `other` column or scalar with this column.
 
         Parameters
@@ -377,7 +377,7 @@ def __mul__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __truediv__(self, other: Self | Scalar) -> Self:
+    def __truediv__(self, other: Self | AnyScalar) -> Self:
         """Divide this column by `other` column or scalar. True division, returns floats.
 
         Parameters
@@ -393,7 +393,7 @@ def __truediv__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __floordiv__(self, other: Self | Scalar) -> Self:
+    def __floordiv__(self, other: Self | AnyScalar) -> Self:
         """Floor-divide `other` column or scalar to this column.
 
         Parameters
@@ -409,7 +409,7 @@ def __floordiv__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __pow__(self, other: Self | Scalar) -> Self:
+    def __pow__(self, other: Self | AnyScalar) -> Self:
         """Raise this column to the power of `other`.
 
         Integer dtype to the power of non-negative integer dtype is integer dtype.
@@ -429,7 +429,7 @@ def __pow__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __mod__(self, other: Self | Scalar) -> Self:
+    def __mod__(self, other: Self | AnyScalar) -> Self:
         """Return modulus of this column by `other` (`%` operator).
 
         Parameters
@@ -445,7 +445,7 @@ def __mod__(self, other: Self | Scalar) -> Self:
         """
         ...
 
-    def __divmod__(self, other: Self | Scalar) -> tuple[Column, Column]:
+    def __divmod__(self, other: Self | AnyScalar) -> tuple[Column, Column]:
         """Return quotient and remainder of integer division. See `divmod` builtin.
 
         Parameters
@@ -461,16 +461,16 @@ def __divmod__(self, other: Self | Scalar) -> tuple[Column, Column]:
         """
         ...
 
-    def __radd__(self, other: Self | Scalar) -> Self:
+    def __radd__(self, other: Self | AnyScalar) -> Self:
         ...
 
-    def __rsub__(self, other: Self | Scalar) -> Self:
+    def __rsub__(self, other: Self | AnyScalar) -> Self:
         ...
 
-    def __rmul__(self, other: Self | Scalar) -> Self:
+    def __rmul__(self, other: Self | AnyScalar) -> Self:
         ...
 
-    def __rtruediv__(self, other: Self | Scalar) -> Self:
+    def __rtruediv__(self, other: Self | AnyScalar) -> Self:
         ...
 
     def __rand__(self, other: Self | bool) -> Self:
@@ -479,13 +479,13 @@ def __rand__(self, other: Self | bool) -> Self:
     def __ror__(self, other: Self | bool) -> Self:
         ...
 
-    def __rfloordiv__(self, other: Self | Scalar) -> Self:
+    def __rfloordiv__(self, other: Self | AnyScalar) -> Self:
         ...
 
-    def __rpow__(self, other: Self | Scalar) -> Self:
+    def __rpow__(self, other: Self | AnyScalar) -> Self:
         ...
 
-    def __rmod__(self, other: Self | Scalar) -> Self:
+    def __rmod__(self, other: Self | AnyScalar) -> Self:
         ...
 
     def __invert__(self) -> Self:
@@ -718,7 +718,7 @@ def is_in(self, values: Self) -> Self:
         """
         ...
 
-    def unique_indices(self, *, skip_nulls: bool = True) -> Self:
+    def unique_indices(self, *, skip_nulls: AnyScalar = True) -> Self:
         """Return indices corresponding to unique values in Column.
 
         Returns
@@ -738,7 +738,7 @@ def unique_indices(self, *, skip_nulls: bool = True) -> Self:
         """
         ...
 
-    def fill_nan(self, value: Scalar, /) -> Self:
+    def fill_nan(self, value: AnyScalar, /) -> Self:
         """Fill floating point ``nan`` values with the given fill value.
 
         Parameters
@@ -751,7 +751,7 @@ def fill_nan(self, value: Scalar, /) -> Self:
         """
         ...
 
-    def fill_null(self, value: Scalar, /) -> Self:
+    def fill_null(self, value: AnyScalar, /) -> Self:
         """Fill null values with the given fill value.
 
         Parameters
@@ -797,7 +797,7 @@ def to_array(self) -> Any:
         """
         ...
 
-    def rename(self, name: str) -> Self:
+    def rename(self, name: AnyScalar) -> Self:
         """Rename column.
 
         Parameters
@@ -886,7 +886,7 @@ def iso_weekday(self) -> Self:
         """
         ...
 
-    def unix_timestamp(self, *, time_unit: Literal["s", "ms", "us"] = "s") -> Self:
+    def unix_timestamp(self, *, time_unit: AnyScalar = "s") -> Self:
         """Return number of seconds / milliseconds / microseconds since the Unix epoch.
 
         The Unix epoch is 00:00:00 UTC on 1 January 1970.
diff --git a/spec/API_specification/dataframe_api/dataframe_object.py b/spec/API_specification/dataframe_api/dataframe_object.py
index 5454893e..58fd5439 100644
--- a/spec/API_specification/dataframe_api/dataframe_object.py
+++ b/spec/API_specification/dataframe_api/dataframe_object.py
@@ -9,7 +9,7 @@
 
     from .column_object import Column
     from .groupby_object import GroupBy
-    from .typing import DType, Namespace, Scalar, SupportsDataFrameAPI
+    from .typing import AnyScalar, DType, Namespace, SupportsDataFrameAPI
 
 
 __all__ = ["DataFrame"]
@@ -335,7 +335,7 @@ def sorted_indices(
         """
         ...
 
-    def __eq__(self, other: Scalar) -> Self:  # type: ignore[override]
+    def __eq__(self, other: AnyScalar) -> Self:  # type: ignore[override]
         """Compare for equality.
 
         Nulls should follow Kleene Logic.
@@ -352,7 +352,7 @@ def __eq__(self, other: Scalar) -> Self:  # type: ignore[override]
         """
         ...
 
-    def __ne__(self, other: Scalar) -> Self:  # type: ignore[override]
+    def __ne__(self, other: AnyScalar) -> Self:  # type: ignore[override]
         """Compare for non-equality.
 
         Nulls should follow Kleene Logic.
@@ -369,7 +369,7 @@ def __ne__(self, other: Scalar) -> Self:  # type: ignore[override]
         """
         ...
 
-    def __ge__(self, other: Scalar) -> Self:
+    def __ge__(self, other: AnyScalar) -> Self:
         """Compare for "greater than or equal to" `other`.
 
         Parameters
@@ -384,7 +384,7 @@ def __ge__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __gt__(self, other: Scalar) -> Self:
+    def __gt__(self, other: AnyScalar) -> Self:
         """Compare for "greater than" `other`.
 
         Parameters
@@ -399,7 +399,7 @@ def __gt__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __le__(self, other: Scalar) -> Self:
+    def __le__(self, other: AnyScalar) -> Self:
         """Compare for "less than or equal to" `other`.
 
         Parameters
@@ -414,7 +414,7 @@ def __le__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __lt__(self, other: Scalar) -> Self:
+    def __lt__(self, other: AnyScalar) -> Self:
         """Compare for "less than" `other`.
 
         Parameters
@@ -469,7 +469,7 @@ def __or__(self, other: bool) -> Self:  # noqa: FBT001
         """
         ...
 
-    def __add__(self, other: Scalar) -> Self:
+    def __add__(self, other: AnyScalar) -> Self:
         """Add `other` scalar to this dataframe.
 
         Parameters
@@ -484,7 +484,7 @@ def __add__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __sub__(self, other: Scalar) -> Self:
+    def __sub__(self, other: AnyScalar) -> Self:
         """Subtract `other` scalar from this dataframe.
 
         Parameters
@@ -499,7 +499,7 @@ def __sub__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __mul__(self, other: Scalar) -> Self:
+    def __mul__(self, other: AnyScalar) -> Self:
         """Multiply  `other` scalar with this dataframe.
 
         Parameters
@@ -514,7 +514,7 @@ def __mul__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __truediv__(self, other: Scalar) -> Self:
+    def __truediv__(self, other: AnyScalar) -> Self:
         """Divide  this dataframe by `other` scalar. True division, returns floats.
 
         Parameters
@@ -529,7 +529,7 @@ def __truediv__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __floordiv__(self, other: Scalar) -> Self:
+    def __floordiv__(self, other: AnyScalar) -> Self:
         """Floor-divide (returns integers) this dataframe by `other` scalar.
 
         Parameters
@@ -544,7 +544,7 @@ def __floordiv__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __pow__(self, other: Scalar) -> Self:
+    def __pow__(self, other: AnyScalar) -> Self:
         """Raise this dataframe to the power of `other`.
 
