0cochains

0-cochains are the degree-zero elements of a cochain complex used in algebraic topology. For a topological space X and an abelian group A (or a ring), the zeroth cochain group C^0(X; A) consists of all functions that assign to every 0-simplex a coefficient in A. In singular cohomology this corresponds to functions f: X → A, since the 0-simplices are just points of X; in simplicial cohomology, C^0 is the direct product over the vertex set V of A, i.e., functions f: V → A.

The cochains form an abelian group under pointwise addition. There is a coboundary operator δ: C^0 → C^1, defined on a 1-simplex σ by (δf)(σ) = f(σ(0)) − f(σ(1)) (sign conventions may vary). Thus δf is a 1-cochain measuring the difference of f’s values at the endpoints of edges.

A 0-cochain f is a 0-cocycle if δf = 0, which means f assigns the same value to the endpoints of every edge, i.e., f is locally constant and constant on each connected component of X.

In standard (non-reduced) cohomology, B^0 = δ(C^{-1}) is zero because there are no negative-degree cochains, so H^0(X; A) ≅ Z^0(X; A). This group can be described as the group of locally constant A-valued functions on X, equivalently the set of functions from the set of connected components π0(X) to A.

0-cochains provide the starting point for defining higher cochains and the full cohomology theory, and their structure reflects the decomposition of X into connected components.