From b19a1ffe144f09f01aea3134630a2a8262c41c32 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Wed, 17 Aug 2022 11:46:48 -0700
Subject: [PATCH 01/10] src/sage/sets/set.py: Accept and handle 'category'
 arguments

---
 src/sage/sets/set.py | 50 ++++++++++++++++++++++++++------------------
 1 file changed, 30 insertions(+), 20 deletions(-)

diff --git a/src/sage/sets/set.py b/src/sage/sets/set.py
index 96e3b2ad7a4..3e034bd7125 100644
--- a/src/sage/sets/set.py
+++ b/src/sage/sets/set.py
@@ -48,6 +48,7 @@
 
 from sage.categories.sets_cat import Sets
 from sage.categories.enumerated_sets import EnumeratedSets
+from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
 
 import sage.rings.infinity
 
@@ -85,7 +86,7 @@ def has_finite_length(obj):
         return True
 
 
-def Set(X=None):
+def Set(X=None, category=None):
     r"""
     Create the underlying set of ``X``.
 
@@ -187,12 +188,12 @@ def Set(X=None):
     if X is None:
         X = []
     elif isinstance(X, CategoryObject):
-        if isinstance(X, Set_generic):
+        if isinstance(X, Set_generic) and category is None:
             return X
         elif X in Sets().Finite():
-            return Set_object_enumerated(X)
+            return Set_object_enumerated(X, category=category)
         else:
-            return Set_object(X)
+            return Set_object(X, category=category)
 
     if isinstance(X, Element) and not isinstance(X, Set_base):
         raise TypeError("Element has no defined underlying set")
@@ -200,9 +201,9 @@ def Set(X=None):
     try:
         X = frozenset(X)
     except TypeError:
-        return Set_object(X)
+        return Set_object(X, category=category)
     else:
-        return Set_object_enumerated(X)
+        return Set_object_enumerated(X, category=category)
 
 
 class Set_base():
@@ -474,7 +475,7 @@ def __init__(self, X, category=None):
             sage: type(Set(QQ))
             <class 'sage.sets.set.Set_object_with_category'>
             sage: Set(QQ).category()
-            Category of sets
+            Category of infinite sets
 
         TESTS::
 
@@ -492,6 +493,15 @@ def __init__(self, X, category=None):
 
         if category is None:
             category = Sets()
+
+        if isinstance(X, CategoryObject):
+            if X in Sets().Finite():
+                category = category.Finite()
+            elif X in Sets().Infinite():
+                category = category.Infinite()
+            if X in Sets().Enumerated():
+                category = category.Enumerated()
+
         Parent.__init__(self, category=category)
         self.__object = X
 
@@ -830,7 +840,7 @@ class Set_object_enumerated(Set_object):
     """
     A finite enumerated set.
     """
-    def __init__(self, X):
+    def __init__(self, X, category=None):
         r"""
         Initialize ``self``.
 
@@ -839,12 +849,12 @@ def __init__(self, X):
             sage: S = Set(GF(19)); S
             {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}
             sage: S.category()
-            Category of finite sets
+            Category of finite enumerated sets
             sage: print(latex(S))
             \left\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18\right\}
             sage: TestSuite(S).run()
         """
-        Set_object.__init__(self, X, category=Sets().Finite())
+        Set_object.__init__(self, X, category=FiniteEnumeratedSets().or_subcategory(category))
 
     def random_element(self):
         r"""
@@ -1267,7 +1277,7 @@ def __classcall__(cls, X, Y, *args, **kwds):
             Y = Set(Y)
         return type.__call__(cls, X, Y, *args, **kwds)
 
-    def __init__(self, X, Y, op, latex_op):
+    def __init__(self, X, Y, op, latex_op, category=None):
         r"""
         Initialization.
 
@@ -1284,7 +1294,7 @@ def __init__(self, X, Y, op, latex_op):
         self._Y = Y
         self._op = op
         self._latex_op = latex_op
-        Set_object.__init__(self, self)
+        Set_object.__init__(self, self, category=category)
 
     def _repr_(self):
         r"""
@@ -1338,7 +1348,7 @@ class Set_object_union(Set_object_binary):
     """
     A formal union of two sets.
     """
-    def __init__(self, X, Y):
+    def __init__(self, X, Y, category=None):
         r"""
         Initialize ``self``.
 