         Integer dtype to the power of non-negative integer dtype is integer dtype.
@@ -563,7 +563,7 @@ def __pow__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __mod__(self, other: Scalar) -> Self:
+    def __mod__(self, other: AnyScalar) -> Self:
         """Return modulus of this dataframe by `other` (`%` operator).
 
         Parameters
@@ -578,7 +578,7 @@ def __mod__(self, other: Scalar) -> Self:
         """
         ...
 
-    def __divmod__(self, other: Scalar) -> tuple[DataFrame, DataFrame]:
+    def __divmod__(self, other: AnyScalar) -> tuple[DataFrame, DataFrame]:
         """Return quotient and remainder of integer division. See `divmod` builtin.
 
         Parameters
@@ -593,31 +593,31 @@ def __divmod__(self, other: Scalar) -> tuple[DataFrame, DataFrame]:
         """
         ...
 
-    def __radd__(self, other: Scalar) -> Self:
+    def __radd__(self, other: AnyScalar) -> Self:
         ...
 
-    def __rsub__(self, other: Scalar) -> Self:
+    def __rsub__(self, other: AnyScalar) -> Self:
         ...
 
-    def __rmul__(self, other: Scalar) -> Self:
+    def __rmul__(self, other: AnyScalar) -> Self:
         ...
 
-    def __rtruediv__(self, other: Scalar) -> Self:
+    def __rtruediv__(self, other: AnyScalar) -> Self:
         ...
 
-    def __rand__(self, other: Scalar) -> Self:
+    def __rand__(self, other: AnyScalar) -> Self:
         ...
 
-    def __ror__(self, other: Scalar) -> Self:
+    def __ror__(self, other: AnyScalar) -> Self:
         ...
 
-    def __rfloordiv__(self, other: Scalar) -> Self:
+    def __rfloordiv__(self, other: AnyScalar) -> Self:
         ...
 
-    def __rpow__(self, other: Scalar) -> Self:
+    def __rpow__(self, other: AnyScalar) -> Self:
         ...
 
-    def __rmod__(self, other: Scalar) -> Self:
+    def __rmod__(self, other: AnyScalar) -> Self:
         ...
 
     def __invert__(self) -> Self:
@@ -803,7 +803,7 @@ def unique_indices(self, *keys: str, skip_nulls: bool = True) -> Column:
         """
         ...
 
-    def fill_nan(self, value: Scalar, /) -> Self:
+    def fill_nan(self, value: AnyScalar, /) -> Self:
         """Fill ``nan`` values with the given fill value.
 
         The fill operation will apply to all columns with a floating-point
@@ -821,7 +821,7 @@ def fill_nan(self, value: Scalar, /) -> Self:
 
     def fill_null(
         self,
-        value: Scalar,
+        value: AnyScalar,
         /,
         *,
         column_names: list[str] | None = None,
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index ca5f12b6..01a26ab2 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -164,6 +164,10 @@ def __column_consortium_standard__(
         ...
 
 
+PythonScalar = Union[str, int, float]
+AnyScalar = Union[PythonScalar, Scalar]
+
+
 __all__ = [
     "Column",
     "DataFrame",
@@ -171,6 +175,8 @@ def __column_consortium_standard__(
     "GroupBy",
     "Namespace",
     "Scalar",
+    "PythonScalar",
+    "AnyScalar",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
 ]
diff --git a/spec/API_specification/examples/05_scalars_example.py b/spec/API_specification/examples/05_scalars_example.py
index b6b5f41b..9dd95461 100644
--- a/spec/API_specification/examples/05_scalars_example.py
+++ b/spec/API_specification/examples/05_scalars_example.py
@@ -9,11 +9,11 @@
 def main(df_raw: SupportsDataFrameAPI) -> SupportsDataFrameAPI:
     df = df_raw.__dataframe_consortium_standard__(api_version="2023-11.beta")
 
-    # We can fill nulls using a Scalar object.
+    # `DataFrame.fill_null` accepts `AnyScalar` objections.
+    # This means we can fill nulls using a Standard Scalar object...
     df = df.fill_null(df.col("a").mean())
 
-    # Python Scalars also implement the Scalar Protocol (indeed, the Scalar
-    # Protocol is designed to be a subset of the Python Scalar types), so we
-    # can pass Python scalars too.
+    # ... but also Python scalars:
     df = df.fill_null(3)
+    df = df.fill_null("3")
     return df.dataframe
diff --git a/spec/conf.py b/spec/conf.py
index 054c012d..d4fc0bbe 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -82,6 +82,7 @@
     ('py:class', 'enum.Enum'),
     ('py:class', 'ellipsis'),
     ('py:class', 'Scalar'),
+    ('py:class', 'AnyScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),

From b879f31ac596451aa2986d82fd0c934758e1f84d Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 12:32:54 +0000
Subject: [PATCH 12/26] add Scalar.dtype and Scalar.persist

---
 .../dataframe_api/scalar_object.py            | 65 +++++++++++++------
 .../examples/05_scalars_example.py            | 15 +++++
 2 files changed, 59 insertions(+), 21 deletions(-)

diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
index a15db639..100be8ab 100644
--- a/spec/API_specification/dataframe_api/scalar_object.py
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -1,6 +1,9 @@
 from __future__ import annotations
 
-from typing import Any, Protocol
+from typing import TYPE_CHECKING, Protocol
+
+if TYPE_CHECKING:
+    from dataframe_api.typing import AnyScalar, DType
 
 __all__ = ["Scalar"]
 
@@ -13,64 +16,66 @@ class Scalar(Protocol):
     as `:meth:`Column.sum`.
     """
 
-    def __lt__(self, other: Any) -> Scalar:
+    def __lt__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __le__(self, other: Any) -> Scalar:
+    def __le__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __eq__(self, other: object) -> Scalar:  # type: ignore[override]
+    def __eq__(self, other: AnyScalar) -> Scalar:  # type: ignore[override]
         ...
 
-    def __ne__(self, other: object) -> Scalar:  # type: ignore[override]
+    def __ne__(self, other: AnyScalar) -> Scalar:  # type: ignore[override]
         ...
 
-    def __gt__(self, other: Any) -> Scalar:
+    def __gt__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __ge__(self, other: Any) -> Scalar:
+    def __ge__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __add__(self, other: Any) -> Scalar:
+    def __add__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __radd__(self, other: Any) -> Scalar:
+    def __radd__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __sub__(self, other: Any) -> Scalar:
+    def __sub__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __rsub__(self, other: Any) -> Scalar:
+    def __rsub__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __mul__(self, other: Any) -> Scalar:
+    def __mul__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __rmul__(self, other: Any) -> Scalar:
+    def __rmul__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __mod__(self, other: Any) -> Scalar:
+    def __mod__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __rmod__(self, other: Any) -> Scalar:
+    # Signatures of "__rmod__" of "Scalar" and "__mod__" of "str | int | float | Scalar"
+    # are unsafely overlapping
+    def __rmod__(self, other: AnyScalar) -> Scalar:  # type: ignore[misc]
         ...
 
-    def __pow__(self, other: Any) -> Scalar:
+    def __pow__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __rpow__(self, other: Any) -> Scalar:
+    def __rpow__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __floordiv__(self, other: Any) -> Scalar:
+    def __floordiv__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __rfloordiv__(self, other: Any) -> Scalar:
+    def __rfloordiv__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __truediv__(self, other: Any) -> Scalar:
+    def __truediv__(self, other: AnyScalar) -> Scalar:
         ...
 
