What are the key properties of functors and how are they used in mathematics?

[jcsd]

Out of idle curiousitywhat is a catergory? What are the 'catergory axioms'? And while we're about it what's a functor? Please try to keep it at as simple as poss. though I'd like a bit more info than mathworld have under their entry.

 

[mathwonk]

in one of my posts on tensors and covariance versus contravraiance i gave what eilenberg and maclane gave as their primordial example of a functor which is naturally equivalent to the identity functor. namely the double dual operation, which when applied to a finite dimensional vector space gives back an isomorphic one.


OK, a category is a bunch of objects of the same kind equipped with structure preserving maps between them.
examples: groups and group homomorphisms, topological spaces and continuous maps, banach spaces and bounded linear transformations, metric spaces and continuous maps, algebraic varieties and regular morphisms, algebraic varieties and rational maps, analytic manifolds and analytic maps, differentiable manifolds and differentiable maps, simplicial complexes and simplicial maps, cw complexes and homotopy classes of maps,...had enough??


i do not know the axioms but they are all trivially obvious, such as: to every object there is an associated identity map. composition of maps is associative. the identity map of any object acts trivially under composition when it is defined.

an isomorphism is by definition a map X-->Y with another inverse map Y-->X defined so that both compositions equal their respective identities.


They are not very interesting, and are only defined so that one can define functors.

a functor is a construction that takes any object of one category and changes it into an object in that or a different category, and ALSO that changes maps between two objects into maps between the two corresponding objects in the other category.


for example, the "dual" space functor, changes a vector space V into its dual V^ = {linear maps from V to the field k of scalars}. the map T:V-->W transforms into the map T^:W^-->V^ taking the map f:W-->k, to the composed map T^(f) = foT:V--W-->k.


The composition of this functor with itself, the double dual functor, takes V to V^^, and T to T^^. Here there is a further concept, the "natural transformation". This is a relation between two functors. a "natural equivalence" is a natural transformation with an inverse, up to isomorphism. in this case the double dual functor is equivalent to the dientity functor. i.e. there is a natural choice, for every (finite dimensional) V, of an isomorphism between V and V^^, such that any map T:V-->W corresponds under that isomorphism to the map T^^:V^^-->W^^.


A functor F has two basic properties: if F(T) goes in the same direction as T, F is called covariant (for example the double dual functor), and one requires: F(ToS) = F(T)oF(S). also one requires F(id(X)) = id(F(X)), i.e. functors take compositions to compositions and identities to identities.

(a contravariant functor reverses the direction of maps (for example the dual functor)

it follows that all functors take isomorphisms to isomorphisms.

big corollary: if there is a functor such that F(T) is not an isomorphism, then T was not an isomorphism.

Fact: there exists a homotopy invariant functor from topological spaces to abelian groups, that vanishes on a point but that does not vanish on spheres.

(examples, homology, cohomology)

corollary: an n disc cannot be retracted onto its boundary:
proof: let F be the functor whose existence was asserted.
then F(disc) = F(point) = {0}. so if the disc S retracts onto its boundary sphere S, via D-->S then the composition S-->D-->S is the identity so F of this composition is non zero, but it factors through F(S-->D) = F(S)-->F(D) = {0}, which must be zero, a contradiction.


if one does not know category theory, one belabors all the details of this prroof over and over for every separate case in which this type of proof is used. for example an old fashioned book such as hocking and young, first proves that homolgy is a functor and then proves that it takes homeomorphisms to group isomorphisms, as if that were not true of every functor in the world.

the derivative is a functor, from non linear maps to lnear maps. the proof of functoriality is the chain rule. corollary: if two manifolds are diffeomorphic then they have the same dimension, since their tangent spaces at corresponding points are linearly isomorphic.

a cute exercise in category theory i like is that an object is always characterized by its family, i.e. functor, of morphisms. i.e. for any object X, there is an associated functor

hom(X, ), from objects to sets (of maps out of X), taking X to hom(X, ), and taking a map T:X-->Y to the composition map from hom(Y, )-->hom(X, ) sending the map f:Y-->Z in hom(Y,Z) to the map foT:X-->Y-->Z in hom(X,Z). then if two such functors hom(X, ) and hom(Y, ) are naturally equivalent, then the two objects X,Y are isomorphic.

