diff --git a/src/sage/categories/__init__.py b/src/sage/categories/__init__.py
new file mode 100644
index 00000000000..3f293d43eac
--- /dev/null
+++ b/src/sage/categories/__init__.py
@@ -0,0 +1 @@
+import sage.structure.element # resolve a cyclic import (categories.map > structure.element > ... > categories.map)
diff --git a/src/sage/categories/all__sagemath_objects.py b/src/sage/categories/all__sagemath_objects.py
index d87b4a1d20e..671fb0dfd16 100644
--- a/src/sage/categories/all__sagemath_objects.py
+++ b/src/sage/categories/all__sagemath_objects.py
@@ -2,15 +2,10 @@
 
 from sage.misc.lazy_import import lazy_import
 
-# Resolve a circular import so that "import sage.categories.all" can succeed
-# in initializing the category system.
-import sage.structure.category_object  # imports sage.categories.category
-
 # Small part of "from sage.categories.basic import *":
 from sage.categories.objects import Objects
 from sage.categories.sets_cat import Sets, EmptySetError
 
-
 from sage.categories.category import Category
 
 from sage.categories.category_types import Elements
diff --git a/src/sage/categories/category.py b/src/sage/categories/category.py
index b71331c5ab5..8520225b2b4 100644
--- a/src/sage/categories/category.py
+++ b/src/sage/categories/category.py
@@ -107,7 +107,6 @@
 from sage.misc.cachefunc import cached_method, cached_function
 from sage.misc.c3_controlled import _cmp_key, _cmp_key_named, C3_sorted_merge
 from sage.misc.lazy_attribute import lazy_attribute
-from sage.misc.unknown import Unknown
 from sage.misc.weak_dict import WeakValueDictionary
 
 from sage.structure.sage_object import SageObject
@@ -1802,6 +1801,8 @@ def is_subcategory(self, c):
             sage: PoV3.is_subcategory(PoA3)
             False
         """
+        from sage.misc.unknown import Unknown
+
         if c is self:
             return True
         subcat_hook = c._subcategory_hook_(self)
@@ -2888,6 +2889,8 @@ def _subcategory_hook_(self, C):
             sage: Bimodules(QQ,QQ)._subcategory_hook_(Rings())
             False
         """
+        from sage.misc.unknown import Unknown
+
         if not issubclass(C.parent_class, self.parent_class):
             return False
         return Unknown
@@ -3053,6 +3056,8 @@ def is_subcategory(self, C):
             sage: Category.join([Rings(),Modules(QQ)]).is_subcategory(Category.join([Rngs(),Bimodules(QQ,QQ)]))
             True
         """
+        from sage.misc.unknown import Unknown
+
         if C is self:
             return True
         hook = C._subcategory_hook_(self)
diff --git a/src/sage/categories/category_singleton.pyx b/src/sage/categories/category_singleton.pyx
index 543ce0375f7..2b63a5c7cf9 100644
--- a/src/sage/categories/category_singleton.pyx
+++ b/src/sage/categories/category_singleton.pyx
@@ -9,7 +9,6 @@ Singleton categories
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.misc.constant_function import ConstantFunction
 from sage.misc.lazy_attribute import lazy_class_attribute
 from sage.categories.category import Category
 from sage.structure.category_object cimport CategoryObject
@@ -314,6 +313,8 @@ class Category_singleton(Category):
             ...
             AssertionError: <class '__main__.MySubStuff'> is not a direct subclass of <class 'sage.categories.category_singleton.Category_singleton'>
         """
+        from sage.misc.constant_function import ConstantFunction
+
         if isinstance(cls, DynamicMetaclass):  # cls is something like Rings_with_category
             cls = cls.__base__
         # TODO: find a better way to check that cls is an abstract class
diff --git a/src/sage/categories/map.pyx b/src/sage/categories/map.pyx
index 134b88ef1ba..78cdef54d5d 100644
--- a/src/sage/categories/map.pyx
+++ b/src/sage/categories/map.pyx
@@ -26,7 +26,6 @@ import weakref
 from sage.ext.stdsage cimport HAS_DICTIONARY
 from sage.arith.power cimport generic_power
 from sage.sets.pythonclass cimport Set_PythonType
-from sage.misc.constant_function import ConstantFunction
 from sage.structure.element cimport parent
 from cpython.object cimport PyObject_RichCompare
 