-    def __rtruediv__(self, other: Any) -> Scalar:
+    def __rtruediv__(self, other: AnyScalar) -> Scalar:
         ...
 
     def __neg__(self) -> Scalar:
@@ -85,3 +90,21 @@ def __bool__(self) -> bool:
         Depending on the implementation, this may raise or trigger computation.
         """
         ...
+    
+    @property
+    def dtype(self) -> DType:
+        """Return data type of column."""
+        ...
+
+    def persist(self) -> object:
+        """Hint that computation prior to this point should not be repeated.
+
+        This is intended as a hint, rather than as a directive. Implementations
+        which do not separate lazy vs eager execution may ignore this method and
+        treat it as a no-op.
+
+        .. note::
+            This may trigger computation and so should be used with care.
+            See `execution_model` page for more details.
+        """
+        ...
diff --git a/spec/API_specification/examples/05_scalars_example.py b/spec/API_specification/examples/05_scalars_example.py
index 9dd95461..acf001a8 100644
--- a/spec/API_specification/examples/05_scalars_example.py
+++ b/spec/API_specification/examples/05_scalars_example.py
@@ -3,6 +3,9 @@
 from typing import TYPE_CHECKING
 
 if TYPE_CHECKING:
+    from dataframe_api.column_object import Column
+    from dataframe_api.dataframe_object import DataFrame
+    from dataframe_api.scalar_object import Scalar
     from dataframe_api.typing import SupportsDataFrameAPI
 
 
@@ -16,4 +19,16 @@ def main(df_raw: SupportsDataFrameAPI) -> SupportsDataFrameAPI:
     # ... but also Python scalars:
     df = df.fill_null(3)
     df = df.fill_null("3")
+
+    # Scalars can be used in arithmetic expressions with other scalars, columns,
+    # or DataFrames
+    value: Scalar = df.col("a").mean()
+    col: Column = df.col("a")
+    _res1: Column = value - col
+    _res2: Scalar = value - 3
+    _res3: Scalar = 3 - value
+    _res4: Column = df.col("a") - 3
+    _res5: DataFrame = df - value
+    _res6: DataFrame = value - df
+
     return df.dataframe

From b8011c74d3729aad822e450b0e54379548d00dc4 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 12:38:02 +0000
Subject: [PATCH 13/26] update shift arg

---
 spec/API_specification/dataframe_api/column_object.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index a37df40e..c5ecd55d 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -812,7 +812,7 @@ def rename(self, name: AnyScalar) -> Self:
         """
         ...
 
-    def shift(self, offset: int) -> Self:
+    def shift(self, offset: AnyScalar) -> Self:
         """Shift values by `offset` positions, filling missing values with `null`.
 
         For example, if the original column contains values `[1, 4, 2]`, then:
@@ -822,7 +822,7 @@ def shift(self, offset: int) -> Self:
     
         Parameters
         ----------
-        offset
+        offset : int
             How many positions to shift by.
         """
         ...

From 97d8f9a172cc8196126eece9f6bf3ddff5e54b1f Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 14:33:32 +0000
Subject: [PATCH 14/26] use BoolScalar

---
 .../dataframe_api/column_object.py            | 30 +++++------
 .../dataframe_api/dataframe_object.py         | 28 +++++-----
 .../dataframe_api/groupby_object.py           | 51 +++++++++++--------
 .../dataframe_api/scalar_object.py            |  2 +-
 .../API_specification/dataframe_api/typing.py | 10 +++-
 spec/conf.py                                  |  1 +
 spec/design_topics/python_builtin_types.md    |  2 +-
 7 files changed, 72 insertions(+), 52 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index c5ecd55d..a506f1f2 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -6,7 +6,7 @@
     from typing_extensions import Self
 
     from .scalar_object import Scalar
-    from .typing import AnyScalar, DType, Namespace
+    from .typing import AnyScalar, BoolScalar, DType, Namespace
 
 
 __all__ = ["Column"]
@@ -287,7 +287,7 @@ def __lt__(self, other: Self | AnyScalar) -> Self:
         """
         ...
 
-    def __and__(self, other: Self | AnyScalar) -> Self:
+    def __and__(self, other: Self | BoolScalar) -> Self:
         """Apply logical 'and' to `other` Column (or scalar) and this Column.
 
         Nulls should follow Kleene Logic.
@@ -308,7 +308,7 @@ def __and__(self, other: Self | AnyScalar) -> Self:
         """
         ...
 
-    def __or__(self, other: Self | AnyScalar) -> Self:
+    def __or__(self, other: Self | BoolScalar) -> Self:
         """Apply logical 'or' to `other` Column (or scalar) and this column.
 
         Nulls should follow Kleene Logic.
@@ -498,7 +498,7 @@ def __invert__(self) -> Self:
         """
         ...
 
-    def any(self, *, skip_nulls: bool = True) -> Scalar:
+    def any(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a bool.
 
         Raises
@@ -508,7 +508,7 @@ def any(self, *, skip_nulls: bool = True) -> Scalar:
         """
         ...
 
-    def all(self, *, skip_nulls: bool = True) -> Scalar:
+    def all(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a bool.
 
         Raises
@@ -518,7 +518,7 @@ def all(self, *, skip_nulls: bool = True) -> Scalar:
         """
         ...
 
-    def min(self, *, skip_nulls: bool = True) -> Scalar:
+    def min(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Any data type that supports comparisons
@@ -526,7 +526,7 @@ def min(self, *, skip_nulls: bool = True) -> Scalar:
         """
         ...
 
-    def max(self, *, skip_nulls: bool = True) -> Scalar:
+    def max(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Any data type that supports comparisons
@@ -534,7 +534,7 @@ def max(self, *, skip_nulls: bool = True) -> Scalar:
         """
         ...
 
-    def sum(self, *, skip_nulls: bool = True) -> Scalar:
+    def sum(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -543,7 +543,7 @@ def sum(self, *, skip_nulls: bool = True) -> Scalar:
         """
         ...
 
-    def prod(self, *, skip_nulls: bool = True) -> Scalar:
+    def prod(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical data types.
@@ -551,7 +551,7 @@ def prod(self, *, skip_nulls: bool = True) -> Scalar:
         """
         ...
 
-    def median(self, *, skip_nulls: bool = True) -> Scalar:
+    def median(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -561,7 +561,7 @@ def median(self, *, skip_nulls: bool = True) -> Scalar:
         """
         ...
 
-    def mean(self, *, skip_nulls: bool = True) -> Scalar:
+    def mean(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -575,7 +575,7 @@ def std(
         self,
         *,
         correction: int | float = 1,
-        skip_nulls: bool = True,
+        skip_nulls: BoolScalar = True,
     ) -> Scalar:
         """Reduction returns a scalar.
 
@@ -608,7 +608,7 @@ def var(
         self,
         *,
         correction: int | float = 1,
-        skip_nulls: bool = True,
+        skip_nulls: BoolScalar = True,
     ) -> Scalar:
         """Reduction returns a scalar.
 
@@ -718,7 +718,7 @@ def is_in(self, values: Self) -> Self:
         """
         ...
 
-    def unique_indices(self, *, skip_nulls: AnyScalar = True) -> Self:
+    def unique_indices(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Return indices corresponding to unique values in Column.
 
         Returns
@@ -819,7 +819,7 @@ def shift(self, offset: AnyScalar) -> Self:
 
         - `.shift(1)` will return `[null, 1, 4]`,
         - `.shift(-1)` will return `[4, 2, null]`,
-    
+
         Parameters
         ----------
         offset : int
diff --git a/spec/API_specification/dataframe_api/dataframe_object.py b/spec/API_specification/dataframe_api/dataframe_object.py
index 58fd5439..c6e91f3e 100644
--- a/spec/API_specification/dataframe_api/dataframe_object.py
+++ b/spec/API_specification/dataframe_api/dataframe_object.py
@@ -9,7 +9,7 @@
 
     from .column_object import Column
     from .groupby_object import GroupBy
-    from .typing import AnyScalar, DType, Namespace, SupportsDataFrameAPI
+    from .typing import AnyScalar, BoolScalar, DType, Namespace, SupportsDataFrameAPI
 
 
 __all__ = ["DataFrame"]
@@ -641,7 +641,7 @@ def __iter__(self) -> NoReturn:
         """
         raise NotImplementedError("'__iter__' is intentionally not implemented.")
 
-    def any(self, *, skip_nulls: bool = True) -> Self:
+    def any(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Raises
@@ -651,7 +651,7 @@ def any(self, *, skip_nulls: bool = True) -> Self:
         """
         ...
 
-    def all(self, *, skip_nulls: bool = True) -> Self:
+    def all(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Raises
@@ -661,7 +661,7 @@ def all(self, *, skip_nulls: bool = True) -> Self:
         """
         ...
 
-    def any_rowwise(self, *, skip_nulls: bool = True) -> Column:
+    def any_rowwise(self, *, skip_nulls: BoolScalar = True) -> Column:
         """Reduction returns a Column.
 
         Differs from ``DataFrame.any`` and that the reduction happens
@@ -674,7 +674,7 @@ def any_rowwise(self, *, skip_nulls: bool = True) -> Column:
         """
         ...
 
-    def all_rowwise(self, *, skip_nulls: bool = True) -> Column:
+    def all_rowwise(self, *, skip_nulls: BoolScalar = True) -> Column:
         """Reduction returns a Column.
 
         Differs from ``DataFrame.all`` and that the reduction happens
@@ -687,31 +687,31 @@ def all_rowwise(self, *, skip_nulls: bool = True) -> Column:
         """
         ...
 
