diff --git a/src/sage/categories/action.pxd b/src/sage/categories/action.pxd
index 444fcc1f38f..a22a37cc22a 100644
--- a/src/sage/categories/action.pxd
+++ b/src/sage/categories/action.pxd
@@ -1,7 +1,7 @@
 from sage.structure.element cimport Element
-from .morphism cimport Morphism
-from .map cimport Map
-from .functor cimport Functor
+from sage.categories.morphism cimport Morphism
+from sage.categories.map cimport Map
+from sage.categories.functor cimport Functor
 
 cdef class Action(Functor):
     cdef readonly G
diff --git a/src/sage/categories/action.pyx b/src/sage/categories/action.pyx
index 46effeecc42..b164e9213ce 100644
--- a/src/sage/categories/action.pyx
+++ b/src/sage/categories/action.pyx
@@ -56,13 +56,13 @@ AUTHOR:
 
 from cpython.tuple cimport PyTuple_GET_ITEM
 
-from .functor cimport Functor
-from .morphism cimport Morphism
-from .map cimport Map
+from sage.categories.functor cimport Functor
+from sage.categories.morphism cimport Morphism
+from sage.categories.map cimport Map
 from sage.structure.element cimport parent
 from sage.structure.parent cimport Parent
 
-from . import homset
+from sage.categories import homset
 from weakref import ref
 
 
diff --git a/src/sage/categories/algebra_ideals.py b/src/sage/categories/algebra_ideals.py
index 1d6e3775e7b..c33fa0ef7b0 100644
--- a/src/sage/categories/algebra_ideals.py
+++ b/src/sage/categories/algebra_ideals.py
@@ -10,10 +10,10 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from .algebra_modules import AlgebraModules
-from .algebras import Algebras
-from .rings import Rings
-from .category_types import Category_ideal
+from sage.categories.algebra_modules import AlgebraModules
+from sage.categories.algebras import Algebras
+from sage.categories.category_types import Category_ideal
+from sage.categories.rings import Rings
 
 
 class AlgebraIdeals(Category_ideal):
diff --git a/src/sage/categories/algebra_modules.py b/src/sage/categories/algebra_modules.py
index 7be7de2c471..e827cc7130d 100644
--- a/src/sage/categories/algebra_modules.py
+++ b/src/sage/categories/algebra_modules.py
@@ -10,10 +10,11 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
-from .category_types import Category_module
-from .commutative_algebras import CommutativeAlgebras
-from .commutative_rings import CommutativeRings
-from .modules import Modules
+from sage.categories.category_types import Category_module
+from sage.categories.commutative_algebras import CommutativeAlgebras
+from sage.categories.commutative_rings import CommutativeRings
+from sage.categories.modules import Modules
+
 
 class AlgebraModules(Category_module):
     """
diff --git a/src/sage/categories/algebras_with_basis.py b/src/sage/categories/algebras_with_basis.py
index 28177a1ff45..dfd693a9520 100644
--- a/src/sage/categories/algebras_with_basis.py
+++ b/src/sage/categories/algebras_with_basis.py
@@ -9,13 +9,13 @@
 #                  https://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.categories.cartesian_product import CartesianProductsCategory
+from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
+from sage.categories.tensor import TensorProductsCategory, tensor
+from sage.categories.unital_algebras import UnitalAlgebras
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.misc.lazy_import import LazyImport
-from sage.categories.tensor import TensorProductsCategory, tensor
-from sage.categories.cartesian_product import CartesianProductsCategory
-from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
-from .unital_algebras import UnitalAlgebras
 
 
 class AlgebrasWithBasis(CategoryWithAxiom_over_base_ring):
diff --git a/src/sage/categories/all.py b/src/sage/categories/all.py
index 110a9954da3..9c2b8bd6518 100644
--- a/src/sage/categories/all.py
+++ b/src/sage/categories/all.py
@@ -27,128 +27,128 @@
 from sage.misc.namespace_package import install_doc
 install_doc(__package__, __doc__)
 
-from . import primer
+from sage.categories import primer
 
 from sage.misc.lazy_import import lazy_import
 
-from .all__sagemath_objects import *
+from sage.categories.all__sagemath_objects import *
 
-from .basic import *
+from sage.categories.basic import *
 
-from .chain_complexes import ChainComplexes, HomologyFunctor
+from sage.categories.chain_complexes import ChainComplexes, HomologyFunctor
 
-from .simplicial_complexes import SimplicialComplexes
+from sage.categories.simplicial_complexes import SimplicialComplexes
 
-from .tensor import tensor
-from .signed_tensor import tensor_signed
+from sage.categories.tensor import tensor
+from sage.categories.signed_tensor import tensor_signed
 
-from .g_sets import GSets
-from .pointed_sets import PointedSets
+from sage.categories.g_sets import GSets
+from sage.categories.pointed_sets import PointedSets
 
-from .sets_with_grading import SetsWithGrading
+from sage.categories.sets_with_grading import SetsWithGrading
 
-from .groupoid import Groupoid
-from .permutation_groups import PermutationGroups
+from sage.categories.groupoid import Groupoid
+from sage.categories.permutation_groups import PermutationGroups
 
 # enumerated sets
-from .finite_sets import FiniteSets
-from .enumerated_sets import EnumeratedSets
-from .finite_enumerated_sets import FiniteEnumeratedSets
-from .infinite_enumerated_sets import InfiniteEnumeratedSets
+from sage.categories.finite_sets import FiniteSets
+from sage.categories.enumerated_sets import EnumeratedSets
+from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
+from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
 
