  What is the role of topos theory in the topos models for quantum theory as
used by Isham, Butterfield, Doring, Heunen, Landsman, Spitters and others? In
other words, what is the interplay between physical motivation for the models
and the mathematical framework used in these models? Concretely, we show that
the presheaf topos model of Butterfield, Isham and Doring resembles classical
physics when viewed from the internal language of the presheaf topos, similar
to the copresheaf topos model of Heunen, Landsman and Spitters. Both the
presheaf and copresheaf models provide a `quantum logic' in the form of a
complete Heyting algebra. Although these algebras are natural from a topos
theoretic stance, we seek a physical interpretation for the logical operations.
Finally, we investigate dynamics. In particular we describe how an automorphism
on the operator algebra induces a homeomorphism (or isomorphism of locales) on
the associated state spaces of the topos models, and how elementary
propositions and truth values transform under the action of this homeomorphism.
Also with dynamics the focus is on the internal perspective of the topos.