-    def min(self, *, skip_nulls: bool = True) -> Self:
+    def min(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def max(self, *, skip_nulls: bool = True) -> Self:
+    def max(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def sum(self, *, skip_nulls: bool = True) -> Self:
+    def sum(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def prod(self, *, skip_nulls: bool = True) -> Self:
+    def prod(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def median(self, *, skip_nulls: bool = True) -> Self:
+    def median(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def mean(self, *, skip_nulls: bool = True) -> Self:
+    def mean(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def std(self, *, correction: int | float = 1, skip_nulls: bool = True) -> Self:
+    def std(self, *, correction: int | float = 1, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Parameters
@@ -725,7 +725,7 @@ def std(self, *, correction: int | float = 1, skip_nulls: bool = True) -> Self:
         """
         ...
 
-    def var(self, *, correction: int | float = 1, skip_nulls: bool = True) -> Self:
+    def var(self, *, correction: int | float = 1, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Parameters
@@ -777,7 +777,7 @@ def is_nan(self) -> Self:
         """
         ...
 
-    def unique_indices(self, *keys: str, skip_nulls: bool = True) -> Column:
+    def unique_indices(self, *keys: str, skip_nulls: BoolScalar = True) -> Column:
         """Return indices corresponding to unique values across selected columns.
 
         Parameters
diff --git a/spec/API_specification/dataframe_api/groupby_object.py b/spec/API_specification/dataframe_api/groupby_object.py
index adecb8aa..6e8e0d3b 100644
--- a/spec/API_specification/dataframe_api/groupby_object.py
+++ b/spec/API_specification/dataframe_api/groupby_object.py
@@ -4,6 +4,7 @@
 
 if TYPE_CHECKING:
     from .dataframe_object import DataFrame
+    from .typing import BoolScalar
 
 
 __all__ = [
@@ -22,34 +23,44 @@ class GroupBy(Protocol):
 
     """
 
-    def any(self, *, skip_nulls: bool = True) -> DataFrame:
+    def any(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def all(self, *, skip_nulls: bool = True) -> DataFrame:
+    def all(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def min(self, *, skip_nulls: bool = True) -> DataFrame:
+    def min(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def max(self, *, skip_nulls: bool = True) -> DataFrame:
+    def max(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def sum(self, *, skip_nulls: bool = True) -> DataFrame:
+    def sum(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def prod(self, *, skip_nulls: bool = True) -> DataFrame:
+    def prod(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def median(self, *, skip_nulls: bool = True) -> DataFrame:
+    def median(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def mean(self, *, skip_nulls: bool = True) -> DataFrame:
+    def mean(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
         ...
 
-    def std(self, *, correction: int | float = 1, skip_nulls: bool = True) -> DataFrame:
+    def std(
+        self,
+        *,
+        correction: int | float = 1,
+        skip_nulls: BoolScalar = True,
+    ) -> DataFrame:
         ...
 
-    def var(self, *, correction: int | float = 1, skip_nulls: bool = True) -> DataFrame:
+    def var(
+        self,
+        *,
+        correction: int | float = 1,
+        skip_nulls: BoolScalar = True,
+    ) -> DataFrame:
         ...
 
     def size(self) -> DataFrame:
@@ -82,35 +93,35 @@ def rename(self, name: str) -> Aggregation:
         ...
 
     @classmethod
-    def any(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def any(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def all(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def all(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def min(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def min(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def max(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def max(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def sum(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def sum(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def prod(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def prod(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def median(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def median(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def mean(cls, column: str, *, skip_nulls: bool = True) -> Aggregation:
+    def mean(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
         ...
 
     @classmethod
@@ -119,7 +130,7 @@ def std(
         column: str,
         *,
         correction: int | float = 1,
-        skip_nulls: bool = True,
+        skip_nulls: BoolScalar = True,
     ) -> Aggregation:
         ...
 
@@ -129,7 +140,7 @@ def var(
         column: str,
         *,
         correction: int | float = 1,
-        skip_nulls: bool = True,
+        skip_nulls: BoolScalar = True,
     ) -> Aggregation:
         ...
 
diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
index 100be8ab..2f0012a0 100644
--- a/spec/API_specification/dataframe_api/scalar_object.py
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -90,7 +90,7 @@ def __bool__(self) -> bool:
         Depending on the implementation, this may raise or trigger computation.
         """
         ...
-    
+
     @property
     def dtype(self) -> DType:
         """Return data type of column."""
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index 01a26ab2..68501060 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -164,7 +164,11 @@ def __column_consortium_standard__(
         ...
 
 
-PythonScalar = Union[str, int, float]
+PythonScalar = Union[str, int, float, bool]
+BoolScalar = Union[bool, Scalar]
+FloatScalar = Union[float, Scalar]
+IntScalar = Union[int, Scalar]
+StringScalar = Union[str, Scalar]
 AnyScalar = Union[PythonScalar, Scalar]
 
 
@@ -176,6 +180,10 @@ def __column_consortium_standard__(
     "Namespace",
     "Scalar",
     "PythonScalar",
+    "FloatScalar",
+    "IntScalar",
+    "BoolScalar",
+    "StringScalar",
     "AnyScalar",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
diff --git a/spec/conf.py b/spec/conf.py
index d4fc0bbe..589e6c0e 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -83,6 +83,7 @@
     ('py:class', 'ellipsis'),
     ('py:class', 'Scalar'),
     ('py:class', 'AnyScalar'),
+    ('py:class', 'BoolScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),
diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 11411267..9e9fc323 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -18,7 +18,7 @@ class DataFrame:
         ...
 
 class Column:
-    def mean(self, skip_nulls: bool = True) -> Scalar:
+    def mean(self, skip_nulls: BoolScalar = True) -> Scalar:
         ...
 
 larger = df2 > df1.col('foo').mean()

From 3b7bcb6ae6962856ef0be2e6687f0f37961c7ff8 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 14:43:38 +0000
Subject: [PATCH 15/26] use float scalar in some parts

---
 .../API_specification/dataframe_api/column_object.py |  4 ++--
 .../dataframe_api/dataframe_object.py                | 12 ++++++++++--
 spec/API_specification/dataframe_api/typing.py       |  8 ++++++--
 spec/conf.py                                         |  1 +
 4 files changed, 19 insertions(+), 6 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index a506f1f2..03adc7af 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -6,7 +6,7 @@
     from typing_extensions import Self
 
     from .scalar_object import Scalar
-    from .typing import AnyScalar, BoolScalar, DType, Namespace
+    from .typing import AnyScalar, BoolScalar, DType, FloatScalar, Namespace, NullType
 
 
 __all__ = ["Column"]
@@ -738,7 +738,7 @@ def unique_indices(self, *, skip_nulls: BoolScalar = True) -> Self:
         """
         ...
 
-    def fill_nan(self, value: AnyScalar, /) -> Self:
+    def fill_nan(self, value: FloatScalar | NullType, /) -> Self:
         """Fill floating point ``nan`` values with the given fill value.
 
         Parameters
diff --git a/spec/API_specification/dataframe_api/dataframe_object.py b/spec/API_specification/dataframe_api/dataframe_object.py
index c6e91f3e..8bd11228 100644
--- a/spec/API_specification/dataframe_api/dataframe_object.py
+++ b/spec/API_specification/dataframe_api/dataframe_object.py
@@ -9,7 +9,15 @@
 
     from .column_object import Column
     from .groupby_object import GroupBy
-    from .typing import AnyScalar, BoolScalar, DType, Namespace, SupportsDataFrameAPI
+    from .typing import (
+        AnyScalar,
+        BoolScalar,
+        DType,
+        FloatScalar,
+        Namespace,
+        NullType,
+        SupportsDataFrameAPI,
+    )
 
 
 __all__ = ["DataFrame"]
@@ -803,7 +811,7 @@ def unique_indices(self, *keys: str, skip_nulls: BoolScalar = True) -> Column:
         """
         ...
 
-    def fill_nan(self, value: AnyScalar, /) -> Self:
+    def fill_nan(self, value: FloatScalar | NullType, /) -> Self:
         """Fill ``nan`` values with the given fill value.
 
         The fill operation will apply to all columns with a floating-point
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index 68501060..d5552605 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -63,7 +63,10 @@ class Bool:
     class Date:
         ...
 