 # posets
-from .posets import Posets
-from .finite_posets import FinitePosets
-from .lattice_posets import LatticePosets
-from .finite_lattice_posets import FiniteLatticePosets
+from sage.categories.posets import Posets
+from sage.categories.finite_posets import FinitePosets
+from sage.categories.lattice_posets import LatticePosets
+from sage.categories.finite_lattice_posets import FiniteLatticePosets
 
 # finite groups/...
-from .finite_semigroups import FiniteSemigroups
-from .finite_monoids import FiniteMonoids
-from .finite_groups import FiniteGroups
-from .finite_permutation_groups import FinitePermutationGroups
+from sage.categories.finite_semigroups import FiniteSemigroups
+from sage.categories.finite_monoids import FiniteMonoids
+from sage.categories.finite_groups import FiniteGroups
+from sage.categories.finite_permutation_groups import FinitePermutationGroups
 
 # fields
-from .number_fields import NumberFields
-from .function_fields import FunctionFields
+from sage.categories.number_fields import NumberFields
+from sage.categories.function_fields import FunctionFields
 
 # modules
-from .left_modules import LeftModules
-from .right_modules import RightModules
-from .bimodules import Bimodules
+from sage.categories.left_modules import LeftModules
+from sage.categories.right_modules import RightModules
+from sage.categories.bimodules import Bimodules
 
-from .modules import Modules
+from sage.categories.modules import Modules
 RingModules = Modules
-from .vector_spaces import VectorSpaces
+from sage.categories.vector_spaces import VectorSpaces
 
 # (hopf) algebra structures
-from .algebras import Algebras
-from .commutative_algebras import CommutativeAlgebras
-from .coalgebras import Coalgebras
-from .bialgebras import Bialgebras
-from .hopf_algebras import HopfAlgebras
-from .lie_algebras import LieAlgebras
+from sage.categories.algebras import Algebras
+from sage.categories.commutative_algebras import CommutativeAlgebras
+from sage.categories.coalgebras import Coalgebras
+from sage.categories.bialgebras import Bialgebras
+from sage.categories.hopf_algebras import HopfAlgebras
+from sage.categories.lie_algebras import LieAlgebras
 
 # specific algebras
-from .monoid_algebras import MonoidAlgebras
-from .group_algebras import GroupAlgebras
-from .matrix_algebras import MatrixAlgebras
+from sage.categories.monoid_algebras import MonoidAlgebras
+from sage.categories.group_algebras import GroupAlgebras
+from sage.categories.matrix_algebras import MatrixAlgebras
 
 # ideals
-from .ring_ideals import RingIdeals
+from sage.categories.ring_ideals import RingIdeals
 Ideals = RingIdeals
-from .commutative_ring_ideals import CommutativeRingIdeals
-from .algebra_modules import AlgebraModules
-from .algebra_ideals import AlgebraIdeals
-from .commutative_algebra_ideals import CommutativeAlgebraIdeals
+from sage.categories.commutative_ring_ideals import CommutativeRingIdeals
+from sage.categories.algebra_modules import AlgebraModules
+from sage.categories.algebra_ideals import AlgebraIdeals
+from sage.categories.commutative_algebra_ideals import CommutativeAlgebraIdeals
 
 # schemes and varieties
-from .modular_abelian_varieties import ModularAbelianVarieties
-from .schemes import Schemes
+from sage.categories.modular_abelian_varieties import ModularAbelianVarieties
+from sage.categories.schemes import Schemes
 
 # * with basis
-from .modules_with_basis import ModulesWithBasis
+from sage.categories.modules_with_basis import ModulesWithBasis
 FreeModules = ModulesWithBasis
-from .hecke_modules            import HeckeModules
-from .algebras_with_basis      import AlgebrasWithBasis
-from .coalgebras_with_basis    import CoalgebrasWithBasis
-from .bialgebras_with_basis    import BialgebrasWithBasis
-from .hopf_algebras_with_basis import HopfAlgebrasWithBasis
+from sage.categories.hecke_modules            import HeckeModules
+from sage.categories.algebras_with_basis      import AlgebrasWithBasis
+from sage.categories.coalgebras_with_basis    import CoalgebrasWithBasis
+from sage.categories.bialgebras_with_basis    import BialgebrasWithBasis
+from sage.categories.hopf_algebras_with_basis import HopfAlgebrasWithBasis
 
 # finite dimensional * with basis
-from .finite_dimensional_modules_with_basis       import FiniteDimensionalModulesWithBasis
-from .finite_dimensional_algebras_with_basis      import FiniteDimensionalAlgebrasWithBasis
-from .finite_dimensional_coalgebras_with_basis    import FiniteDimensionalCoalgebrasWithBasis
-from .finite_dimensional_bialgebras_with_basis    import FiniteDimensionalBialgebrasWithBasis
-from .finite_dimensional_hopf_algebras_with_basis import FiniteDimensionalHopfAlgebrasWithBasis
+from sage.categories.finite_dimensional_modules_with_basis       import FiniteDimensionalModulesWithBasis
+from sage.categories.finite_dimensional_algebras_with_basis      import FiniteDimensionalAlgebrasWithBasis
+from sage.categories.finite_dimensional_coalgebras_with_basis    import FiniteDimensionalCoalgebrasWithBasis
+from sage.categories.finite_dimensional_bialgebras_with_basis    import FiniteDimensionalBialgebrasWithBasis
+from sage.categories.finite_dimensional_hopf_algebras_with_basis import FiniteDimensionalHopfAlgebrasWithBasis
 
 # graded *
-from .graded_modules       import GradedModules
-from .graded_algebras      import GradedAlgebras
-from .graded_coalgebras    import GradedCoalgebras
-from .graded_bialgebras    import GradedBialgebras
-from .graded_hopf_algebras import GradedHopfAlgebras
+from sage.categories.graded_modules       import GradedModules
+from sage.categories.graded_algebras      import GradedAlgebras
+from sage.categories.graded_coalgebras    import GradedCoalgebras
+from sage.categories.graded_bialgebras    import GradedBialgebras
+from sage.categories.graded_hopf_algebras import GradedHopfAlgebras
 