@@ -120,6 +119,8 @@ cdef class Map(Element):
               From: Univariate Polynomial Ring in x over Rational Field
               To:   Symmetric group of order 6! as a permutation group
         """
+        from sage.misc.constant_function import ConstantFunction
+
         if codomain is not None:
             if isinstance(parent, type):
                 parent = Set_PythonType(parent)
@@ -304,6 +305,8 @@ cdef class Map(Element):
             sage: phi
             Defunct map
         """
+        from sage.misc.constant_function import ConstantFunction
+
         if not isinstance(self.domain, ConstantFunction):
             return
         self.domain = weakref.ref(self.domain())
@@ -380,6 +383,8 @@ cdef class Map(Element):
             ...
             ValueError: This map is in an invalid state, the domain has been garbage collected
         """
+        from sage.misc.constant_function import ConstantFunction
+
         if isinstance(self.domain, ConstantFunction):
             return
         D = self.domain()
@@ -418,6 +423,8 @@ cdef class Map(Element):
             sage: f._repr_type_str
             'bla'
         """
+        from sage.misc.constant_function import ConstantFunction
+
         # todo: the following can break during unpickling of complex
         # objects with circular references! In that case, _slots might
         # contain incomplete objects.
@@ -584,6 +591,8 @@ cdef class Map(Element):
             sage: phi
             Defunct map
         """
+        from sage.misc.constant_function import ConstantFunction
+
         D = self.domain()
         if D is None:
             return "Defunct map"
diff --git a/src/sage/categories/morphism.pyx b/src/sage/categories/morphism.pyx
index d4b412dd126..74362a2fa87 100644
--- a/src/sage/categories/morphism.pyx
+++ b/src/sage/categories/morphism.pyx
@@ -39,8 +39,6 @@ AUTHORS:
 
 from cpython.object cimport *
 
-from sage.misc.constant_function import ConstantFunction
-
 from sage.structure.element cimport Element, ModuleElement
 from sage.structure.richcmp cimport richcmp_not_equal, rich_to_bool
 from sage.structure.parent cimport Parent
@@ -90,6 +88,8 @@ cdef class Morphism(Map):
             sage: phi
             Defunct morphism
         """
+        from sage.misc.constant_function import ConstantFunction
+
         D = self.domain()
         if D is None:
             return "Defunct morphism"
diff --git a/src/sage/misc/__init__.py b/src/sage/misc/__init__.py
new file mode 100644
index 00000000000..f5961558d9f
--- /dev/null
+++ b/src/sage/misc/__init__.py
@@ -0,0 +1,5 @@
+try:
+    import sage.structure.element  # to break a cyclic import (misc.constant_function > structure.element > ... > misc.constant_function)
+except ModuleNotFoundError:
+    # Ignore, the import above may fail while collecting metadata info (at which point sage.structure is not yet installed)
+    pass
diff --git a/src/sage/misc/all__sagemath_objects.py b/src/sage/misc/all__sagemath_objects.py
index 30a76f38fab..ed0ff500949 100644
--- a/src/sage/misc/all__sagemath_objects.py
+++ b/src/sage/misc/all__sagemath_objects.py
@@ -1,7 +1,5 @@
 # Subset of sage.misc.all that is made available by the sage-objects distribution
 
-import sage.structure.all   # to break a cyclic import
-
 from sage.misc.lazy_attribute import lazy_attribute, lazy_class_attribute
 from sage.misc.lazy_import import lazy_import
 
diff --git a/src/sage/misc/c3_controlled.pyx b/src/sage/misc/c3_controlled.pyx
index befaa7b5b32..0526e64764c 100644
--- a/src/sage/misc/c3_controlled.pyx
+++ b/src/sage/misc/c3_controlled.pyx
@@ -365,7 +365,6 @@ AUTHOR:
 from sage.misc.classcall_metaclass import ClasscallMetaclass, typecall
 from sage.misc.cachefunc import cached_function, cached_method
 from sage.misc.lazy_attribute import lazy_attribute
-from sage.structure.dynamic_class import dynamic_class
 
 ##############################################################################
 # Implementation of the total order between categories
@@ -1349,6 +1348,8 @@ class HierarchyElement(object, metaclass=ClasscallMetaclass):
             sage: x.cls.mro()
             [<class '30.cls'>, <class '15.cls'>, <class '10.cls'>, <class '6.cls'>, <class '5.cls'>, <class '3.cls'>, <class '2.cls'>, <class '1.cls'>, <... 'object'>]
         """
+        from sage.structure.dynamic_class import dynamic_class
+
         super_classes = tuple(self._from_value(base).cls for base in self._bases_controlled)
         if not super_classes:
             super_classes = (object,)
diff --git a/src/sage/misc/fast_methods.pyx b/src/sage/misc/fast_methods.pyx
index ce7f918cccd..522efc1e201 100644
--- a/src/sage/misc/fast_methods.pyx
+++ b/src/sage/misc/fast_methods.pyx
@@ -29,7 +29,6 @@ AUTHOR:
 #*****************************************************************************
 