@@ -1354,7 +1364,7 @@ def __init__(self, X, Y):
 
             sage: TestSuite(X).run()
         """
-        Set_object_binary.__init__(self, X, Y, "union", "\\cup")
+        Set_object_binary.__init__(self, X, Y, "union", "\\cup", category=category)
 
     def is_finite(self):
         r"""
@@ -1481,7 +1491,7 @@ class Set_object_intersection(Set_object_binary):
     """
     Formal intersection of two sets.
     """
-    def __init__(self, X, Y):
+    def __init__(self, X, Y, category=None):
         r"""
         Initialize ``self``.
 
@@ -1499,7 +1509,7 @@ def __init__(self, X, Y):
             True
             sage: TestSuite(X).run()
         """
-        Set_object_binary.__init__(self, X, Y, "intersection", "\\cap")
+        Set_object_binary.__init__(self, X, Y, "intersection", "\\cap", category=category)
 
     def is_finite(self):
         r"""
@@ -1643,7 +1653,7 @@ class Set_object_difference(Set_object_binary):
     """
     Formal difference of two sets.
     """
-    def __init__(self, X, Y):
+    def __init__(self, X, Y, category=None):
         r"""
         Initialize ``self``.
 
@@ -1658,7 +1668,7 @@ def __init__(self, X, Y):
 
             sage: TestSuite(X).run()
         """
-        Set_object_binary.__init__(self, X, Y, "difference", "-")
+        Set_object_binary.__init__(self, X, Y, "difference", "-", category=category)
 
     def is_finite(self):
         r"""
@@ -1807,7 +1817,7 @@ class Set_object_symmetric_difference(Set_object_binary):
     """
     Formal symmetric difference of two sets.
     """
-    def __init__(self, X, Y):
+    def __init__(self, X, Y, category=None):
         r"""
         Initialize ``self``.
 
@@ -1822,7 +1832,7 @@ def __init__(self, X, Y):
 
             sage: TestSuite(X).run()
         """
-        Set_object_binary.__init__(self, X, Y, "symmetric difference", "\\bigtriangleup")
+        Set_object_binary.__init__(self, X, Y, "symmetric difference", "\\bigtriangleup", category=category)
 
     def is_finite(self):
         r"""

From d461411617f8bca18f3e9575daaa3adf3bed1551 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Wed, 17 Aug 2022 13:35:07 -0700
Subject: [PATCH 02/10] Update doctest outputs

---
 src/sage/categories/cartesian_product.py | 2 +-
 src/sage/categories/enumerated_sets.py   | 2 +-
 src/sage/categories/homset.py            | 2 +-
 src/sage/combinat/sidon_sets.py          | 4 ++--
 4 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/sage/categories/cartesian_product.py b/src/sage/categories/cartesian_product.py
index 33f5c5b9b02..e6fbf670750 100644
--- a/src/sage/categories/cartesian_product.py
+++ b/src/sage/categories/cartesian_product.py
@@ -153,7 +153,7 @@ def __call__(self, args, **kwds):
             sage: cartesian_product([set([0,1,2]), [0,1]])
             The Cartesian product of ({0, 1, 2}, {0, 1})
             sage: _.category()
-            Category of Cartesian products of sets
+            Category of Cartesian products of finite enumerated sets
 
         Check that the empty product is handled correctly:
 
diff --git a/src/sage/categories/enumerated_sets.py b/src/sage/categories/enumerated_sets.py
index 563a8025e6d..e1dcbd86839 100644
--- a/src/sage/categories/enumerated_sets.py
+++ b/src/sage/categories/enumerated_sets.py
@@ -137,7 +137,7 @@ def _call_(self, X):
             sage: S = EnumeratedSets()(Set([1, 2, 3])); S
             {1, 2, 3}
             sage: S.category()
-            Category of facade finite enumerated sets
+            Category of finite enumerated sets
 
         Also Python3 range are now accepted::
 
diff --git a/src/sage/categories/homset.py b/src/sage/categories/homset.py
index bfd78e8d8a5..219286af8b9 100644
--- a/src/sage/categories/homset.py
+++ b/src/sage/categories/homset.py
@@ -828,7 +828,7 @@ def _element_constructor_(self, x, check=None, **options):
             sage: H = Hom(Set([1,2,3]), Set([1,2,3]))
             sage: f = H( lambda x: 4-x )
             sage: f.parent()
-            Set of Morphisms from {1, 2, 3} to {1, 2, 3} in Category of finite sets
+            Set of Morphisms from {1, 2, 3} to {1, 2, 3} in Category of finite enumerated sets
             sage: f(1), f(2), f(3) # todo: not implemented
 
             sage: H = Hom(ZZ, QQ, Sets())
diff --git a/src/sage/combinat/sidon_sets.py b/src/sage/combinat/sidon_sets.py
index 3f5cfcb3173..ec5038eb2c4 100644
--- a/src/sage/combinat/sidon_sets.py
+++ b/src/sage/combinat/sidon_sets.py
@@ -46,12 +46,12 @@ def sidon_sets(N, g = 1):
         sage: S.cardinality()
         8
         sage: S.category()
-        Category of finite sets
+        Category of finite enumerated sets
         sage: sid = S.an_element()
         sage: sid
         {2}
         sage: sid.category()
-        Category of finite sets
+        Category of finite enumerated sets
 