 # graded * with basis
-from .graded_modules_with_basis       import GradedModulesWithBasis
-from .graded_algebras_with_basis      import GradedAlgebrasWithBasis
-from .graded_coalgebras_with_basis    import GradedCoalgebrasWithBasis
-from .graded_bialgebras_with_basis    import GradedBialgebrasWithBasis
-from .graded_hopf_algebras_with_basis import GradedHopfAlgebrasWithBasis
+from sage.categories.graded_modules_with_basis       import GradedModulesWithBasis
+from sage.categories.graded_algebras_with_basis      import GradedAlgebrasWithBasis
+from sage.categories.graded_coalgebras_with_basis    import GradedCoalgebrasWithBasis
+from sage.categories.graded_bialgebras_with_basis    import GradedBialgebrasWithBasis
+from sage.categories.graded_hopf_algebras_with_basis import GradedHopfAlgebrasWithBasis
 
 # Coxeter groups
-from .coxeter_groups import CoxeterGroups
+from sage.categories.coxeter_groups import CoxeterGroups
 lazy_import('sage.categories.finite_coxeter_groups', 'FiniteCoxeterGroups')
-from .weyl_groups import WeylGroups
-from .finite_weyl_groups import FiniteWeylGroups
-from .affine_weyl_groups import AffineWeylGroups
+from sage.categories.weyl_groups import WeylGroups
+from sage.categories.finite_weyl_groups import FiniteWeylGroups
+from sage.categories.affine_weyl_groups import AffineWeylGroups
 
 # crystal bases
-from .crystals import Crystals
-from .highest_weight_crystals import HighestWeightCrystals
-from .regular_crystals import RegularCrystals
-from .finite_crystals import FiniteCrystals
-from .classical_crystals import ClassicalCrystals
+from sage.categories.crystals import Crystals
+from sage.categories.highest_weight_crystals import HighestWeightCrystals
+from sage.categories.regular_crystals import RegularCrystals
+from sage.categories.finite_crystals import FiniteCrystals
+from sage.categories.classical_crystals import ClassicalCrystals
 
 # polyhedra
 lazy_import('sage.categories.polyhedra', 'PolyhedralSets')
diff --git a/src/sage/categories/all__sagemath_objects.py b/src/sage/categories/all__sagemath_objects.py
index dee0881d787..d87b4a1d20e 100644
--- a/src/sage/categories/all__sagemath_objects.py
+++ b/src/sage/categories/all__sagemath_objects.py
@@ -4,28 +4,25 @@
 
 # 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
+import sage.structure.category_object  # imports sage.categories.category
 
-# Small part of "from .basic import *":
-from .objects import Objects
-from .sets_cat import Sets, EmptySetError
+# Small part of "from sage.categories.basic import *":
+from sage.categories.objects import Objects
+from sage.categories.sets_cat import Sets, EmptySetError
 
 
-from .category import Category
+from sage.categories.category import Category
 
-from .category_types import Elements
+from sage.categories.category_types import Elements
 
-from .cartesian_product import cartesian_product
+from sage.categories.cartesian_product import cartesian_product
 
-from .functor  import (ForgetfulFunctor,
-                      IdentityFunctor)
+from sage.categories.functor import ForgetfulFunctor, IdentityFunctor
 
-from .homset   import (Hom, hom,
-                      End, end,
-                      Homset, HomsetWithBase)
+from sage.categories.homset import Hom, hom, End, end, Homset, HomsetWithBase
 
-from .morphism import Morphism
+from sage.categories.morphism import Morphism
 
-from .realizations import Realizations
+from sage.categories.realizations import Realizations
 
-from .sets_with_partial_maps import SetsWithPartialMaps
+from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
diff --git a/src/sage/categories/basic.py b/src/sage/categories/basic.py
index b3cf1d3a29c..f3574db469d 100644
--- a/src/sage/categories/basic.py
+++ b/src/sage/categories/basic.py
@@ -2,49 +2,59 @@
 A subset of sage.categories.all with just the basic categories needed
 for sage startup (i.e. to define ZZ, QQ, ...).
 """
-#*****************************************************************************
+# *****************************************************************************
 #  Copyright (C) 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #                  http://www.gnu.org/licenses/
-#******************************************************************************
+# ******************************************************************************
+
+from sage.categories.objects import Objects
+from sage.categories.sets_cat import Sets, EmptySetError
+from sage.categories.posets import Posets
 
-from .objects import Objects
-from .sets_cat import Sets, EmptySetError
-from .posets import Posets
 # For backward compatibility; will be deprecated at some point
 PartiallyOrderedSets = Posets
 OrderedSets = Posets
 
-from .additive_magmas import AdditiveMagmas
-from .commutative_additive_semigroups import CommutativeAdditiveSemigroups
-from .commutative_additive_monoids import CommutativeAdditiveMonoids
-from .commutative_additive_groups import CommutativeAdditiveGroups
+from sage.categories.additive_magmas import AdditiveMagmas
+from sage.categories.commutative_additive_semigroups import (
+    CommutativeAdditiveSemigroups,
+)
+from sage.categories.commutative_additive_monoids import CommutativeAdditiveMonoids
+from sage.categories.commutative_additive_groups import CommutativeAdditiveGroups
+
+from sage.categories.magmas import Magmas
+from sage.categories.semigroups import Semigroups
+from sage.categories.monoids import Monoids
+from sage.categories.groups import Groups
+from sage.categories.partially_ordered_monoids import PartiallyOrderedMonoids
 
