Trac #31703: first attempt

src/sage/manifolds/sheaf.py

@@ -0,0 +1,102 @@
+r"""
+Presheaves and Sheaves over Manifolds
+
+AUTHORS:
+
+- Michael Jung (2021): initial version
+
+"""
+
+#******************************************************************************
+#       Copyright (C) 2021 Michael Jung <m.jung at vu.nl>
+#
+#  Distributed under the terms of the GNU General Public License (GPL)
+#  as published by the Free Software Foundation; either version 2 of
+#  the License, or (at your option) any later version.
+#                  http://www.gnu.org/licenses/
+#******************************************************************************
+
+from sage.structure.sage_object import SageObject
+from sage.misc.abstract_method import abstract_method
+
+class Presheaf(SageObject):
+    r"""
+
+    """
+    def __init__(self, manifold, section_set_constructor):
+        r"""
+
+        EXAMPLES::
+
+            sage: from sage.manifolds.sheaf import Presheaf
+            sage: M = Manifold(2, 'M')
+            sage: F = Presheaf(M, lambda d: d.scalar_field_algebra()); F
+            ...
+            sage: U = M.open_subset()
+            sage: F(U)
+            ...
+
+        """
+        if manifold is not self._manifold:
+            raise ValueError
+        self._manifold = manifold
+        self._section_set_constructor = section_set_constructor
+
+    def __call__(self, open_subset):
+        r"""
+        Return the set of sections of ``self`` w.r.t. ``open_subset``.
+
+        """
+        if not open_subset.is_subset(self._manifold):
+            raise ValueError
+        return self._section_set_constructor(open_subset)
+
+    def section_set(self, open_subset):
+        r"""
+        Return the set of sections of ``self`` w.r.t. ``open_subset``.
+
+        """
+        return self(open_subset)
+
+    def restriction_morphism(self, from_open_subset, to_open_subset):
+        r"""
+        Return the restriction morphism from ``from_open_subset`` to
+        ``to_open_subset``.
+
+        """
+        if not from_open_subset.is_subset(self._manifold) or not to_open_subset.is_subset(self._manifold):
+            raise ValueError
+        from sage.categories.morphism import SetMorphism
+        from sage.categories.homset import Hom
+        from sage.categories.sets_cat import Sets
+        sec_dom = self.section_set(from_open_subset)
+        sec_codom = self.section_set(to_open_subset)
+        return SetMorphism(Hom(sec_dom, sec_codom, Sets()),
+                           lambda x: x.restrict(to_open_subset))
+
+class PresheafSection(SageObject):
+    r"""
+
+    """
+    def __init__(self, domain):
+        r"""
+
+        """
+        self._domain = domain
+
+    def restrict(self, open_subset):
+        r"""
+        Return the restriction of ``self`` to ``open_subset``.
+
+        """
+        # try to get restriction from restriction graph
+        # ...
+        # if that fails, create restriction from scratch
+        return self._construct_restriction_(open_subset)
+
+    @abstract_method
+    def _construct_restriction_(self, open_subset):
+        r"""
+        Construct a restriction of ``self`` to ``open_subset`` from scratch.
+
+        """