gh-35279: sage.categories: Modularization fixes for imports

    
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### 📚 Description

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We should be able to import Python modules from `sage.categories` even
if implementation modules are not installed; that's the premise of the
distribution **sagemath-categories**.
Here we make changes to avoid module-level dependencies on some higher-
level functionality, such as symbolic functions, linear algebra, etc.
<!-- Why is this change required? What problem does it solve? -->
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example "Closes #1337" -->

#35275 takes care of some other imports.

Part of
- #29705

### 📝 Checklist

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- [ ] I have linked an issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies
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URL: #35279
Reported by: Matthias Köppe
Reviewer(s): Kwankyu Lee

src/sage/categories/finite_complex_reflection_groups.py

@@ -16,7 +16,6 @@
 from sage.misc.cachefunc import cached_method
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.coxeter_groups import CoxeterGroups
-from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet
 
 
 class FiniteComplexReflectionGroups(CategoryWithAxiom):
@@ -758,6 +757,8 @@ def absolute_order_ideal(self, gens=None,
                     [1] 1
                     [] 0
                 """
+                from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet
+
                 if gens is None:
                     seeds = [(self.coxeter_element(), self.rank())]
                 else:

src/sage/categories/finite_dimensional_lie_algebras_with_basis.py

@@ -24,9 +24,7 @@
 from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
 from sage.categories.lie_algebras import LieAlgebras
 from sage.categories.subobjects import SubobjectsCategory
-from sage.algebras.free_algebra import FreeAlgebra
 from sage.sets.family import Family
-from sage.matrix.constructor import matrix
 
 
 class FiniteDimensionalLieAlgebrasWithBasis(CategoryWithAxiom_over_base_ring):
@@ -97,6 +95,8 @@ def _construct_UEA(self):
                  The 6-Witt Lie algebra over Ring of integers modulo 6
                  in the Poincare-Birkhoff-Witt basis
             """
+            from sage.algebras.free_algebra import FreeAlgebra
+
             # Create the UEA relations
             # We need to get names for the basis elements, not just the generators
             I = self._basis_ordering
@@ -321,6 +321,8 @@ def killing_form_matrix(self):
                 sage: parent(m)
                 Full MatrixSpace of 0 by 0 dense matrices over Rational Field
             """
+            from sage.matrix.constructor import matrix
+
             B = self.basis()
             m = matrix(self.base_ring(),
                        [[self.killing_form(x, y) for x in B] for y in B])
@@ -420,6 +422,8 @@ def centralizer_basis(self, S):
                  D{1} + D{1, 2} + D{2, 3} + D{3},
                  D{1, 2, 3} + D{1, 3} + D{2})
             """
+            from sage.matrix.constructor import matrix
+
             #from sage.algebras.lie_algebras.subalgebra import LieSubalgebra
             #if isinstance(S, LieSubalgebra) or S is self:
             if S is self:
@@ -541,6 +545,7 @@ def derivations_basis(self):
             :wikipedia:`Derivation_(differential_algebra)`
             """
             from sage.matrix.constructor import matrix
+
             R = self.base_ring()
             B = self.basis()
             keys = list(B.keys())
@@ -581,6 +586,8 @@ def inner_derivations_basis(self):
                 [1 0 0], [0 1 0]
                 )
             """
+            from sage.matrix.constructor import matrix
+
             R = self.base_ring()
             IDer = matrix(R, [b.adjoint_matrix().list() for b in self.basis()])
             N = self.dimension()
@@ -691,6 +698,9 @@ def is_ideal(self, A):
             if A not in LieAlgebras(self.base_ring()).FiniteDimensional().WithBasis():
                 raise NotImplementedError("A must be a finite dimensional"
                                           " Lie algebra with basis")
+
+            from sage.matrix.constructor import matrix
+
             B = self.basis()
             AB = A.basis()
             try:
@@ -798,6 +808,8 @@ def product_space(self, L, submodule=False):
                 Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
                 ()
             """
+            from sage.matrix.constructor import matrix
+
             # Make sure we lift everything to the ambient space
             if self in LieAlgebras(self.base_ring()).Subobjects():
                 A = self.ambient()
@@ -1373,6 +1385,8 @@ def as_finite_dimensional_algebra(self):
                 sage: X * Y
                 Z
             """
+            from sage.matrix.constructor import matrix
+
             K = self._basis_ordering
             mats = []
             R = self.base_ring()
@@ -1606,6 +1620,8 @@ def adjoint_matrix(self, sparse=False): # In #11111 (more or less) by using matr
                 sage: E1 * E2 - E2 * E1 == e12.adjoint_matrix()
                 True
             """
+            from sage.matrix.constructor import matrix
+
             P = self.parent()
             basis = P.basis()
             return matrix(self.base_ring(),

src/sage/categories/loop_crystals.py

@@ -19,8 +19,7 @@
 from sage.categories.regular_crystals import RegularCrystals
 from sage.categories.tensor import TensorProductsCategory
 from sage.categories.map import Map
-from sage.graphs.dot2tex_utils import have_dot2tex
-from sage.functions.other import ceil
+
 
 class LoopCrystals(Category_singleton):
     r"""
@@ -117,6 +116,8 @@ def digraph(self, subset=None, index_set=None):
                 {...'edge_options': <function ... at ...>...}
                 sage: view(G, tightpage=True)  # optional - dot2tex graphviz, not tested (opens external window)
             """
+            from sage.graphs.dot2tex_utils import have_dot2tex
+
             G = Crystals().parent_class.digraph(self, subset, index_set)
             if have_dot2tex():
                 def eopt(u_v_label):
@@ -959,6 +960,8 @@ def energy_function(self, algorithm=None):
                     ....:     for b in hw)
                     True
                 """
+                from sage.functions.other import ceil
+
                 C = self.parent().crystals[0]
                 ell = ceil(C.s()/C.cartan_type().c()[C.r()])
                 is_perfect = all(ell == K.s()/K.cartan_type().c()[K.r()]
@@ -1090,6 +1093,8 @@ def e_string_to_ground_state(self):
                     ....:     for elt in hw)
                     True
                 """
+                from sage.functions.other import ceil
+
                 ell = max(ceil(K.s()/K.cartan_type().c()[K.r()])
                           for K in self.parent().crystals)
                 if self.cartan_type().dual().type() == 'BC':

src/sage/categories/pushout.py

@@ -3463,8 +3463,8 @@ def merge(self, other):
             # nothing else helps, hence, we move to the pushout of the codomains of the embeddings
             try:
                 P = pushout(self.embeddings[0].parent(), other.embeddings[0].parent())
-                from sage.rings.number_field.number_field import is_NumberField
-                if is_NumberField(P):
+                from sage.rings.number_field.number_field_base import NumberField
+                if isinstance(P, NumberField):
                     return P.construction()[0]
             except CoercionException:
                 return None

src/sage/categories/rings.py

@@ -1159,7 +1159,10 @@ def normalize_arg(arg):
 
             if isinstance(arg, tuple):
                 from sage.categories.morphism import Morphism
-                from sage.rings.derivation import RingDerivation
+                try:
+                    from sage.rings.derivation import RingDerivation
+                except ImportError:
+                    RingDerivation = ()
                 if len(arg) == 2 and isinstance(arg[1], (Morphism, RingDerivation)):
                     from sage.rings.polynomial.ore_polynomial_ring import OrePolynomialRing
                     return OrePolynomialRing(self, arg[1], names=arg[0])