-from .magmas import Magmas
-from .semigroups import Semigroups
-from .monoids import Monoids
-from .groups import Groups
-from .partially_ordered_monoids import PartiallyOrderedMonoids
 # For backward compatibility; might be deprecated at some point
 OrderedMonoids = PartiallyOrderedMonoids
 
-from .rngs import Rngs
-from .semirings import Semirings
-from .rings import Rings
-from .domains import Domains
-from .division_rings import DivisionRings
-
-from .commutative_rings import CommutativeRings
-from .integral_domains import IntegralDomains
-from .gcd_domains import GcdDomains
-from .principal_ideal_domains import PrincipalIdealDomains
-from .euclidean_domains import EuclideanDomains
-from .unique_factorization_domains import UniqueFactorizationDomains
-from .complete_discrete_valuation import CompleteDiscreteValuationRings
-
-from .fields import Fields
-from .quotient_fields import QuotientFields
-from .finite_fields import FiniteFields
-from .discrete_valuation import DiscreteValuationRings, DiscreteValuationFields
-from .complete_discrete_valuation import CompleteDiscreteValuationRings, CompleteDiscreteValuationFields
+from sage.categories.rngs import Rngs
+from sage.categories.semirings import Semirings
+from sage.categories.rings import Rings
+from sage.categories.domains import Domains
+from sage.categories.division_rings import DivisionRings
+
+from sage.categories.commutative_rings import CommutativeRings
+from sage.categories.integral_domains import IntegralDomains
+from sage.categories.gcd_domains import GcdDomains
+from sage.categories.principal_ideal_domains import PrincipalIdealDomains
+from sage.categories.euclidean_domains import EuclideanDomains
+from sage.categories.unique_factorization_domains import UniqueFactorizationDomains
+from sage.categories.complete_discrete_valuation import CompleteDiscreteValuationRings
+
+from sage.categories.fields import Fields
+from sage.categories.quotient_fields import QuotientFields
+from sage.categories.finite_fields import FiniteFields
+from sage.categories.discrete_valuation import (
+    DiscreteValuationRings,
+    DiscreteValuationFields,
+)
+from sage.categories.complete_discrete_valuation import (
+    CompleteDiscreteValuationRings,
+    CompleteDiscreteValuationFields,
+)
diff --git a/src/sage/categories/category_types.py b/src/sage/categories/category_types.py
index 3c1aae55c5a..4ead4f23530 100644
--- a/src/sage/categories/category_types.py
+++ b/src/sage/categories/category_types.py
@@ -14,9 +14,10 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from sage.misc.unknown import Unknown
-from .category import JoinCategory, Category, CategoryWithParameters
+from sage.categories.category import Category, CategoryWithParameters, JoinCategory
 from sage.misc.lazy_import import lazy_import
+from sage.misc.unknown import Unknown
+
 lazy_import('sage.categories.objects', 'Objects')
 lazy_import('sage.misc.latex', 'latex')
 
@@ -198,6 +199,7 @@ def _test_category_over_bases(self, **options):
         """
         tester = self._tester(**options)
         from sage.categories.category_singleton import Category_singleton
+
         from .bimodules import Bimodules
         from .schemes import Schemes
         for cat in self.super_categories():
diff --git a/src/sage/categories/chain_complexes.py b/src/sage/categories/chain_complexes.py
index db9c5d03f56..9616aa608c4 100644
--- a/src/sage/categories/chain_complexes.py
+++ b/src/sage/categories/chain_complexes.py
@@ -13,11 +13,12 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from .category_types import Category_module
-from .commutative_additive_groups import CommutativeAdditiveGroups
-from .functor import Functor
+from sage.categories.category_types import Category_module
+from sage.categories.commutative_additive_groups import CommutativeAdditiveGroups
+from sage.categories.functor import Functor
 from sage.misc.abstract_method import abstract_method
 
+
 #############################################################
 # ChainComplex
 #############################################################
@@ -259,8 +260,8 @@ def _apply_functor_to_morphism(self, f):
             sage: id_star(one)
             [one]
         """
-        from .morphism import SetMorphism
-        from .homset import Hom
+        from sage.categories.homset import Hom
+        from sage.categories.morphism import SetMorphism
 
         domain = f.domain()
         codomain = f.codomain()
diff --git a/src/sage/categories/coalgebras.py b/src/sage/categories/coalgebras.py
index abe36aef0bb..b69fdf29784 100644
--- a/src/sage/categories/coalgebras.py
+++ b/src/sage/categories/coalgebras.py
@@ -10,14 +10,14 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from .category_types import Category_over_base_ring
-from sage.categories.modules import Modules
+from sage.categories.category_types import Category_over_base_ring
 from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
-from sage.categories.tensor import TensorProductsCategory
 from sage.categories.dual import DualObjectsCategory
 from sage.categories.filtered_modules import FilteredModulesCategory
-from sage.categories.super_modules import SuperModulesCategory
+from sage.categories.modules import Modules
 from sage.categories.realizations import RealizationsCategory
+from sage.categories.super_modules import SuperModulesCategory
+from sage.categories.tensor import TensorProductsCategory
 from sage.categories.with_realizations import WithRealizationsCategory
 from sage.misc.abstract_method import abstract_method
 from sage.misc.cachefunc import cached_method
diff --git a/src/sage/categories/commutative_algebra_ideals.py b/src/sage/categories/commutative_algebra_ideals.py
index 070503ae6c8..e47e1c4f87e 100644
--- a/src/sage/categories/commutative_algebra_ideals.py
+++ b/src/sage/categories/commutative_algebra_ideals.py
@@ -10,10 +10,10 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
-from .algebra_ideals import AlgebraIdeals
-from .category_types import Category_ideal, Category_in_ambient
-from .commutative_algebras import CommutativeAlgebras
-from .commutative_rings import CommutativeRings
+from sage.categories.algebra_ideals import AlgebraIdeals
+from sage.categories.category_types import Category_ideal, Category_in_ambient
+from sage.categories.commutative_algebras import CommutativeAlgebras
+from sage.categories.commutative_rings import CommutativeRings
 
