Adjoint Functors: Explaining What They ARE

[daveyp225]

Hey all,

Can anyone give me a low-down on functors. More specifically, adjoint functors. I do know what they are, and by that I know their definition. However, after consulting more than a few books, I'm still at a loss for what they ARE.

I would say naively, that a functor is adjoint of another if it "undoes" what the other one "does". But when facing an actual problem I'm not sure what to do. It seems like the authors hint that, for example, the "abelianization functor" undoes a loss of information (forgetful functor) going from the category of abelian groups to the category of groups. I.e., they make up whatever kind of adjoint functor they need without giving details on its construction.

Can anyone set me straight?
Thanks!

 

[Landau]

In short, I would describe them as 'conceptual inverses'. I found this summary helpful when while struggling with the concept myself.

Good examples to keep in mind are the left adjoints to a forgetful functor, which are the functors that assign to a set the free object on that set: group, vector space, ...
Or adjoint functors between posets, called a Galois connection.

 

[micromass]

I would say that it is pretty hard to see what an adjoint functor does. This is because adjoint functors are a generalization of a lot of things. Let me give some examples:

Modification of structure:
- abelianization of a group
- making a relation symmetric
- Making a topological space T0

Extending objects:
- completion of metric spaces
- compactification of a topological space.

Other uses:
- Free objects
- Function spaces

These are three classes of examples that don't seem to be related, but which can both be described by adjoint functors.

The way I intuitively understand adjointness is the following: Given categories C and D. If A is an object in C, what object in D ressembles most closely to A. For example, given a set X, what is the group that we can make out of X. Obviously, this is the free group. Or, given a metric space X, what is the complete metric space that most ressembles X? Obviously, this is the completion.

Adjointness is simply a very good result to have and it arises everywhere. You should just be aware of many examples, then the understanding will follow immediately.

 

[Deveno]

let's say you have a (pick a category type) X, and a functor F which naturally makes X into a (pick another category type). so we have X-->FX from Cat1--->Cat2.

the question naturally arises, is there some other functor G:Cat2--->Cat1 that is "essentially" related to F? in general, you can't just find a G such that GFX = X. for example, if F is the functor that makes a free group out of a set X, the underlying set of FX is much bigger than X (words are a larger set than the alphabet), so just "forgetting" the group structure would wind up giving you GF:X-->GFX.

but, there is a natural relationship between X and GFX, GFX is a superset of X. hmm...what hapens if we start with a group Y, forget the group structure, and then form the free group? we get: FGY:Y--->FGY. and there is a natural group homomorphism from FGY--->Y (notice the arrow goes the other direction here), which is uniquely determined by the function (not homomorphism) i:GY-->Y i(x) = x.

so we have these two "contrary" maps: X-->FGX and GFY-->Y. note that the set map (X-->FGX) is an injection, and that the group map is a surjection (any g in Y has the pre-image in FGY consisting of the word "g", as well as any other words that reduce to g under the homomorphism). these two contrary maps aren't direct inverses, but they have a certain duality to them, we have a correspondence between supersets of X and quotient groups of GFY.

note that this correspondence had nothing to do with the set X, or the group Y. X could be any set we like, and Y could be any group we like, we'd still get the same relationship (the constructions were "universal"). the equivalence isn't between F and G directly, but between the set of set-maps X-->GFX, and the set of group homomorphisms FGY -->Y. in other words:

Hom(-,G-) (this is a family of functions) has a bijection with Hom(F-,-) (this is a family of group homomorphisms).

now, it is very easy to make a set out of a group. take away the multiplication...poof! it's a set. but what is the most efficient (and therefore the most general) way to make a group out of a set?

in general, there are many categorical problems where one has a functor F that takes one kind of structure to another. it may well be that the "target category" is in some sense "simpler" than the source category. if F has an adjoint, G, we can "lift" information in the simple category, back to the "complicated" category, providing, in some sense, the "easiest" solution to a "hard" problem.