  We develop a theory of internal Hochschild cohomology in a ringed topos. We
construct it via the internal Hochschild cochain complex, as well as through
derived functor/topos cohomology theory, and discuss its relationship to the
absolute Hochschild cohomology.
  By specialising to the topos of difference sets, we obtain a theory of
internal difference Hochschild cohomology, and compare it to the absolute
Hochschild cohomology through the Grothendieck and hypercohomology spectral
sequences.
  We provide a systematic and detailed treatment of tensor products in suitable
toposes in hope to complete the existing literature.