 
 class CommutativeAlgebraIdeals(Category_ideal):
diff --git a/src/sage/categories/commutative_ring_ideals.py b/src/sage/categories/commutative_ring_ideals.py
index 5aef6d7d31d..8659b0c793b 100644
--- a/src/sage/categories/commutative_ring_ideals.py
+++ b/src/sage/categories/commutative_ring_ideals.py
@@ -10,9 +10,10 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
-from .category_types import Category_ideal
+from sage.categories.category_types import Category_ideal
 from sage.categories.commutative_rings import CommutativeRings
-from .ring_ideals import RingIdeals
+from sage.categories.ring_ideals import RingIdeals
+
 
 class CommutativeRingIdeals(Category_ideal):
     """
diff --git a/src/sage/categories/complete_discrete_valuation.py b/src/sage/categories/complete_discrete_valuation.py
index 442360465a1..7376d947530 100644
--- a/src/sage/categories/complete_discrete_valuation.py
+++ b/src/sage/categories/complete_discrete_valuation.py
@@ -9,10 +9,13 @@
 #**************************************************************************
 
 
+from sage.categories.category_singleton import Category_singleton
+from sage.categories.discrete_valuation import (
+    DiscreteValuationFields,
+    DiscreteValuationRings,
+)
 from sage.misc.abstract_method import abstract_method
 
-from sage.categories.category_singleton import Category_singleton
-from .discrete_valuation import DiscreteValuationRings, DiscreteValuationFields
 #from sage.misc.cachefunc import cached_method
 
 
diff --git a/src/sage/categories/finite_dimensional_semisimple_algebras_with_basis.py b/src/sage/categories/finite_dimensional_semisimple_algebras_with_basis.py
index 531ee7145a4..da4b297ddde 100644
--- a/src/sage/categories/finite_dimensional_semisimple_algebras_with_basis.py
+++ b/src/sage/categories/finite_dimensional_semisimple_algebras_with_basis.py
@@ -9,10 +9,11 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.categories.algebras import Algebras
 from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
+from sage.categories.semisimple_algebras import SemisimpleAlgebras
 from sage.misc.cachefunc import cached_method
-from .algebras import Algebras
-from .semisimple_algebras import SemisimpleAlgebras
+
 
 class FiniteDimensionalSemisimpleAlgebrasWithBasis(CategoryWithAxiom_over_base_ring):
     """
diff --git a/src/sage/categories/functor.pyx b/src/sage/categories/functor.pyx
index c25e99f2164..32bd079fc73 100644
--- a/src/sage/categories/functor.pyx
+++ b/src/sage/categories/functor.pyx
@@ -33,7 +33,7 @@ AUTHORS:
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from . import category
+from sage.categories import category
 
 
 def _Functor_unpickle(Cl, D, domain, codomain):
diff --git a/src/sage/categories/g_sets.py b/src/sage/categories/g_sets.py
index 7fd09bafea4..5967fcf7d5f 100644
--- a/src/sage/categories/g_sets.py
+++ b/src/sage/categories/g_sets.py
@@ -11,7 +11,8 @@
 #******************************************************************************
 
 from sage.categories.category import Category
-from .sets_cat import Sets
+from sage.categories.sets_cat import Sets
+
 
 #############################################################
 # GSets
diff --git a/src/sage/categories/groupoid.py b/src/sage/categories/groupoid.py
index 850e63a2c09..95af5df164c 100644
--- a/src/sage/categories/groupoid.py
+++ b/src/sage/categories/groupoid.py
@@ -12,7 +12,8 @@
 #******************************************************************************
 
 from sage.categories.category import CategoryWithParameters
-from .sets_cat import Sets
+from sage.categories.sets_cat import Sets
+
 
 class Groupoid(CategoryWithParameters):
     """
diff --git a/src/sage/categories/homset.py b/src/sage/categories/homset.py
index 2a8b8aeedeb..d2203299512 100644
--- a/src/sage/categories/homset.py
+++ b/src/sage/categories/homset.py
@@ -65,12 +65,9 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
+from sage.categories import morphism
 from sage.categories.category import Category, JoinCategory
-from . import morphism
-from sage.structure.parent import Parent, Set_generic
 from sage.misc.fast_methods import WithEqualityById
-from sage.structure.dynamic_class import dynamic_class
-from sage.structure.unique_representation import UniqueRepresentation
 from sage.misc.lazy_attribute import lazy_attribute
 
