Functor
A functor is a type of map between two categories.
More precisely, a functor is a mapping which
- sends every object
of
to and object
of
.
- sends every morphism
of
to a morphism
of
.
Which satisfies the conditions:
for all
.
for all morphisms
and
of
.
A contravariant functor a mapping satisfying the same properties as above, except that is a morphism from
to
, and instead of having
we have
. Alternatively, we can define a contravariant functor as an ordinary functor
, where
is the opposite category of
. We sometimes call our original type of functors covariant functor to distinguish them from contravariant functors.
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