In mathematics, The fundamental theorem of topos theory states that the slice
which preserves exponentials and the subobject classifier.
there is an associated "pullback functor"
which is key in the proof of the theorem.
which shares the same codomain as f, their product
is the diagonal of their pullback square, and the morphism which goes from the domain of
to the domain of f is opposite to g in the pullback square, so it is the pullback of g along f, which can be denoted as
Note that a topos
is isomorphic to the slice over its own terminal object, i.e.
and thereby a pullback functor
denote an object of it, where X is an object of the base category.
is a functor which maps:
This yields so this is how the pullback functor
maps objects of
Furthermore, note that any element C of the base topos is isomorphic to
is indeed a functor from the base topos
Consider a pair of ground formulas
ϕ
ϕ ]
(where the underscore here denotes the null context) are objects of the base topos.
ϕ
ϕ ]
If these are the case then, by theorem, the formula
is true in the slice
ϕ ]
, because the terminal object
of the slice factors through its extension
In logical terms, this could be expressed as so that slicing
would correspond to assuming
Then the theorem would say that making a logical assumption does not change the rules of topos logic.