 ###################################
@@ -78,8 +75,11 @@
 # introduced in github issue #715
 # with weak values, as introduced in
 # github issue #14159
-
 from sage.structure.coerce_dict import TripleDict
+from sage.structure.dynamic_class import dynamic_class
+from sage.structure.parent import Parent, Set_generic
+from sage.structure.unique_representation import UniqueRepresentation
+
 _cache = TripleDict(weak_values=True)
 
 
diff --git a/src/sage/categories/hopf_algebras.py b/src/sage/categories/hopf_algebras.py
index 41e349690fe..218abc1ca27 100644
--- a/src/sage/categories/hopf_algebras.py
+++ b/src/sage/categories/hopf_algebras.py
@@ -8,14 +8,14 @@
 #  Distributed under the terms of the GNU General Public License (GPL)
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
-from sage.misc.lazy_import import LazyImport
-from .category import Category
-from .category_types import Category_over_base_ring
 from sage.categories.bialgebras import Bialgebras
-from sage.categories.tensor import TensorProductsCategory  # tensor
+from sage.categories.category import Category
+from sage.categories.category_types import Category_over_base_ring
 from sage.categories.realizations import RealizationsCategory
 from sage.categories.super_modules import SuperModulesCategory
+from sage.categories.tensor import TensorProductsCategory  # tensor
 from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_import import LazyImport
 
 
 class HopfAlgebras(Category_over_base_ring):
diff --git a/src/sage/categories/l_trivial_semigroups.py b/src/sage/categories/l_trivial_semigroups.py
index 53f03701e6f..9c483dcd3d1 100644
--- a/src/sage/categories/l_trivial_semigroups.py
+++ b/src/sage/categories/l_trivial_semigroups.py
@@ -13,8 +13,9 @@
 #*****************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
-from .magmas import Magmas
-from .semigroups import Semigroups
+from sage.categories.magmas import Magmas
+from sage.categories.semigroups import Semigroups
+
 
 class LTrivialSemigroups(CategoryWithAxiom):
     def extra_super_categories(self):
diff --git a/src/sage/categories/lambda_bracket_algebras.py b/src/sage/categories/lambda_bracket_algebras.py
index 40cb5ed70c4..16dbd7c7fa9 100644
--- a/src/sage/categories/lambda_bracket_algebras.py
+++ b/src/sage/categories/lambda_bracket_algebras.py
@@ -16,13 +16,14 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from .category_types import Category_over_base_ring
-from sage.misc.abstract_method import abstract_method
+from sage.categories.category_types import Category_over_base_ring
+from sage.categories.commutative_rings import CommutativeRings
 from sage.categories.modules import Modules
-from sage.structure.element import coerce_binop
+from sage.misc.abstract_method import abstract_method
 from sage.misc.cachefunc import cached_method
-from sage.categories.commutative_rings import CommutativeRings
 from sage.misc.lazy_import import LazyImport
+from sage.structure.element import coerce_binop
+
 _CommutativeRings = CommutativeRings()
 
 
@@ -53,7 +54,7 @@ def __classcall_private__(cls, R, check=True):
             Category of Lie conformal algebras over Integer Ring
         """
         if check:
-            if not (R in _CommutativeRings):
+            if R not in _CommutativeRings:
                     raise ValueError("base must be a commutative ring got {}".format(R))
         return super().__classcall__(cls, R)
 
diff --git a/src/sage/categories/left_modules.py b/src/sage/categories/left_modules.py
index c92a13f7979..51de3497a53 100644
--- a/src/sage/categories/left_modules.py
+++ b/src/sage/categories/left_modules.py
@@ -8,9 +8,10 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
-from .category_types import Category_over_base_ring
+from sage.categories.category_types import Category_over_base_ring
 from sage.categories.commutative_additive_groups import CommutativeAdditiveGroups
 
+
 #?class LeftModules(Category_over_base_rng):
 class LeftModules(Category_over_base_ring):
     """
diff --git a/src/sage/categories/lie_conformal_algebras.py b/src/sage/categories/lie_conformal_algebras.py
index ee48889dd0b..8ade3811845 100644
--- a/src/sage/categories/lie_conformal_algebras.py
+++ b/src/sage/categories/lie_conformal_algebras.py
@@ -125,11 +125,12 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from .category_types import Category_over_base_ring
-from sage.misc.cachefunc import cached_method
+from sage.categories.category_types import Category_over_base_ring
 from sage.categories.lambda_bracket_algebras import LambdaBracketAlgebras
+from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_import import LazyImport
 
+
 class LieConformalAlgebras(Category_over_base_ring):
     r"""
     The category of Lie conformal algebras.
@@ -224,8 +225,9 @@ def example(self):
             sage: LieConformalAlgebras(QQ).example()                                    # needs sage.combinat sage.modules
             The Virasoro Lie conformal algebra over Rational Field
         """
-        from sage.algebras.lie_conformal_algebras.virasoro_lie_conformal_algebra\
-                                            import VirasoroLieConformalAlgebra
+        from sage.algebras.lie_conformal_algebras.virasoro_lie_conformal_algebra import (
+            VirasoroLieConformalAlgebra,
+        )
         return VirasoroLieConformalAlgebra(self.base_ring())
 
     def _repr_object_names(self):
@@ -284,8 +286,8 @@ def _test_jacobi(self, **options):
             """
             tester = self._tester(**options)
             S = tester.some_elements()
-            from sage.misc.misc import some_tuples
             from sage.arith.misc import binomial
+            from sage.misc.misc import some_tuples
             pz = tester._instance.zero()
             for x,y,z in some_tuples(S, 3, tester._max_runs):
                 brxy = x.bracket(y)
diff --git a/src/sage/categories/map.pyx b/src/sage/categories/map.pyx
index 8f09e3aa00d..fadd066d7f4 100644
--- a/src/sage/categories/map.pyx
+++ b/src/sage/categories/map.pyx
@@ -21,7 +21,7 @@ AUTHORS:
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from . import homset
+from sage.categories import homset
 import weakref
 from sage.ext.stdsage cimport HAS_DICTIONARY
 from sage.arith.power cimport generic_power
diff --git a/src/sage/categories/matrix_algebras.py b/src/sage/categories/matrix_algebras.py
index 90eef738be6..08b102d7451 100644
--- a/src/sage/categories/matrix_algebras.py
+++ b/src/sage/categories/matrix_algebras.py
@@ -10,8 +10,9 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
-from .category_types import Category_over_base_ring
-from .algebras import Algebras
+from sage.categories.algebras import Algebras
+from sage.categories.category_types import Category_over_base_ring
+
 