to see how to get at least a map from X to Y, from the hypothesis, we have
hom(X,X) bijectively equivalent to hom(Y,X) so the identity map X-->X correspodns top some map Y-->X. doing the same to get a map X-->Y, gives its inverse.

the functor above taking X to hom(X, ), is contravariant, and the other one taking X to hom( ,X) is covariant. note these functors take objects to functors. in particular, the set of functors between two given categories is itself a category with natural transformations as morphisms. note too that the dual functor is the functor hom( ,k) associated in this way to the object k.


a class of categories called "abelian categories" have more axioms, like exact sequences, and kernels and so on, but turn out to always be equivalent to subcategories of abelian groups, hence largely lose their independent interest. still the concept is useful.

 

[matt grime]

I would like to add some comments on Functors, and other ways of thinking of things.

A category can equally be thought of as a directed graph, albeit a huge great big one, with nodes for the objects of the category, and the maps (morphisms) the arrows. There is no limit placed on how many objects there are but (usually) we require that for any two objects, the morphisms (arrows) form a set. [some people do not make this a rule, and instead call such categories "small".] The arrows satisfy some composition laws.

To any category there is the oppposite category where all the arrows are reversed. Any contravariant functor on a category C is also a covariant functor on the the opposite category, and the distinction be co and contra variant is blurred.


There is also the idea of adjoint which you may have met before.

Suppose X is an inner product space and T a linear map from X to itself. Then the adjoint of T is the (unique if the inner product is non-degenerate etc...) map denoted T* such that (Tx,y)=(x,T*y) for all x,y.

Functors can have adjoints too. Though they don't always exist - in fact it's a very large theorem of Bousfield that the existence of certain adjoints is very important in topology.

Anyway the right adjoint of F:C->D is a functor from G:D->C such that

hom(Fx,y)=hom(x,G,y)

homs on the left being in D and on the right in C.

Adjoints may not exist.

H is a left adjoint if hom(x,Fy)=hom(Hx,y) for all x,y.

Left and right adjoints may not agree.

Mathwonk's ^ has a left adjoint, and a right adjoint, and in fact ^ again is the adjoint.

 

[jcsd]

I think I need a little bit of time to work my way throught thta and digets it before I ask any more question, thanks for the info!

 

[mathwonk]

examples are probably easier tham definitions.

i.e. everyone knows about categories in practice, but not everyone has thought about their proeprties abstractly.

one of my professors responded to one of my questions by saying: "functors are prehistoric!"

again the first encounter with a functor is the chain rule for derivatives.

At the fundamental average user level, the whole value of functors is just in the basic property that a functor takes maps of one kind to maps of another kind, preserving isomorphisms.

thus one tries to use them to show objects of one kind are not isomorphic, by translating the question into one of showing some easier objects are not isomorphic.

a nice example is in galois theory, i.e. why does one make the definition of normal field extensions? my answer is that it is because these are the ones that make the galois group into a functor.

i.e. it is no problem to define the galois group of any field extension, but if one thinks categorically, one wants also to define a group homomorphism associated to each field homomorphism. This is not generally possible, unless one restricts to norrmal extensions, which is exactly why those are crucial.

i.e. a finite field extension of Q is a complicated, infinite, object. galois' idea was to study it by looking at the group of its automorphisms, a finite group.

But how to compare the groups of different extensions? If you have a map from one field extension to another, (i.e. an inclusion), when is there an associated group homomorphism from one group to the other?

first we have to decide which way the group map should go. if from the group of the smaller field to the group of the larger field, we are talking about extending a field automorphism to an automorpohism of a larger field. this can be done in many ways, so an element of the galois group of the smaller field does not give a unique element of the galois group of the larger field.

in the other direction we are talking about restricting an automorphism of a larger field to an automorphism of a smalller field. The problem here is that the restriction may not leave the smaller field invariant. when it does so it defines an automorphism of the smaller field. hence we do get a group homomorphism from the galois group of a larger field extension to the galois group of any smaller extension which remains invariant under all automorphisms of the larger field.

this is exactly the case when the smaller field extension is a splitting field, i.e. a normal extension. so normal extensions are necessary in order to compare galois groups. hence the galois group is a contravariant functor from normal field extensions to groups.