-    null: Scalar
+    class NullType:
+        ...
+
+    null: NullType
 
     class Datetime:
         def __init__(  # noqa: ANN204
@@ -170,6 +173,7 @@ def __column_consortium_standard__(
 IntScalar = Union[int, Scalar]
 StringScalar = Union[str, Scalar]
 AnyScalar = Union[PythonScalar, Scalar]
+NullType = Namespace.NullType
 
 
 __all__ = [
@@ -184,7 +188,7 @@ def __column_consortium_standard__(
     "IntScalar",
     "BoolScalar",
     "StringScalar",
-    "AnyScalar",
+    "NullType",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
 ]
diff --git a/spec/conf.py b/spec/conf.py
index 589e6c0e..711f5569 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -84,6 +84,7 @@
     ('py:class', 'Scalar'),
     ('py:class', 'AnyScalar'),
     ('py:class', 'BoolScalar'),
+    ('py:class', 'FloatScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),

From d598a8dddc04fe46cad45d21cd9a7fe54cd26ffb Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 14:44:01 +0000
Subject: [PATCH 16/26] use float scalar in some parts

---
 spec/conf.py | 1 +
 1 file changed, 1 insertion(+)

diff --git a/spec/conf.py b/spec/conf.py
index 711f5569..e67f7884 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -85,6 +85,7 @@
     ('py:class', 'AnyScalar'),
     ('py:class', 'BoolScalar'),
     ('py:class', 'FloatScalar'),
+    ('py:class', 'NullType'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),

From a12585b8f5b5712f7acc3b4d8669e91756ebbbb7 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 14:46:44 +0000
Subject: [PATCH 17/26] string scalar for rename

---
 .../API_specification/dataframe_api/column_object.py | 12 ++++++++++--
 .../dataframe_api/groupby_object.py                  |  4 ++--
 spec/conf.py                                         |  1 +
 3 files changed, 13 insertions(+), 4 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 03adc7af..03a22822 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -6,7 +6,15 @@
     from typing_extensions import Self
 
     from .scalar_object import Scalar
-    from .typing import AnyScalar, BoolScalar, DType, FloatScalar, Namespace, NullType
+    from .typing import (
+        AnyScalar,
+        BoolScalar,
+        DType,
+        FloatScalar,
+        Namespace,
+        NullType,
+        StringScalar,
+    )
 
 
 __all__ = ["Column"]
@@ -797,7 +805,7 @@ def to_array(self) -> Any:
         """
         ...
 
-    def rename(self, name: AnyScalar) -> Self:
+    def rename(self, name: StringScalar) -> Self:
         """Rename column.
 
         Parameters
diff --git a/spec/API_specification/dataframe_api/groupby_object.py b/spec/API_specification/dataframe_api/groupby_object.py
index 6e8e0d3b..7637f571 100644
--- a/spec/API_specification/dataframe_api/groupby_object.py
+++ b/spec/API_specification/dataframe_api/groupby_object.py
@@ -4,7 +4,7 @@
 
 if TYPE_CHECKING:
     from .dataframe_object import DataFrame
-    from .typing import BoolScalar
+    from .typing import BoolScalar, StringScalar
 
 
 __all__ = [
@@ -85,7 +85,7 @@ def aggregate(self, *aggregation: Aggregation) -> DataFrame:
 
 
 class Aggregation(Protocol):
-    def rename(self, name: str) -> Aggregation:
+    def rename(self, name: StringScalar) -> Aggregation:
         """Assign given name to output of aggregation.
 
         If not called, the column's name will be used as the output name.
diff --git a/spec/conf.py b/spec/conf.py
index e67f7884..c04881b6 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -86,6 +86,7 @@
     ('py:class', 'BoolScalar'),
     ('py:class', 'FloatScalar'),
     ('py:class', 'NullType'),
+    ('py:class', 'StringScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),

From 35cd4ed3bbf72383f277b044d6502fe24f312fbb Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 14:47:31 +0000
Subject: [PATCH 18/26] intscalar for shift

---
 spec/API_specification/dataframe_api/column_object.py | 3 ++-
 spec/API_specification/dataframe_api/typing.py        | 1 +
 spec/conf.py                                          | 1 +
 3 files changed, 4 insertions(+), 1 deletion(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 03a22822..5415cba5 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -11,6 +11,7 @@
         BoolScalar,
         DType,
         FloatScalar,
+        IntScalar,
         Namespace,
         NullType,
         StringScalar,
@@ -820,7 +821,7 @@ def rename(self, name: StringScalar) -> Self:
         """
         ...
 
-    def shift(self, offset: AnyScalar) -> Self:
+    def shift(self, offset: IntScalar) -> Self:
         """Shift values by `offset` positions, filling missing values with `null`.
 
         For example, if the original column contains values `[1, 4, 2]`, then:
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index d5552605..65fb4b3f 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -188,6 +188,7 @@ def __column_consortium_standard__(
     "IntScalar",
     "BoolScalar",
     "StringScalar",
+    "IntScalar",
     "NullType",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
diff --git a/spec/conf.py b/spec/conf.py
index c04881b6..05fa2104 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -87,6 +87,7 @@
     ('py:class', 'FloatScalar'),
     ('py:class', 'NullType'),
     ('py:class', 'StringScalar'),
+    ('py:class', 'IntScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),

From fade164de9434c7a318b1345157578b4f0b5b946 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 14:51:52 +0000
Subject: [PATCH 19/26] numeric scalar for correction

---
 spec/API_specification/dataframe_api/column_object.py  |  7 ++++---
 .../dataframe_api/dataframe_object.py                  |  9 +++++++--
 spec/API_specification/dataframe_api/groupby_object.py | 10 +++++-----
 spec/API_specification/dataframe_api/typing.py         |  2 ++
 spec/conf.py                                           |  1 +
 5 files changed, 19 insertions(+), 10 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 5415cba5..16f9bddb 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -14,6 +14,7 @@
         IntScalar,
         Namespace,
         NullType,
+        NumericScalar,
         StringScalar,
     )
 
@@ -583,7 +584,7 @@ def mean(self, *, skip_nulls: BoolScalar = True) -> Scalar:
     def std(
         self,
         *,
-        correction: int | float = 1,
+        correction: float = 1,
         skip_nulls: BoolScalar = True,
     ) -> Scalar:
         """Reduction returns a scalar.
@@ -616,7 +617,7 @@ def std(
     def var(
         self,
         *,
-        correction: int | float = 1,
+        correction: NumericScalar = 1,
         skip_nulls: BoolScalar = True,
     ) -> Scalar:
         """Reduction returns a scalar.
@@ -912,7 +913,7 @@ def iso_weekday(self) -> Self:
         """
         ...
 
-    def unix_timestamp(self, *, time_unit: AnyScalar = "s") -> Self:
+    def unix_timestamp(self, *, time_unit: StringScalar = "s") -> Self:
         """Return number of seconds / milliseconds / microseconds since the Unix epoch.
 
         The Unix epoch is 00:00:00 UTC on 1 January 1970.
diff --git a/spec/API_specification/dataframe_api/dataframe_object.py b/spec/API_specification/dataframe_api/dataframe_object.py
index 8bd11228..199c0bd2 100644
--- a/spec/API_specification/dataframe_api/dataframe_object.py
+++ b/spec/API_specification/dataframe_api/dataframe_object.py
@@ -16,6 +16,7 @@
         FloatScalar,
         Namespace,
         NullType,
+        NumericScalar,
         SupportsDataFrameAPI,
     )
 
@@ -719,7 +720,9 @@ def mean(self, *, skip_nulls: BoolScalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def std(self, *, correction: int | float = 1, skip_nulls: BoolScalar = True) -> Self:
+    def std(
+        self, *, correction: NumericScalar = 1, skip_nulls: BoolScalar = True,
+    ) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Parameters
@@ -733,7 +736,9 @@ def std(self, *, correction: int | float = 1, skip_nulls: BoolScalar = True) ->
         """
         ...
 