 class MatrixAlgebras(Category_over_base_ring):
     """
diff --git a/src/sage/categories/modular_abelian_varieties.py b/src/sage/categories/modular_abelian_varieties.py
index a1337e700cd..804361a7f99 100644
--- a/src/sage/categories/modular_abelian_varieties.py
+++ b/src/sage/categories/modular_abelian_varieties.py
@@ -10,11 +10,12 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
-from .category_types import Category_over_base
-from .category_with_axiom import CategoryWithAxiom
-from .homsets import HomsetsCategory
-from .rings import Rings
-from .sets_cat import Sets
+from sage.categories.category_types import Category_over_base
+from sage.categories.category_with_axiom import CategoryWithAxiom
+from sage.categories.homsets import HomsetsCategory
+from sage.categories.rings import Rings
+from sage.categories.sets_cat import Sets
+
 
 class ModularAbelianVarieties(Category_over_base):
     """
diff --git a/src/sage/categories/modules.py b/src/sage/categories/modules.py
index e65bab4faf8..c4f5127306f 100644
--- a/src/sage/categories/modules.py
+++ b/src/sage/categories/modules.py
@@ -11,21 +11,22 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.misc.cachefunc import cached_method
-from sage.misc.abstract_method import abstract_method
-from sage.misc.lazy_import import LazyImport
+from sage.categories.bimodules import Bimodules
+from sage.categories.cartesian_product import CartesianProductsCategory
+from sage.categories.category import Category
+from sage.categories.category_types import Category_module
 from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
-from sage.categories.morphism import SetMorphism
-from sage.categories.homsets import HomsetsCategory
+from sage.categories.dual import DualObjectsCategory
+from sage.categories.fields import Fields
 from sage.categories.homset import Hom
-from .category import Category
-from .category_types import Category_module
-from sage.categories.tensor import TensorProductsCategory, TensorProductFunctor, tensor
-from .dual import DualObjectsCategory
-from sage.categories.cartesian_product import CartesianProductsCategory
+from sage.categories.homsets import HomsetsCategory
+from sage.categories.morphism import SetMorphism
 from sage.categories.sets_cat import Sets
-from sage.categories.bimodules import Bimodules
-from sage.categories.fields import Fields
+from sage.categories.tensor import TensorProductFunctor, TensorProductsCategory, tensor
+from sage.misc.abstract_method import abstract_method
+from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_import import LazyImport
+
 _Fields = Fields()
 
 
diff --git a/src/sage/categories/morphism.pxd b/src/sage/categories/morphism.pxd
index 1a941b62b33..e5befc8207e 100644
--- a/src/sage/categories/morphism.pxd
+++ b/src/sage/categories/morphism.pxd
@@ -1,5 +1,5 @@
 from sage.structure.element cimport Element
-from .map cimport Map
+from sage.categories.map cimport Map
 
 
 cdef class Morphism(Map):
diff --git a/src/sage/categories/pointed_sets.py b/src/sage/categories/pointed_sets.py
index fd91b3b07f3..27376140dbe 100644
--- a/src/sage/categories/pointed_sets.py
+++ b/src/sage/categories/pointed_sets.py
@@ -11,7 +11,8 @@
 #******************************************************************************
 
 from sage.categories.category_singleton import Category_singleton
-from .sets_cat import Sets
+from sage.categories.sets_cat import Sets
+
 
 class PointedSets(Category_singleton):
     """
diff --git a/src/sage/categories/pushout.py b/src/sage/categories/pushout.py
index 3b7b8238d52..a23557037b0 100644
--- a/src/sage/categories/pushout.py
+++ b/src/sage/categories/pushout.py
@@ -27,9 +27,9 @@
 
 import operator
 
+from sage.categories.functor import Functor, IdentityFunctor_generic
 from sage.misc.lazy_import import lazy_import
 from sage.structure.coerce_exceptions import CoercionException
-from .functor import Functor, IdentityFunctor_generic
 
 lazy_import('sage.categories.commutative_additive_groups', 'CommutativeAdditiveGroups')
 lazy_import('sage.categories.commutative_rings', 'CommutativeRings')
@@ -902,7 +902,9 @@ def _apply_functor_to_morphism(self, f):
                       From: Integer Ring
                       To:   Finite Field of size 3
         """
-        from sage.rings.polynomial.polynomial_ring_homomorphism import PolynomialRingHomomorphism_from_base
+        from sage.rings.polynomial.polynomial_ring_homomorphism import (
+            PolynomialRingHomomorphism_from_base,
+        )
         R = self._apply_functor(f.domain())
         S = self._apply_functor(f.codomain())
         return PolynomialRingHomomorphism_from_base(R.Hom(S), f)
@@ -1329,7 +1331,9 @@ def _apply_functor(self, R):
             Infinite polynomial ring in a, b, x over Univariate Polynomial Ring in t over Rational Field
 
         """
-        from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing
+        from sage.rings.polynomial.infinite_polynomial_ring import (
+            InfinitePolynomialRing,
+        )
         return InfinitePolynomialRing(R, self._gens, order=self._order, implementation=self._imple)
 
     def _repr_(self):
@@ -1795,7 +1799,9 @@ def _apply_functor(self, R):
 
