For any two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, we show that if the n-th valued hyperfields of K1
Nearby in the stack
and
K2
are isomorphic over
p
for each
n≥1
, then
K1
and
K2
are isomorphic. More generally, for
n1,n2≥1
, if
n2
is large enough, then any homomorphism, which is over
p
, from the
n1
-th valued hyperfield of
K1
to the
n2
-th valued hyperfield of
K2
can be lifted to a homomorphism from
K1
to
K2
. We compute such
n2
effectively, which depends only on the ramification indices of
K1
and
K2
. Moreover, if
K1
is tamely ramified, then any homomorphism over
p
between the first valued hyperfields is induced from a unique homomorphism of valued fields. Using this lifting result, we deduce a relative completeness theorem of AKE-style in terms of valued hyperfields. We also study some relationships between valued hyperfields, truncated discrete valuation rings, and complete discrete valued fields of mixed characteristic. For a prime number
p
and a positive integer
e
and for large enough
n
, we show that a certain category of valued hyperfields is equivalent to the category of truncated discrete valuation rings of length
n
and the ramification indices
e
having perfect residue fields of characteristic
p
. Furthermore, in the tamely ramified case, we show that a subcategory of this category of valued hyperfields is equivalent to the category of complete discrete valued rings of mixed characteristic