     TESTS::
 

From df327b9d5d76bc5b9807a937b7bb04ed334b60ae Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Wed, 17 Aug 2022 17:26:27 -0700
Subject: [PATCH 03/10] src/sage/sets/set.py: Refine categories of unions,
 intersections, differences

---
 src/sage/sets/set.py | 63 ++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 63 insertions(+)

diff --git a/src/sage/sets/set.py b/src/sage/sets/set.py
index 3e034bd7125..5d2380b3d24 100644
--- a/src/sage/sets/set.py
+++ b/src/sage/sets/set.py
@@ -1358,12 +1358,22 @@ def __init__(self, X, Y, category=None):
             sage: T = Set(ZZ)
             sage: X = S.union(T); X
             Set-theoretic union of Set of elements of Vector space of dimension 2 over Rational Field and Set of elements of Integer Ring
+            sage: X.category()
+            Category of infinite sets
 
             sage: latex(X)
             \Bold{Q}^{2} \cup \Bold{Z}
 
             sage: TestSuite(X).run()
         """
+        if category is None:
+            category = Sets()
+        if all(S in Sets().Finite() for S in (X, Y)):
+            category = category.Finite()
+        if all(S in Sets().Enumerated() for S in (X, Y)):
+            category = category.Enumerated()
+        if any(S in Sets().Infinite() for S in (X, Y)):
+            category = category.Infinite()
         Set_object_binary.__init__(self, X, Y, "union", "\\cup", category=category)
 
     def is_finite(self):
@@ -1501,16 +1511,47 @@ def __init__(self, X, Y, category=None):
             sage: T = Set(ZZ)
             sage: X = S.intersection(T); X
             Set-theoretic intersection of Set of elements of Vector space of dimension 2 over Rational Field and Set of elements of Integer Ring
+            sage: X.category()
+            Category of enumerated sets
             sage: latex(X)
             \Bold{Q}^{2} \cap \Bold{Z}
 
             sage: X = Set(IntegerRange(100)).intersection(Primes())
             sage: X.is_finite()
             True
+            sage: X.category()
+            Category of finite enumerated sets
             sage: TestSuite(X).run()
         """
+        if category is None:
+            category = Sets()
+        if any(S in Sets().Finite() for S in (X, Y)):
+            category = category.Finite()
+        if any(S in Sets().Enumerated() for S in (X, Y)):
+            category = category.Enumerated()
         Set_object_binary.__init__(self, X, Y, "intersection", "\\cap", category=category)
 
+    def cardinality(self):
+        r"""
+        Return the cardinality of this set.
+
+        EXAMPLES::
+
+            sage: X = Set(IntegerRange(100)).intersection(Primes())
+            sage: X.cardinality()
+            25
+
+            sage: X = Set(Primes(), category=Sets()).intersection(Set(IntegerRange(200)))
+            sage: X.cardinality()
+            46
+        """
+        if self in Sets().Infinite():
+            return Infinity
+        if self in Sets().Finite():
+            from sage.rings.integer import Integer
+            return Integer(len(list(iter(self))))
+        return super().cardinality()
+
     def is_finite(self):
         r"""
         Return whether this set is finite.
@@ -1663,11 +1704,21 @@ def __init__(self, X, Y, category=None):
             sage: T = Set(ZZ)
             sage: X = S.difference(T); X
             Set-theoretic difference of Set of elements of Rational Field and Set of elements of Integer Ring
+            sage: X.category()
+            Category of sets
             sage: latex(X)
             \Bold{Q} - \Bold{Z}
 
             sage: TestSuite(X).run()
         """
+        if category is None:
+            category = Sets()
+        if X in Sets().Finite():
+            category = category.Finite()
+        if X in Sets().Enumerated():
+            category = category.Enumerated()
+        if X in Sets().Infinite() and Y in Sets().Finite():
+            category = category.Infinite()
         Set_object_binary.__init__(self, X, Y, "difference", "-", category=category)
 
     def is_finite(self):
@@ -1797,6 +1848,8 @@ def _sympy_(self):
             Set-theoretic difference of
              Set of elements of Rational Field and
              Set of elements of Integer Ring
+            sage: X.category()
+            Category of sets
             sage: X._sympy_()
             Complement(Rationals, Integers)
 