-    def var(self, *, correction: int | float = 1, skip_nulls: BoolScalar = True) -> Self:
+    def var(
+        self, *, correction: NumericScalar = 1, skip_nulls: BoolScalar = True,
+    ) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Parameters
diff --git a/spec/API_specification/dataframe_api/groupby_object.py b/spec/API_specification/dataframe_api/groupby_object.py
index 7637f571..e01b477a 100644
--- a/spec/API_specification/dataframe_api/groupby_object.py
+++ b/spec/API_specification/dataframe_api/groupby_object.py
@@ -4,7 +4,7 @@
 
 if TYPE_CHECKING:
     from .dataframe_object import DataFrame
-    from .typing import BoolScalar, StringScalar
+    from .typing import BoolScalar, NumericScalar, StringScalar
 
 
 __all__ = [
@@ -50,7 +50,7 @@ def mean(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
     def std(
         self,
         *,
-        correction: int | float = 1,
+        correction: NumericScalar = 1,
         skip_nulls: BoolScalar = True,
     ) -> DataFrame:
         ...
@@ -58,7 +58,7 @@ def std(
     def var(
         self,
         *,
-        correction: int | float = 1,
+        correction: NumericScalar = 1,
         skip_nulls: BoolScalar = True,
     ) -> DataFrame:
         ...
@@ -129,7 +129,7 @@ def std(
         cls,
         column: str,
         *,
-        correction: int | float = 1,
+        correction: NumericScalar = 1,
         skip_nulls: BoolScalar = True,
     ) -> Aggregation:
         ...
@@ -139,7 +139,7 @@ def var(
         cls,
         column: str,
         *,
-        correction: int | float = 1,
+        correction: NumericScalar = 1,
         skip_nulls: BoolScalar = True,
     ) -> Aggregation:
         ...
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index 65fb4b3f..10c111a6 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -171,6 +171,7 @@ def __column_consortium_standard__(
 BoolScalar = Union[bool, Scalar]
 FloatScalar = Union[float, Scalar]
 IntScalar = Union[int, Scalar]
+NumericScalar = Union[FloatScalar, IntScalar]
 StringScalar = Union[str, Scalar]
 AnyScalar = Union[PythonScalar, Scalar]
 NullType = Namespace.NullType
@@ -189,6 +190,7 @@ def __column_consortium_standard__(
     "BoolScalar",
     "StringScalar",
     "IntScalar",
+    "NumericScalar",
     "NullType",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
diff --git a/spec/conf.py b/spec/conf.py
index 05fa2104..8f9dd5ff 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -88,6 +88,7 @@
     ('py:class', 'NullType'),
     ('py:class', 'StringScalar'),
     ('py:class', 'IntScalar'),
+    ('py:class', 'NumericScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),

From 29ceed2467da0106a975f9c46b5b7a1458283e85 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 20:03:56 +0000
Subject: [PATCH 20/26] simplify

---
 .../dataframe_api/column_object.py            | 41 +++++++--------
 .../dataframe_api/dataframe_object.py         | 38 ++++++++------
 .../dataframe_api/groupby_object.py           | 52 +++++++++----------
 .../dataframe_api/scalar_object.py            |  2 +-
 .../API_specification/dataframe_api/typing.py | 13 +----
 spec/conf.py                                  |  6 +--
 spec/design_topics/python_builtin_types.md    |  2 +-
 7 files changed, 71 insertions(+), 83 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 16f9bddb..7a7bdb99 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -8,14 +8,9 @@
     from .scalar_object import Scalar
     from .typing import (
         AnyScalar,
-        BoolScalar,
         DType,
-        FloatScalar,
-        IntScalar,
         Namespace,
         NullType,
-        NumericScalar,
-        StringScalar,
     )
 
 
@@ -297,7 +292,7 @@ def __lt__(self, other: Self | AnyScalar) -> Self:
         """
         ...
 
-    def __and__(self, other: Self | BoolScalar) -> Self:
+    def __and__(self, other: Self | bool | Scalar) -> Self:
         """Apply logical 'and' to `other` Column (or scalar) and this Column.
 
         Nulls should follow Kleene Logic.
@@ -318,7 +313,7 @@ def __and__(self, other: Self | BoolScalar) -> Self:
         """
         ...
 
-    def __or__(self, other: Self | BoolScalar) -> Self:
+    def __or__(self, other: Self | bool | Scalar) -> Self:
         """Apply logical 'or' to `other` Column (or scalar) and this column.
 
         Nulls should follow Kleene Logic.
@@ -508,7 +503,7 @@ def __invert__(self) -> Self:
         """
         ...
 
-    def any(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def any(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a bool.
 
         Raises
@@ -518,7 +513,7 @@ def any(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """
         ...
 
-    def all(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def all(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a bool.
 
         Raises
@@ -528,7 +523,7 @@ def all(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """
         ...
 
-    def min(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def min(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Any data type that supports comparisons
@@ -536,7 +531,7 @@ def min(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """
         ...
 
-    def max(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def max(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Any data type that supports comparisons
@@ -544,7 +539,7 @@ def max(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """
         ...
 
-    def sum(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def sum(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -553,7 +548,7 @@ def sum(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """
         ...
 
-    def prod(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def prod(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical data types.
@@ -561,7 +556,7 @@ def prod(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """
         ...
 
-    def median(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def median(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -571,7 +566,7 @@ def median(self, *, skip_nulls: BoolScalar = True) -> Scalar:
         """
         ...
 
-    def mean(self, *, skip_nulls: BoolScalar = True) -> Scalar:
+    def mean(self, *, skip_nulls: bool | Scalar = True) -> Scalar:
         """Reduction returns a scalar.
 
         Must be supported for numerical and
@@ -585,7 +580,7 @@ def std(
         self,
         *,
         correction: float = 1,
-        skip_nulls: BoolScalar = True,
+        skip_nulls: bool | Scalar = True,
     ) -> Scalar:
         """Reduction returns a scalar.
 
@@ -617,8 +612,8 @@ def std(
     def var(
         self,
         *,
-        correction: NumericScalar = 1,
-        skip_nulls: BoolScalar = True,
+        correction: float | Scalar = 1,
+        skip_nulls: bool | Scalar = True,
     ) -> Scalar:
         """Reduction returns a scalar.
 
@@ -728,7 +723,7 @@ def is_in(self, values: Self) -> Self:
         """
         ...
 
-    def unique_indices(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def unique_indices(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Return indices corresponding to unique values in Column.
 
         Returns
@@ -748,7 +743,7 @@ def unique_indices(self, *, skip_nulls: BoolScalar = True) -> Self:
         """
         ...
 
-    def fill_nan(self, value: FloatScalar | NullType, /) -> Self:
+    def fill_nan(self, value: float | NullType | Scalar, /) -> Self:
         """Fill floating point ``nan`` values with the given fill value.
 
         Parameters
@@ -807,7 +802,7 @@ def to_array(self) -> Any:
         """
         ...
 
-    def rename(self, name: StringScalar) -> Self:
+    def rename(self, name: str | Scalar) -> Self:
         """Rename column.
 
         Parameters
@@ -822,7 +817,7 @@ def rename(self, name: StringScalar) -> Self:
         """
         ...
 
-    def shift(self, offset: IntScalar) -> Self:
+    def shift(self, offset: int | Scalar) -> Self:
         """Shift values by `offset` positions, filling missing values with `null`.
 
         For example, if the original column contains values `[1, 4, 2]`, then:
@@ -913,7 +908,7 @@ def iso_weekday(self) -> Self:
         """
         ...
 
-    def unix_timestamp(self, *, time_unit: StringScalar = "s") -> Self:
+    def unix_timestamp(self, *, time_unit: str | Scalar = "s") -> Self:
         """Return number of seconds / milliseconds / microseconds since the Unix epoch.
 
         The Unix epoch is 00:00:00 UTC on 1 January 1970.
diff --git a/spec/API_specification/dataframe_api/dataframe_object.py b/spec/API_specification/dataframe_api/dataframe_object.py
index 199c0bd2..842c05b2 100644
--- a/spec/API_specification/dataframe_api/dataframe_object.py
+++ b/spec/API_specification/dataframe_api/dataframe_object.py
@@ -11,12 +11,10 @@
     from .groupby_object import GroupBy
     from .typing import (
         AnyScalar,
-        BoolScalar,
         DType,
-        FloatScalar,
         Namespace,
         NullType,
-        NumericScalar,
+        Scalar,
         SupportsDataFrameAPI,
     )
 
@@ -650,7 +648,7 @@ def __iter__(self) -> NoReturn:
         """
         raise NotImplementedError("'__iter__' is intentionally not implemented.")
 
-    def any(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def any(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Raises
@@ -660,7 +658,7 @@ def any(self, *, skip_nulls: BoolScalar = True) -> Self:
         """
         ...
 
-    def all(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def all(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame.
 
         Raises
@@ -670,7 +668,7 @@ def all(self, *, skip_nulls: BoolScalar = True) -> Self:
         """
         ...
 
-    def any_rowwise(self, *, skip_nulls: BoolScalar = True) -> Column:
+    def any_rowwise(self, *, skip_nulls: bool | Scalar = True) -> Column:
         """Reduction returns a Column.
 
         Differs from ``DataFrame.any`` and that the reduction happens
@@ -683,7 +681,7 @@ def any_rowwise(self, *, skip_nulls: BoolScalar = True) -> Column:
         """
         ...
 
-    def all_rowwise(self, *, skip_nulls: BoolScalar = True) -> Column:
+    def all_rowwise(self, *, skip_nulls: bool | Scalar = True) -> Column:
         """Reduction returns a Column.
 
