Let us denote by the set of all differentiable maps from to . A so-called ``de Rham cohomology'' of is computed as a cohomology of a complex
We see that:
Actually, the dimension of the 0 -th cohomology is related to a number of the connected component of . The dimension of the -st cohomology is related to a number of the `hole' of . Cohomology is then a good tool to obtain numbers (``invariants'') of geometric objects.
Cohomology also arises as ``obstructions''. Indeed, the de Rham cohomology of the tells us a hint about ``which functions are integrable'', etc.
In this talk we give a definition and explain some basic properties of cohomologies. But before that, we first deal with some category theory.
holds for any ( ).
In this text we always assume the following.
For any set , there always exists a universe such that .