# why is power set functor contravariant?

• September 12th 2012, 09:53 AM
why is power set functor contravariant?
I have read that the powerset constuction when viewed as an endofunctor is contravariant but I cant find any explanation of why it is contravariant and not covariant. Any clues? :confused:

ie: if f is an arrow A->B why is the arrow P(f) an arrow PB->PA

I did wonder if the author was saying if you view it as contravariant as opposed to covariant then various other things follow. So is it necessarily contravariant?

Thanks,
• September 12th 2012, 10:44 AM
emakarov
Re: why is power set functor contravariant?
If P(f)(A) is defined as f[A], then P is covariant. If P(f)(A) is defined as $f^{-1}[A]$, then it is contravariant.
• September 12th 2012, 11:00 AM
Re: why is power set functor contravariant?
So it can be either covariant or contravariant depending on which way you want to define it...?
• September 12th 2012, 11:18 AM
emakarov
Re: why is power set functor contravariant?
Yes.
• September 13th 2012, 08:16 AM
Deveno
Re: why is power set functor contravariant?
the covariant power-set functor arrow P(f) is not a lattice isomorphism (considering meet as intersection, and join as union) from P(A) to P(B), whereas the contravariant P(f) is from P(B) to P(A). naively, you can think of this as: "functions may lose information" (they may identify distinct elements), whereas "pre-images preserve information" (more precisely, they respect the partition of A induced by f).

this is why it is more natural to consider pre-images of topologies, than images of topologies (topologies are lattices, defined by bases...if we wish say that continuous functions are "nearness-morphisms" we have to account for the fact that two points "not near" in A may indeed be "near" in the image set f(A), for example: constant functions).

monomorphisms (or as functors: faithful functors) are "nicer" than regular morphisms. often these have special names in any given category (injection, embedding, sub-thingy, full rank) to indicate their distinguished status.

the contravariant power-set functor is an example of a Hom-functor: since we can regard P(A) as 2A = Hom(A,2) (where "2" is any two-element set, typically
{ {},{{}} }, or its more "usual" name {0,1}) and Hom is contravariant in the first argument.

some people use opposite (or dual) categories to make everything "covariant". for example, when creating a partial order, we get to choose what "up" ("bigger", "join") means, usually we do so to aid our intuition, or following convention (in a complemented lattice, you can see how arbitrary this choice actually is: for example, "smaller open set" means "bigger closed complement", and vice-versa).