  The aim of this paper is to compare the two topos-theoretic approaches to
quantum mechanics that may be found in the literature to date. The first
approach, which we will call the contravariant approach, was originally
proposed by Isham and Butterfield, and was later extended by Doering and Isham.
The second approach, which we will call the covariant approach, was developed
by Heunen, Landsman and Spitters. Motivated by coarse-graining and the
Kochen-Specker theorem, the contravariant approach uses the topos of presheaves
on a specific context category, defined as the poset of commutative von Neumann
subalgebras of some given von Neumann algebra. In particular, the approach uses
the spectral presheaf. The intuitionistic logic of this approach is given by
the (complete) Heyting algebra of closed open subob jects of the spectral
presheaf. We show that this Heyting algebra is, in a natural way, a locale in
the ambient topos, and compare this locale with the internal Gelfand spectrum
of the covariant approach. In the covariant approach, a non-commutative
C*-algebra (in the topos Set) defines a commutative C*-algebra internal to the
topos of covariant functors from the context category to the category of sets.
We give an explicit description of the internal Gelfand spectrum of this
commutative C*-algebra, from which it follows that the external spectrum is
spatial. Using the daseinisation of self-adjoint operators from the
contravariant approach, we give a new definition of the daseinisation arrow in
the covariant approach and compare it with the original version. States and
state-proposition pairing in both approaches are compared. We also investigate
the physical interpretation of the covariant approach.