 from sage.misc.classcall_metaclass import ClasscallMetaclass, typecall
-from sage.misc.constant_function import ConstantFunction
 
 from cpython.object cimport Py_EQ, Py_NE
 
@@ -296,6 +295,8 @@ class Singleton(WithEqualityById, metaclass=ClasscallMetaclass):
             sage: loads(dumps(c)) is copy(c) is C()  # indirect doctest
             True
         """
+        from sage.misc.constant_function import ConstantFunction
+
         assert cls.mro()[1] == Singleton, "{} is not a direct subclass of {}".format(cls, Singleton)
         res = typecall(cls)
         cf = ConstantFunction(res)
diff --git a/src/sage/rings/rational.pyx b/src/sage/rings/rational.pyx
index 92fa6fd8fc3..5f94b8ade9b 100644
--- a/src/sage/rings/rational.pyx
+++ b/src/sage/rings/rational.pyx
@@ -79,7 +79,6 @@ from sage.structure.element cimport Element
 from sage.structure.parent cimport Parent
 from sage.structure.richcmp cimport rich_to_bool_sgn
 
-
 RealNumber_classes = ()
 
 def _register_real_number_class(cls):
@@ -2088,6 +2087,7 @@ cdef class Rational(sage.structure.element.FieldElement):
             ...
             ValueError: cannot take even root of negative number
         """
+        from sage.rings.integer_ring import ZZ
         # TODO -- this could be quicker, by using GMP directly.
         cdef integer.Integer num
         cdef integer.Integer den
@@ -3196,6 +3196,7 @@ cdef class Rational(sage.structure.element.FieldElement):
             sage: (-1/2).log(3)                                                         # needs sage.symbolic
             (I*pi + log(1/2))/log(3)
         """
+        from sage.rings.integer_ring import ZZ
         cdef int self_sgn
         if self.denom().is_one():
             return ZZ(self.numer()).log(m, prec)
diff --git a/src/sage/sets/pythonclass.pyx b/src/sage/sets/pythonclass.pyx
index bfa7f3bc617..40ffbd55a89 100644
--- a/src/sage/sets/pythonclass.pyx
+++ b/src/sage/sets/pythonclass.pyx
@@ -14,7 +14,6 @@ Set of all objects of a given Python class
 
 from cpython.object cimport Py_EQ, Py_NE
 from sage.structure.richcmp cimport rich_to_bool
-from sage.categories.sets_cat import Sets
 
 
 cdef dict _type_set_cache = {}
@@ -83,6 +82,8 @@ cdef class Set_PythonType_class(Set_generic):
             sage: Set_PythonType(float).category()
             Category of sets
         """
+        from sage.categories.sets_cat import Sets
+
         if not isinstance(typ, type):
             raise TypeError(f"must be initialized with a class, not {typ!r}")
         super().__init__(category=Sets())
diff --git a/src/sage/structure/__init__.py b/src/sage/structure/__init__.py
index 8689eb1717f..e69de29bb2d 100644
--- a/src/sage/structure/__init__.py
+++ b/src/sage/structure/__init__.py
@@ -1,2 +0,0 @@
-# Resolve a cyclic import
-import sage.structure.element
diff --git a/src/sage/structure/category_object.pyx b/src/sage/structure/category_object.pyx
index 48cee10c2bc..ea5aaa1393c 100644
--- a/src/sage/structure/category_object.pyx
+++ b/src/sage/structure/category_object.pyx
@@ -58,7 +58,6 @@ This example illustrates generators for a free module over `\ZZ`.
 
 from sage.cpython.getattr import dir_with_other_class
 from sage.cpython.getattr cimport getattr_from_other_class
-from sage.categories.category import Category
 from sage.misc.cachefunc import cached_method
 from sage.structure.dynamic_class import DynamicMetaclass
 
@@ -145,6 +144,8 @@ cdef class CategoryObject(SageObject):
             ...
             TypeError: CategoryObject of type CategoryObject requires a Category, list or tuple, not NoneType
         """
+        from sage.categories.category import Category
+
         if isinstance(category, Category):
             self._category = category
         elif isinstance(category, (list, tuple)):
diff --git a/src/sage/structure/factory.pyx b/src/sage/structure/factory.pyx
index 75a043250c2..7474e310796 100644
--- a/src/sage/structure/factory.pyx
+++ b/src/sage/structure/factory.pyx
@@ -768,7 +768,3 @@ def lookup_global(name):
     else:
         import sage.all as all
     return getattr(all, name)
-
-
-# Old imports required for unpickling old pickles
-from sage.structure.test_factory import test_factory