         """
         from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
-        from sage.rings.polynomial.laurent_polynomial_ring_base import LaurentPolynomialRing_generic
+        from sage.rings.polynomial.laurent_polynomial_ring_base import (
+            LaurentPolynomialRing_generic,
+        )
         if self.multi_variate and isinstance(R, LaurentPolynomialRing_generic):
             return LaurentPolynomialRing(R.base_ring(), list(R.variable_names()) + [self.var])
         else:
@@ -2474,8 +2480,8 @@ def __init__(self):
             sage: F(ZZ['t'])
             Fraction Field of Univariate Polynomial Ring in t over Integer Ring
         """
-        from sage.categories.integral_domains import IntegralDomains
         from sage.categories.fields import Fields
+        from sage.categories.integral_domains import IntegralDomains
         Functor.__init__(self, IntegralDomains(), Fields())
 
     def _apply_functor(self, R):
@@ -3383,8 +3389,8 @@ def _apply_functor(self, R):
             3-adic Eisenstein Extension Field in a defined by a^2 - 3
 
         """
-        from sage.rings.rational_field import QQ
         from sage.rings.integer_ring import ZZ
+        from sage.rings.rational_field import QQ
         if self.cyclotomic:
             from sage.rings.number_field.number_field import CyclotomicField
             if R == QQ:
diff --git a/src/sage/categories/r_trivial_semigroups.py b/src/sage/categories/r_trivial_semigroups.py
index 93ba9e61ca8..7b5c9fea5ed 100644
--- a/src/sage/categories/r_trivial_semigroups.py
+++ b/src/sage/categories/r_trivial_semigroups.py
@@ -13,7 +13,8 @@
 #*****************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
-from .semigroups import Semigroups
+from sage.categories.semigroups import Semigroups
+
 
 class RTrivialSemigroups(CategoryWithAxiom):
     def extra_super_categories(self):
diff --git a/src/sage/categories/right_modules.py b/src/sage/categories/right_modules.py
index 5a7557c4663..47a8081d2a9 100644
--- a/src/sage/categories/right_modules.py
+++ b/src/sage/categories/right_modules.py
@@ -8,9 +8,10 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
-from .category_types import Category_over_base_ring
+from sage.categories.category_types import Category_over_base_ring
 from sage.categories.commutative_additive_groups import CommutativeAdditiveGroups
 
+
 ##?class RightModules(Category_over_base_rng):
 class RightModules(Category_over_base_ring):
     """
diff --git a/src/sage/categories/ring_ideals.py b/src/sage/categories/ring_ideals.py
index 51573635ef0..fd634f1caef 100644
--- a/src/sage/categories/ring_ideals.py
+++ b/src/sage/categories/ring_ideals.py
@@ -10,9 +10,10 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from .category_types import Category_ideal
-from .modules import Modules
+from sage.categories.category_types import Category_ideal
+from sage.categories.modules import Modules
 from sage.categories.rings import Rings
+
 _Rings = Rings()
 
 
diff --git a/src/sage/categories/semirings.py b/src/sage/categories/semirings.py
index cbff0b29a8e..62e6f7d918f 100644
--- a/src/sage/categories/semirings.py
+++ b/src/sage/categories/semirings.py
@@ -9,7 +9,8 @@
 #******************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
-from .magmas_and_additive_magmas import MagmasAndAdditiveMagmas
+from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas
+
 
 class Semirings(CategoryWithAxiom):
     """
diff --git a/src/sage/categories/semisimple_algebras.py b/src/sage/categories/semisimple_algebras.py
index 9a310614290..453b854b7e7 100644
--- a/src/sage/categories/semisimple_algebras.py
+++ b/src/sage/categories/semisimple_algebras.py
@@ -8,12 +8,13 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.categories.algebras import Algebras
+from sage.categories.category_types import Category_over_base_ring
+from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
 from sage.misc.bindable_class import BoundClass
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_import import LazyImport
-from .category_types import Category_over_base_ring
-from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
-from .algebras import Algebras
+
 
 class SemisimpleAlgebras(Category_over_base_ring):
     """
diff --git a/src/sage/categories/sets_with_grading.py b/src/sage/categories/sets_with_grading.py
index b1e321f67f2..a29d8c9c59c 100644
--- a/src/sage/categories/sets_with_grading.py
+++ b/src/sage/categories/sets_with_grading.py
@@ -8,11 +8,11 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.misc.cachefunc import cached_method
-from sage.misc.abstract_method import abstract_method
-from .category_types import Category
-from sage.categories.sets_cat import Sets
+from sage.categories.category_types import Category
 from sage.categories.enumerated_sets import EnumeratedSets
+from sage.categories.sets_cat import Sets
+from sage.misc.abstract_method import abstract_method
+from sage.misc.cachefunc import cached_method
 
 
 class SetsWithGrading(Category):
@@ -220,9 +220,9 @@ def generating_series(self):
 
                  - Very likely, this should always return a lazy power series.
             """
-            from sage.sets.non_negative_integers import NonNegativeIntegers
-            from sage.rings.lazy_series_ring import LazyPowerSeriesRing
             from sage.rings.integer_ring import ZZ
+            from sage.rings.lazy_series_ring import LazyPowerSeriesRing
+            from sage.sets.non_negative_integers import NonNegativeIntegers
             if isinstance(self.grading_set(), NonNegativeIntegers):
                 R = LazyPowerSeriesRing(ZZ, names="z")
                 return R(lambda n: self.graded_component(n).cardinality())
diff --git a/src/sage/categories/sets_with_partial_maps.py b/src/sage/categories/sets_with_partial_maps.py
index 5b2f9b35427..4210548b0fd 100644
--- a/src/sage/categories/sets_with_partial_maps.py
+++ b/src/sage/categories/sets_with_partial_maps.py
@@ -11,7 +11,7 @@
 #******************************************************************************
 
 from sage.categories.category_singleton import Category_singleton
-from .objects import Objects
+from sage.categories.objects import Objects
 
 
 class SetsWithPartialMaps(Category_singleton):