Cellular sheaves, described in Elementary Applied Topology and Justin Curry’s thesis, seem much more intuitive than their more complex brethren defined over general topological spaces, even more so when we consider them with values in vector spaces. But they still keep enough of the interesting behavior of noncellular sheaves to sometimes be confusing.

  • Nontrivial sheaves need not have nontrivial global sections.

A section in EAT implies that this is in fact the case for cellular sheaves with all stalks \(\mathbb{Z}\), but this is not the case. Consider a sheaf defined on the boundary of a 2-simplex, with all stalks \(\mathbb{Z}\) (or equivalently \(\mathbb{R}\)). At each vertex, let the clockwise restriction map be multiplication by -1, and the counterclockwise restriction map be the identity. Note that this forces elements assigned to adjacent vertices by a section to have opposite signs, but since there are only three vertices, this cannot be the case. Therefore only the trivial section exists.

  • Short exact sequences of sheaves valued in vector spaces need not split.

The counterexample is extremely simple. Let the base space be a 1-simplex, and consider the short exact sequence \(0 \to \mathcal{A} \to \mathcal{B} \to \mathcal{C} \to 0,\) where \(\mathcal{A}\) assigns the zero vector space to both vertices and \(\mathbb{R}\) to the edge, \(\mathcal{B}\) assigns \(\mathbb{R}\) to all cells, with identity restriction maps, and \(\mathcal{C}\) assigns \(\mathbb{R}\) to the vertices and 0 to the edge. The sheaf morphisms either restrict to the identity or the zero map on each cell. Note that \(\mathcal{A}\oplus\mathcal{C} \neq \mathcal{B}\) because the restriction maps on the direct sum are the zero map, not the identity.

The hard part in coming up with this example is working out exactly what injective and surjective morphisms of cellular sheaves are supposed to be. In fact, if a morphism is injective or surjective over every cell, then it is an injective/surjective cellular sheaf morphism. This is a special case of a more general fact: that injectivity and surjectivity of sheaf morphisms is determined by their behavior on stalks. It’s not necessary to go all the way to stalks for injectivity, though.

  • The global section functor (\(H^0\)) is not right exact.

Taking global sections of the previous example, we get the sequence \(0 \to 0 \to \mathbb{R} \to \mathbb{R} \oplus \mathbb{R} \to 0,\) which is not exact at the second to last term. However, note that \(H^1\) of \(\mathcal{A}\) is \(\mathbb{R}\), and so we can extend this sequence by the cohomology functor to get a long exact sequence \(0 \to 0 \to \mathbb{R} \to \mathbb{R} \oplus \mathbb{R} \to \mathbb{R} \to 0 \to \cdots.\)