@@ -1804,6 +1857,8 @@ def _sympy_(self):
             Set-theoretic difference of
              Set of elements of Integer Ring and
              Set of elements of Rational Field
+            sage: X.category()
+            Category of enumerated sets
             sage: X._sympy_()
             EmptySet
         """
@@ -1827,11 +1882,19 @@ def __init__(self, X, Y, category=None):
             sage: T = Set(ZZ)
             sage: X = S.symmetric_difference(T); X
             Set-theoretic symmetric difference of Set of elements of Rational Field and Set of elements of Integer Ring
+            sage: X.category()
+            Category of sets
             sage: latex(X)
             \Bold{Q} \bigtriangleup \Bold{Z}
 
             sage: TestSuite(X).run()
         """
+        if category is None:
+            category = Sets()
+        if all(S in Sets().Finite() for S in (X, Y)):
+            category = category.Finite()
+        if all(S in Sets().Enumerated() for S in (X, Y)):
+            category = category.Enumerated()
         Set_object_binary.__init__(self, X, Y, "symmetric difference", "\\bigtriangleup", category=category)
 
     def is_finite(self):

From 8a304931879a0ce4292d284439c8e717540755cf Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Wed, 17 Aug 2022 20:31:44 -0700
Subject: [PATCH 04/10] src/sage/sets/set.py: Simplify category checks as
 suggested

---
 src/sage/sets/set.py | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/sage/sets/set.py b/src/sage/sets/set.py
index 5d2380b3d24..bd94a217e0d 100644
--- a/src/sage/sets/set.py
+++ b/src/sage/sets/set.py
@@ -1368,12 +1368,12 @@ def __init__(self, X, Y, category=None):
         """
         if category is None:
             category = Sets()
-        if all(S in Sets().Finite() for S in (X, Y)):
-            category = category.Finite()
         if all(S in Sets().Enumerated() for S in (X, Y)):
             category = category.Enumerated()
         if any(S in Sets().Infinite() for S in (X, Y)):
             category = category.Infinite()
+        elif all(S in Sets().Finite() for S in (X, Y)):
+            category = category.Finite()
         Set_object_binary.__init__(self, X, Y, "union", "\\cup", category=category)
 
     def is_finite(self):
@@ -1713,11 +1713,11 @@ def __init__(self, X, Y, category=None):
         """
         if category is None:
             category = Sets()
-        if X in Sets().Finite():
-            category = category.Finite()
         if X in Sets().Enumerated():
             category = category.Enumerated()
-        if X in Sets().Infinite() and Y in Sets().Finite():
+        if X in Sets().Finite():
+            category = category.Finite()
+        elif X in Sets().Infinite() and Y in Sets().Finite():
             category = category.Infinite()
         Set_object_binary.__init__(self, X, Y, "difference", "-", category=category)
 

From 141ebf14d80478128540f39ff1bac46e80eaca26 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Wed, 17 Aug 2022 20:39:59 -0700
Subject: [PATCH 05/10] src/sage/categories/sets_cat.py: Update doctest output

---
 src/sage/categories/sets_cat.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/categories/sets_cat.py b/src/sage/categories/sets_cat.py
index 09a57770a3b..4d7e0c11da4 100644
--- a/src/sage/categories/sets_cat.py
+++ b/src/sage/categories/sets_cat.py
@@ -247,7 +247,7 @@ def _call_(self, X, enumerated_set=False):
             {1, 2, 3}
 
             sage: S = Sets()([1, 2, 3]); S.category()
-            Category of finite sets
+            Category of finite enumerated sets
             sage: S = Sets()([1, 2, 3], enumerated_set=True); S.category()
             Category of facade finite enumerated sets
 

From 8aa274ef877e4c987daa7d20fa761d2e5af94567 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Wed, 17 Aug 2022 20:47:41 -0700
Subject: [PATCH 06/10] src/sage/categories/sets_cat.py: Fix doctest

---
 src/sage/categories/sets_cat.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/categories/sets_cat.py b/src/sage/categories/sets_cat.py
index 4d7e0c11da4..b6fc0779716 100644
--- a/src/sage/categories/sets_cat.py
+++ b/src/sage/categories/sets_cat.py
@@ -1361,7 +1361,7 @@ def some_elements(self):
                 sage: S.some_elements()
                 [47]
                 sage: S = Set([])
-                sage: S.some_elements()
+                sage: list(S.some_elements())
                 []
 
             This method should return an iterable, *not* an iterator.

