Editing Locally constant function (section)

== Connection with sheaf theory ==

There are {{em|sheaves}} of locally constant functions on <math>X.</math> To be more definite, the locally constant integer-valued functions on <math>X</math> form a [[Sheaf (mathematics)|sheaf]] in the sense that for each open set <math>U</math> of <math>X</math> we can form the functions of this kind; and then verify that the sheaf {{em|axioms}} hold for this construction, giving us a sheaf of [[abelian group]]s (even [[commutative ring]]s).<ref>{{cite book |last1=Hartshorne |first1=Robin |title=Algebraic Geometry |date=1977 |publisher=Springer |page=62}}</ref> This sheaf could be written <math>Z_X</math>; described by means of {{em|stalks}} we have stalk <math>Z_x,</math> a copy of <math>Z</math> at <math>x,</math> for each <math>x \in X.</math> This can be referred to a {{em|constant sheaf}}, meaning exactly {{em|sheaf of locally constant functions}} taking their values in the (same) group. The typical sheaf of course is not constant in this way; but the construction is useful in linking up [[sheaf cohomology]] with [[homology theory]], and in logical applications of sheaves. The idea of [[local coefficient system]] is that we can have a theory of sheaves that {{em|locally}} look like such 'harmless' sheaves (near any <math>x</math>), but from a global point of view exhibit some 'twisting'.