Trac #31669: add differentials to category

src/sage/categories/chain_complexes.py

@@ -87,6 +87,30 @@ def homology(self, *args, **kwargs):
             """
             raise NotImplementedError
 
+        def differential(self, *args, **kwargs):
+            r"""
+            Return the differentials (or boundary maps) of the chain complex.
+
+            EXAMPLES::
+
+                sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
+                sage: C.differential(0)
+                [3 0 0]
+                [0 0 0]
+
+            ::
+
+                sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=(2, 2, 3))
+                sage: C = A.cdg_algebra({z: x*y})
+                sage: C.differential()
+                Differential of Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (2, 2, 3) over Rational Field
+                  Defn: x --> 0
+                        y --> 0
+                        z --> x*y
+
+            """
+            raise NotImplementedError
+
     class MorphismMethods:
         def induced_homology_morphism(self, *args, **kwargs):
             r"""

src/sage/homology/chain_complex.py

@@ -957,6 +957,7 @@ def differential(self, dim=None):
         except KeyError:
             pass
         # all differentials that are not 0x0 are in self._diff
+        # TODO: turn differentials into morphisms between free modules?
         return matrix(self.base_ring(), 0, 0)
 
     def dual(self):