         Differs from ``DataFrame.all`` and that the reduction happens
@@ -696,32 +694,35 @@ def all_rowwise(self, *, skip_nulls: BoolScalar = True) -> Column:
         """
         ...
 
-    def min(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def min(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def max(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def max(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def sum(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def sum(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def prod(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def prod(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def median(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def median(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
-    def mean(self, *, skip_nulls: BoolScalar = True) -> Self:
+    def mean(self, *, skip_nulls: bool | Scalar = True) -> Self:
         """Reduction returns a 1-row DataFrame."""
         ...
 
     def std(
-        self, *, correction: NumericScalar = 1, skip_nulls: BoolScalar = True,
+        self,
+        *,
+        correction: float | Scalar = 1,
+        skip_nulls: bool | Scalar = True,
     ) -> Self:
         """Reduction returns a 1-row DataFrame.
 
@@ -737,7 +738,10 @@ def std(
         ...
 
     def var(
-        self, *, correction: NumericScalar = 1, skip_nulls: BoolScalar = True,
+        self,
+        *,
+        correction: float | Scalar = 1,
+        skip_nulls: bool | Scalar = True,
     ) -> Self:
         """Reduction returns a 1-row DataFrame.
 
@@ -790,7 +794,7 @@ def is_nan(self) -> Self:
         """
         ...
 
-    def unique_indices(self, *keys: str, skip_nulls: BoolScalar = True) -> Column:
+    def unique_indices(self, *keys: str, skip_nulls: bool | Scalar = True) -> Column:
         """Return indices corresponding to unique values across selected columns.
 
         Parameters
@@ -816,7 +820,7 @@ def unique_indices(self, *keys: str, skip_nulls: BoolScalar = True) -> Column:
         """
         ...
 
-    def fill_nan(self, value: FloatScalar | NullType, /) -> Self:
+    def fill_nan(self, value: float | NullType | Scalar, /) -> Self:
         """Fill ``nan`` values with the given fill value.
 
         The fill operation will apply to all columns with a floating-point
diff --git a/spec/API_specification/dataframe_api/groupby_object.py b/spec/API_specification/dataframe_api/groupby_object.py
index e01b477a..8b26eff3 100644
--- a/spec/API_specification/dataframe_api/groupby_object.py
+++ b/spec/API_specification/dataframe_api/groupby_object.py
@@ -4,7 +4,7 @@
 
 if TYPE_CHECKING:
     from .dataframe_object import DataFrame
-    from .typing import BoolScalar, NumericScalar, StringScalar
+    from .typing import Scalar
 
 
 __all__ = [
@@ -23,43 +23,43 @@ class GroupBy(Protocol):
 
     """
 
-    def any(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def any(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
-    def all(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def all(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
-    def min(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def min(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
-    def max(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def max(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
-    def sum(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def sum(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
-    def prod(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def prod(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
-    def median(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def median(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
-    def mean(self, *, skip_nulls: BoolScalar = True) -> DataFrame:
+    def mean(self, *, skip_nulls: bool | Scalar = True) -> DataFrame:
         ...
 
     def std(
         self,
         *,
-        correction: NumericScalar = 1,
-        skip_nulls: BoolScalar = True,
+        correction: float | Scalar = 1,
+        skip_nulls: bool | Scalar = True,
     ) -> DataFrame:
         ...
 
     def var(
         self,
         *,
-        correction: NumericScalar = 1,
-        skip_nulls: BoolScalar = True,
+        correction: float | Scalar = 1,
+        skip_nulls: bool | Scalar = True,
     ) -> DataFrame:
         ...
 
@@ -85,7 +85,7 @@ def aggregate(self, *aggregation: Aggregation) -> DataFrame:
 
 
 class Aggregation(Protocol):
-    def rename(self, name: StringScalar) -> Aggregation:
+    def rename(self, name: str | Scalar) -> Aggregation:
         """Assign given name to output of aggregation.
 
         If not called, the column's name will be used as the output name.
@@ -93,35 +93,35 @@ def rename(self, name: StringScalar) -> Aggregation:
         ...
 
     @classmethod
-    def any(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def any(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def all(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def all(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def min(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def min(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def max(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def max(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def sum(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def sum(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def prod(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def prod(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def median(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def median(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
-    def mean(cls, column: str, *, skip_nulls: BoolScalar = True) -> Aggregation:
+    def mean(cls, column: str, *, skip_nulls: bool | Scalar = True) -> Aggregation:
         ...
 
     @classmethod
@@ -129,8 +129,8 @@ def std(
         cls,
         column: str,
         *,
-        correction: NumericScalar = 1,
-        skip_nulls: BoolScalar = True,
+        correction: float | Scalar = 1,
+        skip_nulls: bool | Scalar = True,
     ) -> Aggregation:
         ...
 
@@ -139,8 +139,8 @@ def var(
         cls,
         column: str,
         *,
-        correction: NumericScalar = 1,
-        skip_nulls: BoolScalar = True,
+        correction: float | Scalar = 1,
+        skip_nulls: bool | Scalar = True,
     ) -> Aggregation:
         ...
 
diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
index 2f0012a0..3bbd8d0d 100644
--- a/spec/API_specification/dataframe_api/scalar_object.py
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -93,7 +93,7 @@ def __bool__(self) -> bool:
 
     @property
     def dtype(self) -> DType:
-        """Return data type of column."""
+        """Return data type of scalar."""
         ...
 
     def persist(self) -> object:
diff --git a/spec/API_specification/dataframe_api/typing.py b/spec/API_specification/dataframe_api/typing.py
index 10c111a6..574c9480 100644
--- a/spec/API_specification/dataframe_api/typing.py
+++ b/spec/API_specification/dataframe_api/typing.py
@@ -168,11 +168,6 @@ def __column_consortium_standard__(
 
 
 PythonScalar = Union[str, int, float, bool]
-BoolScalar = Union[bool, Scalar]
-FloatScalar = Union[float, Scalar]
-IntScalar = Union[int, Scalar]
-NumericScalar = Union[FloatScalar, IntScalar]
-StringScalar = Union[str, Scalar]
 AnyScalar = Union[PythonScalar, Scalar]
 NullType = Namespace.NullType
 
@@ -183,14 +178,8 @@ def __column_consortium_standard__(
     "DType",
     "GroupBy",
     "Namespace",
+    "AnyScalar",
     "Scalar",
-    "PythonScalar",
-    "FloatScalar",
-    "IntScalar",
-    "BoolScalar",
-    "StringScalar",
-    "IntScalar",
-    "NumericScalar",
     "NullType",
     "SupportsColumnAPI",
     "SupportsDataFrameAPI",
diff --git a/spec/conf.py b/spec/conf.py
index 8f9dd5ff..8690fb84 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -84,10 +84,10 @@
     ('py:class', 'Scalar'),
     ('py:class', 'AnyScalar'),
     ('py:class', 'BoolScalar'),
-    ('py:class', 'FloatScalar'),
+    ('py:class', 'float | Scalar'),
     ('py:class', 'NullType'),
-    ('py:class', 'StringScalar'),
-    ('py:class', 'IntScalar'),
+    ('py:class', 'str | Scalar'),
+    ('py:class', 'int | Scalar'),
     ('py:class', 'NumericScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 9e9fc323..11d5db54 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -18,7 +18,7 @@ class DataFrame:
         ...
 
 class Column:
-    def mean(self, skip_nulls: BoolScalar = True) -> Scalar:
+    def mean(self, skip_nulls: bool | Scalar = True) -> Scalar:
         ...
 
 larger = df2 > df1.col('foo').mean()

From bee402f2335d0202967c4dac5a8ffcdb6c4b741c Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 20:21:53 +0000
Subject: [PATCH 21/26] update python builtin types desc

---
 .../dataframe_api/scalar_object.py            |  8 +++++++
 spec/design_topics/python_builtin_types.md    | 22 ++++++++++---------
 2 files changed, 20 insertions(+), 10 deletions(-)

diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
index 3bbd8d0d..0642c26e 100644
--- a/spec/API_specification/dataframe_api/scalar_object.py
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -14,6 +14,14 @@ class Scalar(Protocol):
     Not meant to be instantiated directly, but rather created via
     `:meth:Column.get_value` or one of the column reductions such
     as `:meth:`Column.sum`.
+
+    Note that, just like how `:class:Column`s can hold null values,
+    a `Scalar` can also be backed by a null value. Given that `Scalar`s
+    aren't instantiated directly, but rather derived from existing
+    `Column`s, `Scalar.dtype` is determined by the parent `Column`.
+    For example, if `column` is `Column` of dtype `Int64`, then
+    `column.get_value(0)` will return a `Scalar` of dtype `Int64`
+    (even if it is backed by a null value).
     """
 
     def __lt__(self, other: AnyScalar) -> Scalar:
diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index 11d5db54..be2cde0c 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -6,9 +6,9 @@ it is also potentially problematic when trying to write performant dataframe
 library code or supporting devices other than CPU.
 
