  The article is devoted to a comparison of the \v{C}ech cohomology with the
coefficients in a presheaf of Abelian groups and the topos cohomology of the
sheaf generated by this presheaf for a poset with the Aleksandrov topology. The
article consists of three parts. The first part provides information from the
theory of cohomology of small categories and cohomology of simplicial sets with
systems of coefficients. The second part is devoted to Laudal's Theorem stating
that covering cohomology for an arbitrary topological space with coefficients
in the presheaf of Abelian groups is isomorphic to the derived limit functors.
The third part presents the main results. The criterion for the invariance of
cohomology groups of small categories when passing to the inverse image leads
to necessary and sufficient isomorphism conditions for the \v{C}ech cohomology
of an arbitrary presheaf and the topos cohomology of the sheaf generated by
this presheaf. In particular, for a finite poset, these conditions reduce to
the acyclicity of the upper secrions of Dedekind-MacNeille cuts having a
non-empty lower section, and the verification of these conditions is
algorithmically computable.
