gh-41175: typing "__contains__" in categories,geometry,groups,monoids
    
similar to #41145

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
    
URL: #41175
Reported by: Frédéric Chapoton
Reviewer(s): Vincent Macri

Author: Release Manager

src/sage/categories/algebras.py

@@ -59,7 +59,7 @@ class Algebras(CategoryWithAxiom_over_base_ring):
     _base_category_class_and_axiom = (AssociativeAlgebras, 'Unital')
 
     # For backward compatibility?
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Membership testing.
 

src/sage/categories/category.py

@@ -684,7 +684,7 @@ def _subcategory_hook_(self, category):
         """
         return issubclass(category.parent_class, self.parent_class)
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Membership testing.
 

src/sage/categories/category_types.py

@@ -491,7 +491,7 @@ def _subcategory_hook_(self, C):
             return C.base() in base_ring
         return False
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Return whether ``x`` is an object of this category.
 
@@ -605,7 +605,7 @@ def ring(self):
         """
         return self.ambient()
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         EXAMPLES::
 

src/sage/categories/commutative_algebras.py

@@ -46,7 +46,7 @@ class CommutativeAlgebras(CategoryWithAxiom_over_base_ring):
      - coproduct ( = tensor product over base ring)
     """
 
-    def __contains__(self, A):
+    def __contains__(self, A) -> bool:
         """
         EXAMPLES::
 

src/sage/categories/examples/finite_coxeter_groups.py

@@ -103,7 +103,7 @@ def __init__(self, n=5):
         Parent.__init__(self, category=FiniteCoxeterGroups())
         self.n = n
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         EXAMPLES::
 
@@ -114,7 +114,7 @@ def _repr_(self):
         """
         return "The %s-th dihedral group of order %s" % (self.n, 2 * self.n)
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         r"""
         Check if the element ``x`` is in the mathematical parent ``self``.
 

src/sage/categories/examples/finite_enumerated_sets.py

@@ -81,7 +81,7 @@ def __init__(self):
         Parent.__init__(self, facade=IntegerRing(),
                         category=FiniteEnumeratedSets())
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         TESTS::
 
@@ -90,7 +90,7 @@ def _repr_(self):
         """
         return "An example of a finite enumerated set: {1,2,3}"
 
-    def __contains__(self, o):
+    def __contains__(self, o) -> bool:
         """
         EXAMPLES::
 
@@ -182,7 +182,7 @@ def retract(self, x):
         """
         return x ** 2
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Membership testing by checking whether the preimage by the
         bijection is in the ambient space.

src/sage/categories/examples/infinite_enumerated_sets.py

@@ -91,7 +91,7 @@ def __init__(self):
         """
         Parent.__init__(self, category=InfiniteEnumeratedSets())
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         TESTS::
 
@@ -100,7 +100,7 @@ def _repr_(self):
         """
         return "An example of an infinite enumerated set: the nonnegative integers"
 
-    def __contains__(self, elt):
+    def __contains__(self, elt) -> bool:
         """
         EXAMPLES::
 

src/sage/categories/examples/sets_cat.py

@@ -115,9 +115,9 @@ def an_element(self):
             sage: x.parent()
             Integer Ring
         """
-        return self(47) # if speed is needed, call: self.element_class(47)
+        return self(47)  # if speed is needed, call: self.element_class(47)
 
-    def __contains__(self, p):
+    def __contains__(self, p) -> bool:
         """
         TESTS::
 
@@ -393,7 +393,7 @@ def __init__(self):
         super().__init__()
         self._populate_coercion_lists_(embedding=IntegerRing())
 
-    def __contains__(self, p):
+    def __contains__(self, p) -> bool:
         """
         TESTS::
 
@@ -502,7 +502,7 @@ def __init__(self):
         self.mor = Hom(self, IntegerRing())(lambda z: z.value)
         self._populate_coercion_lists_(embedding=self.mor)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         TESTS::
 
@@ -511,7 +511,7 @@ def _repr_(self):
         """
         return "Set of prime numbers (wrapper implementation)"
 
-    def __contains__(self, p):
+    def __contains__(self, p) -> bool:
         """
         TESTS::
 
@@ -665,7 +665,7 @@ def __init__(self):
         """
         Parent.__init__(self, facade=IntegerRing(), category=Sets())
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         TESTS::
 
@@ -674,7 +674,7 @@ def _repr_(self):
         """
         return "Set of prime numbers (facade implementation)"
 
-    def __contains__(self, p):
+    def __contains__(self, p) -> bool:
         """
         TESTS::
 

src/sage/categories/facade_sets.py

@@ -180,7 +180,7 @@ def is_parent_of(self, element):
             from sage.structure.element import parent
             return parent(element) in parents
 
-        def __contains__(self, element):
+        def __contains__(self, element) -> bool:
             """
             Membership testing.
 

src/sage/categories/fields.py

@@ -61,7 +61,7 @@ def extra_super_categories(self):
         """
         return [EuclideanDomains(), NoetherianRings()]
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         EXAMPLES::