 This standard specifies the use of Python types in quite a few places, and uses
-them as type annotations. As a concrete example, consider the `mean` method and
-the `float` it is documented to return, in combination with the `__gt__` method
-(i.e., the `>` operator) on the dataframe:
+them as type annotations. As a concrete example, consider the `mean` method,
+the `bool | Scalar` argument it takes, and the `Scalar` it is documented to return,
+in combination with the `__gt__` method (i.e., the `>` operator) on the dataframe:
 
 ```python
 class DataFrame:
@@ -21,15 +21,17 @@ class Column:
     def mean(self, skip_nulls: bool | Scalar = True) -> Scalar:
         ...
 
-larger = df2 > df1.col('foo').mean()
+larger = df2 > df1.col('foo', skip_nulls = True).mean()
 ```
 
-For a GPU dataframe library, it is desirable for all data to reside on the GPU,
-and not incur a performance penalty from synchronizing instances of Python
-builtin types to CPU. In the above example, the `.mean()` call returns a
-`Scalar`. It is likely beneficial though to implement this as a library-specific
-scalar object which (partially) duck types with `float`. The required methods it
-must implement are listed in the spec for class `Scalar`.
+Let's go through these arguments:
+- `skip_nulls: bool | Scalar`. This means we can either pass a Python `bool`, or
+  a `Scalar` object backed by a boolean;
+- the return value of `.mean()` is a `Scalar`
+- the return value of `__gt__` is also a `Scalar`.
+
+This allows scalars to reside on different devices (e.g. GPU), or to stay lazy
+(if a library allows that).
 
 ## Example
 

From 15090ac6e6ebc20cd93ee4241453465d995bbee8 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 20:23:23 +0000
Subject: [PATCH 22/26] fixup

---
 spec/API_specification/dataframe_api/column_object.py | 4 +---
 1 file changed, 1 insertion(+), 3 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 841b8837..7d70eb1b 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -5,11 +5,9 @@
 if TYPE_CHECKING:
     from typing_extensions import Self
 
-    from dataframe_api.dataframe_object import DataFrame
-
-    from .scalar_object import Scalar
     from .typing import (
         AnyScalar,
+        DataFrame,
         DType,
         Namespace,
         NullType,

From 24d2ad83b1ae7ff40fa2881426ff13471b66495c Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 20:35:30 +0000
Subject: [PATCH 23/26] enable extra ruff rule, note AnyScalar

---
 spec/API_specification/pyproject.toml      | 1 -
 spec/design_topics/python_builtin_types.md | 9 ++++++---
 2 files changed, 6 insertions(+), 4 deletions(-)

diff --git a/spec/API_specification/pyproject.toml b/spec/API_specification/pyproject.toml
index 8b88d455..31c5f2ad 100644
--- a/spec/API_specification/pyproject.toml
+++ b/spec/API_specification/pyproject.toml
@@ -47,5 +47,4 @@ ignore = [
     "N999", # invalid-module-name
     "PD901", # pandas-df-variable-name
     "PLR0913",   # too-many-arguments
-    "PYI041",  # redundant-numeric-union
 ]
diff --git a/spec/design_topics/python_builtin_types.md b/spec/design_topics/python_builtin_types.md
index be2cde0c..ee0c3541 100644
--- a/spec/design_topics/python_builtin_types.md
+++ b/spec/design_topics/python_builtin_types.md
@@ -28,10 +28,13 @@ Let's go through these arguments:
 - `skip_nulls: bool | Scalar`. This means we can either pass a Python `bool`, or
   a `Scalar` object backed by a boolean;
 - the return value of `.mean()` is a `Scalar`
-- the return value of `__gt__` is also a `Scalar`.
+- the argument `other` of `__gt__` is typed as `AnyScalar`, meaning that we can
+  compare a `DataFrame` with a Python scalar (e.g. `df > 3`) or with a `Scalar`
+  (e.g. `df > df.col('a').mean()`)
+- the return value of `__gt__` is a `Scalar`
 
-This allows scalars to reside on different devices (e.g. GPU), or to stay lazy
-(if a library allows that).
+Returning values as `Scalar` allows scalars to reside on different devices (e.g. GPU),
+or to stay lazy (if a library allows it).
 
 ## Example
 

From 8360d96da4fe20762f7a92764f47415ea7f0052d Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Wed, 15 Nov 2023 20:36:28 +0000
Subject: [PATCH 24/26] remove some unnecessary nitpick ignores

---
 spec/conf.py | 5 -----
 1 file changed, 5 deletions(-)

diff --git a/spec/conf.py b/spec/conf.py
index 8690fb84..a89cfee5 100644
--- a/spec/conf.py
+++ b/spec/conf.py
@@ -83,12 +83,7 @@
     ('py:class', 'ellipsis'),
     ('py:class', 'Scalar'),
     ('py:class', 'AnyScalar'),
-    ('py:class', 'BoolScalar'),
-    ('py:class', 'float | Scalar'),
     ('py:class', 'NullType'),
-    ('py:class', 'str | Scalar'),
-    ('py:class', 'int | Scalar'),
-    ('py:class', 'NumericScalar'),
     ('py:class', 'Bool'),
     ('py:class', 'optional'),
     ('py:class', 'Aggregation'),

From f69679aa0aa66481d04b87e00b04e00c0e573102 Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Fri, 17 Nov 2023 12:39:21 +0000
Subject: [PATCH 25/26] return Self from Scalar.persist, add column.persist

---
 .../dataframe_api/column_object.py                  | 13 +++++++++++++
 .../dataframe_api/scalar_object.py                  |  4 ++--
 2 files changed, 15 insertions(+), 2 deletions(-)

diff --git a/spec/API_specification/dataframe_api/column_object.py b/spec/API_specification/dataframe_api/column_object.py
index 7d70eb1b..9ae7de4a 100644
--- a/spec/API_specification/dataframe_api/column_object.py
+++ b/spec/API_specification/dataframe_api/column_object.py
@@ -1044,3 +1044,16 @@ def unix_timestamp(self, *, time_unit: str | Scalar = "s") -> Self:
             discarded.
         """
         ...
+
+    def persist(self) -> Self:
+        """Hint that computation prior to this point should not be repeated.
+
+        This is intended as a hint, rather than as a directive. Implementations
+        which do not separate lazy vs eager execution may ignore this method and
+        treat it as a no-op.
+
+        .. note::
+            This method may trigger execution. If necessary, it should be called
+            at most once per dataframe, and as late as possible in the pipeline.
+        """
+        ...
diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
index 0642c26e..d9d8c883 100644
--- a/spec/API_specification/dataframe_api/scalar_object.py
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -1,6 +1,6 @@
 from __future__ import annotations
 
-from typing import TYPE_CHECKING, Protocol
+from typing import TYPE_CHECKING, Protocol, Any
 
 if TYPE_CHECKING:
     from dataframe_api.typing import AnyScalar, DType
@@ -104,7 +104,7 @@ def dtype(self) -> DType:
         """Return data type of scalar."""
         ...
 
-    def persist(self) -> object:
+    def persist(self) -> Self:
         """Hint that computation prior to this point should not be repeated.
 
         This is intended as a hint, rather than as a directive. Implementations

From 216b5e61b44cbbde294c2f95bc676c0d8ec2316e Mon Sep 17 00:00:00 2001
From: MarcoGorelli <33491632+MarcoGorelli@users.noreply.github.com>
Date: Fri, 17 Nov 2023 12:40:01 +0000
Subject: [PATCH 26/26] fixup

---
 spec/API_specification/dataframe_api/scalar_object.py | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/spec/API_specification/dataframe_api/scalar_object.py b/spec/API_specification/dataframe_api/scalar_object.py
index d9d8c883..078cf2a4 100644
--- a/spec/API_specification/dataframe_api/scalar_object.py
+++ b/spec/API_specification/dataframe_api/scalar_object.py
@@ -1,8 +1,10 @@
 from __future__ import annotations
 
-from typing import TYPE_CHECKING, Protocol, Any
+from typing import TYPE_CHECKING, Protocol
 
 if TYPE_CHECKING:
+    from typing_extensions import Self
+
     from dataframe_api.typing import AnyScalar, DType
 
 __all__ = ["Scalar"]