gh-36085: some details in coxeter groups
    
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this is making a few little changes in the file for the category of
Coxeter groups, also fixing a typo

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URL: #36085
Reported by: Frédéric Chapoton
Reviewer(s): David Coudert

Author: Release Manager

src/sage/categories/coxeter_groups.py

@@ -518,9 +518,10 @@ def succ(u):
 
             from sage.categories.finite_coxeter_groups import FiniteCoxeterGroups
             default_category = FiniteEnumeratedSets() if self in FiniteCoxeterGroups() else EnumeratedSets()
+            cat = default_category.or_subcategory(category)
             return RecursivelyEnumeratedSet_forest((self.one(),), succ,
-                algorithm='breadth',
-                category=default_category.or_subcategory(category))
+                                                   algorithm='breadth',
+                                                   category=cat)
 
         @cached_method
         def coxeter_element(self):
@@ -1239,10 +1240,6 @@ def random_element_of_length(self, n):
                 x = x.apply_simple_reflection_right(antiD[rnd])
             return x
 
-        # TODO: Groups() should have inverse() call __invert__
-        # With strong doc stating that this is just a convenience for the user
-        # and links to ~ / __invert__
-
         # parabolic_subgroup
 
         def _test_simple_projections(self, **options):
@@ -2260,7 +2257,6 @@ def binary_factorizations(self, predicate=ConstantFunction(True)):
             if not predicate(W.one()):
                 from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
                 return FiniteEnumeratedSet([])
-            s = W.simple_reflections()
 
             def succ(u_v):
                 u, v = u_v
@@ -2399,7 +2395,7 @@ def bruhat_lower_covers_reflections(self):
             wi = self.apply_simple_reflection(i, side='right')
             return [(u.apply_simple_reflection(i, side='right'),
                      r.apply_conjugation_by_simple_reflection(i))
-                    for u,r in wi.bruhat_lower_covers_reflections()
+                    for u, r in wi.bruhat_lower_covers_reflections()
                     if not u.has_descent(i, side='right')] + [
                 (wi, self.parent().simple_reflection(i))]
 
@@ -3148,7 +3144,7 @@ def kazhdan_lusztig_cell(self, side='left'):
 
             The cell computation uses the optional package ``coxeter3`` in
             the background if available to speed up the computation,
-            even in the different implementations implementations::
+            even in the different implementations::
 
                 sage: # optional - coxeter3, needs sage.combinat sage.groups sage.modules
                 sage: W = WeylGroup('A3', prefix='s')

src/sage/categories/monoids.py

@@ -378,6 +378,8 @@ def inverse(self):
             This is an alias for inversion, which can also be invoked
             by ``~x`` for an element ``x``.
 
+            Nota Bene: Element classes should implement ``__invert__`` only.
+
             EXAMPLES::
 
                 sage: AA(sqrt(~2)).inverse()                                            # needs sage.rings.number_field sage.symbolic