>>11805699Have you tried googling, my friend?
https://en.wikipedia.org/wiki/Representable_functorFunctors can be isomorphic, e.g. in a Cartesian closed category (e.g. a category of sets) consider the adjunction between the function spaces
and
realized by curring.
E.g. being mapped to
The same adjunction is given in logic by
having the same prove value as
(if P and Q together imply R, then also: P implies that Q implies R)
Or on a completely decategorified level in arithmetic
is the same number as
In category theory, you abstract away from all of those, and realize that in the right categories
is the same functor (by a natural iso) as
which is admittedly a bit wild.
Think of the two as "one map" but then with an image gliding between different representations. (Although I guess some people would argue that in some categories you don't want to think on the object level)
The representable functor are those iso to something which maps from a given object into an internalized homset.
>>11805814I don't disagree that all of this is nice, but I also don't see why that would one discourage from that latter?
>>11805821Actually, I was thinking of starting a fast. Salt and water for a few days.