From d711a61aeacde364b4ab76b59e55c2f01d8a90d7 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Wed, 17 Aug 2022 22:50:41 -0700
Subject: [PATCH 07/10] src/sage/structure/parent.pyx: Update doctest output

---
 src/sage/structure/parent.pyx | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/structure/parent.pyx b/src/sage/structure/parent.pyx
index 36b1e45d223..585ec9e559e 100644
--- a/src/sage/structure/parent.pyx
+++ b/src/sage/structure/parent.pyx
@@ -2863,7 +2863,7 @@ cdef class Set_generic(Parent):
     TESTS::
 
         sage: Set(QQ).category()
-        Category of sets
+        Category of infinite sets
 
     """
     def object(self):

From e63e4f0bb4950985b13c4fbcc4c76110efb338c5 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Thu, 18 Aug 2022 18:38:56 -0700
Subject: [PATCH 08/10] Set_object.is_finite: Check category

---
 src/sage/sets/set.py | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/src/sage/sets/set.py b/src/sage/sets/set.py
index bd94a217e0d..707f8887c72 100644
--- a/src/sage/sets/set.py
+++ b/src/sage/sets/set.py
@@ -743,6 +743,10 @@ def is_finite(self):
             sage: Set([1,'a',ZZ]).is_finite()
             True
         """
+        if self in Sets().Finite():
+            return True
+        if self in Sets().Infinite():
+            return False
         obj = self.__object
         try:
             is_finite = obj.is_finite

From 970d46772d84a0d31bc12517b16eb936be59a2c4 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Thu, 18 Aug 2022 18:47:32 -0700
Subject: [PATCH 09/10] Set_object.cardinality: Check category

---
 src/sage/sets/set.py | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/src/sage/sets/set.py b/src/sage/sets/set.py
index 707f8887c72..8c0668724d0 100644
--- a/src/sage/sets/set.py
+++ b/src/sage/sets/set.py
@@ -676,6 +676,9 @@ def cardinality(self):
             sage: Set(GF(5^2,'a')).cardinality()
             25
         """
+        if self in Sets().Infinite():
+            return sage.rings.infinity.infinity
+
         if not self.is_finite():
             return sage.rings.infinity.infinity
 

From e58d2abf9490496346fb4f114d98a9c9393a2b8d Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Thu, 18 Aug 2022 18:48:10 -0700
Subject: [PATCH 10/10] Set_object_intersection.cardinality: Remove; move
 delegation to super to Set_object

---
 src/sage/sets/set.py | 30 ++++++++----------------------
 1 file changed, 8 insertions(+), 22 deletions(-)

diff --git a/src/sage/sets/set.py b/src/sage/sets/set.py
index 8c0668724d0..b76e6b064ef 100644
--- a/src/sage/sets/set.py
+++ b/src/sage/sets/set.py
@@ -693,7 +693,7 @@ def cardinality(self):
             except TypeError:
                 pass
 
-        raise NotImplementedError("computation of cardinality of %s not yet implemented" % self.__object)
+        return super().cardinality()
 
     def is_empty(self):
         """
@@ -1526,9 +1526,16 @@ def __init__(self, X, Y, category=None):
             sage: X = Set(IntegerRange(100)).intersection(Primes())
             sage: X.is_finite()
             True
+            sage: X.cardinality()
+            25
             sage: X.category()
             Category of finite enumerated sets
             sage: TestSuite(X).run()
+
+            sage: X = Set(Primes(), category=Sets()).intersection(Set(IntegerRange(200)))
+            sage: X.cardinality()
+            46
+            sage: TestSuite(X).run()
         """
         if category is None:
             category = Sets()
@@ -1538,27 +1545,6 @@ def __init__(self, X, Y, category=None):
             category = category.Enumerated()
         Set_object_binary.__init__(self, X, Y, "intersection", "\\cap", category=category)
 
-    def cardinality(self):
-        r"""
-        Return the cardinality of this set.
-
-        EXAMPLES::
-
-            sage: X = Set(IntegerRange(100)).intersection(Primes())
-            sage: X.cardinality()
-            25
-
-            sage: X = Set(Primes(), category=Sets()).intersection(Set(IntegerRange(200)))
-            sage: X.cardinality()
-            46
-        """
-        if self in Sets().Infinite():
-            return Infinity
-        if self in Sets().Finite():
-            from sage.rings.integer import Integer
-            return Integer(len(list(iter(self))))
-        return super().cardinality()
-
     def is_finite(self):
         r"""
         Return whether this set is finite.