Constant SheafEdit
Constant sheaves are a cornerstone of sheaf theory, providing the simplest way to package data that is fixed across a space but may be viewed and assembled locally. Given a topological space X and a fixed algebraic object A (for example, an abelian group, a ring, or a set), the constant sheaf with value A captures the idea that every open set carries copies of A, and when you restrict to overlaps the data matches in a straightforward way. The standard notation for this object is the underline{A}, and it sits at the crossroads of topology, algebra, and geometry. In practice, constant sheaves serve as the default coefficient system in many cohomology theories and as a testing ground for ideas about how local data determines global structure.
Construction
Start with a topological space X and a fixed algebraic object A. The constant presheaf underline{A} is defined by assigning to every open set U ⊆ X the set (or group, ring, etc.) A, with restriction maps being the identity on A.
The presheaf underline{A} is usually not a sheaf. The true constant sheaf is obtained by sheafifying this presheaf. The resulting sheaf is denoted underline{A} as well, and it is the sheaf of locally constant functions from open sets into A.
The sections over an open set U are precisely the locally constant functions U → A. In particular, if U is connected, a locally constant function is constant, so Γ(U, underline{A}) ≅ A. If U has several connected components, Γ(U, underline{A}) is the product of copies of A indexed by the connected components of U; equivalently, Γ(U, underline{A}) ≅ A^{π0(U)}, where π0(U) is the set of connected components of U.
The stalks of underline{A} are all naturally isomorphic to A: for any x ∈ X, the stalk underline{A}_x ≅ A. This reflects the idea that the fiber seen at a point does not depend on the position in X.
If A carries extra structure (for instance, A is an abelian group or a ring), underline{A} inherits the corresponding sheaf structure. Thus one may speak of the constant sheaf of abelian groups, constant sheaf of rings, and so on.
Locally constant and trivial monodromy
The constant sheaf is the prototypical locally constant sheaf. A locally constant sheaf on X is one that, around every point, looks like a constant sheaf when restricted to suitably small neighborhoods. The constant sheaf corresponds to the trivial local system with fiber A.
In a broader categorical view, locally constant sheaves relate to representations of the fundamental group. The constant sheaf corresponds to the trivial representation, i.e., monodromy that does not twist the fiber as one moves around loops in X. See also Local system for the general framework.
The relationship with stalks is essential: knowing the stalks and how they glue along overlaps allows one to reconstruct the global behavior of a locally constant sheaf. The constant sheaf is the simplest case where all stalks are the same fixed object A and the gluing is governed by identity-type data.
Examples and special cases
Constant sheaf of integers: underline{Z} is the constant sheaf with value the group of integers. On any connected open U, Γ(U, underline{Z}) ≅ Z, and on a general U with c components, Γ(U, underline{Z}) ≅ Z^{c}.
Constant sheaf of real numbers or complex numbers: underline{R} or underline{C} serves as the standard coefficient system in many topological constructions and in various cohomology theories.
On a connected space X, the global sections of underline{A} recover A: Γ(X, underline{A}) ≅ A. On a disconnected X with components X1, X2, ..., the global sections decompose as Γ(X, underline{A}) ≅ A × A × ... ≅ A^{π0(X)}.
Properties and applications
The constant sheaf is a basic example of a sheaf of algebraic objects. Its behavior is governed by the topology of X through the decomposition into connected components, while its fiber is literally the fixed object A.
In cohomology, underline{A} serves as the canonical coefficient system. Its zeroth cohomology H^0(X, underline{A}) amounts to the space of global sections, equivalently the functions that pick an element of A for each connected component of X. Higher cohomology with constant coefficients detects more subtle global obstructions to gluing local data, and this constant sheaf often provides a concrete starting point for computations in sheaf cohomology and Čech cohomology.
The constant sheaf is closely connected to the distinction between local behavior and global structure. The fact that sections over connected open sets look like A makes underline{A} a natural baseline against which more complicated locally constant or constructible sheaves are measured.
In geometry and topology, constant sheaves appear in various constructions where a fixed amount of data is attached to every patch of space. They also provide intuition for more general coefficient systems used in topological field theories, the study of manifolds, and in the algebraic setting where similar ideas are mirrored in schemes and etale topology. See Topological space and Sheaf (mathematics) for foundational context, and Local system for the broader picture of how fibers and monodromy interact.