Then with a normal extension we can analyze the structure of the galois group, since it consists of all embeddings of the field into its algebraic closure. The naturality implies that any decomposition of the original extension into relatively normal subextensions corresponds to a decomposition of the group into relatively normal pairs of subgroups, i.e. a normal tower.

then the solvability of the field extension, i.e. the fact that each extension is obtained by adjoining a root, corresponds to the abelianness of the quotient of the associated pair of subgroups, i.e. to the tower being abelian. then a group with no abelian tower cannot be the galois group of a solvable field extension. hence Sym(5) is not the group of a solvable extension, hence
X^5 - 80X + 2 is not a solvable polynomial.

so this application is more sophisticated than just saying if the groups are not isomorphic then the fields are not either, but it is related. the point is that the naturalness, i.e. the functoriality of the construction, in the case of normal extensions, alllows one to carry over any field theoretic construction into an associated group theoretic construction. so impossibility of some construction for groups implies impossibility of a construction for fields, provided the assignmkent of group to field is a functor.


This point of view is missing in most treatments of galois theory, because we hate to teach functors to students any more for fear of scaring people. the problem in my opinion is in the name; i.e. "functor" is a terrible name. the good name "natural transformation" has already been used for the more complicated concept of morphism of functors. if we called functors "natural operations" or "pesticide free" or "user friendly" or something like that, people would love them.


[for students of algebraic geometry and sheaf theory, the adjoints matt is talking about come up at the beginning of sheaf theory, in defining and characterizing upper star and lower star as transformations of sheaves under a mapping of spaces. interestingly, although two functors are adjoints of each other, and thus in a sense equivalent, in this case one of them can be much harder than the other to understand or calculate.

or maybe there is a trade - off. upper star is harder to understand but easier to calculate, and vice versa.]

I don't know what constructions you like best jcsd, but most constructions are functors, and most functors are in some sense hom functors. e.g. the dual functor takes V to Hom(V,k) where k is the ground field, and takes a map T to composition with T.

the fundamental group functor takes a space X to a quotient of the functor
Hom(S^1, {0};X, {p}), of certain maps from the circle S^1 into X, which send a fixed point {0} of S^1 to a fixed point {p} of X. the quotient is by the equivalence relation of homotopy.

Ed Brown's representability theorem in homotopy says that in the category of CW complexes, that cohomology is representable as a hom functor. this makes the algebraic object "cohomology" seem more geometric, since a cohomology element of a space is represented by a continuous map into a certain fixed space.

In general, functors that are Hom functors are called "representable", and it is a big question which functors are representable.

In algebraic geometry for example "moduli spaces" are an attempt to represent functors. A big topic in algebraic geometry is to classify all curves. If we study curves by looking at them in families, then we can use continuity and limits and calculus to study them. A family of curves over a base space B is a map to B whose fibres are all curves. the assignment taking a base space B to the set of all families of curves fibered over B is a functor.

An ideal moduli space would be a maximal family of curves, i.e. a big space M with one point for each curve, and a map to that space with the fiber over each point equal to the curve represented by that point.

Then any map from B to this space M would pullback the maximal family over M to a family over B. if every family over B could be obtaioned this way, we would have represented the functor of curves over B by the hom functor
Hom(B,M).


More often the maximal space M exists, but not quite the maximal family over it. Still we get a lot of mileage out of just M. It has an "open cover" but one by open sets that map U-->M to M by maps that are not quite injections. The god part is that over each U there is a good family of curves. But these families do not quite patch together over all of M. just over most of M. These are called "stacks", I think.

Moduli spaces were introduced by Riemann, for algebraic curves, and the dimension of the moduli space of Riemann surfaces (curves over C) of genus g > 1 was computed by him to be 3g-3.

Sorry about going off the deep end after specifically being asked to stop, but this was getting interesting to me.

Thye main point is that the whole value of any construction is in how it transforms. This is why some physicists have escaped understanding what tensors are all these years because in fact they do focus on the key aspect of them, which is how they transform under coordinate changes.

Unfortunately some people with that limited point of view have argued extensively here that covariance and contravariance are not intrinsic, when those properties describe the way the objects transform. (the trick matt refers to of changing the direction of the arrows by fiat does not solve this problem, but merely relabels it.)

So although transformation laws are the key aspect of tensors, the discussions here have revealed that people who think of tensors solely this way have a great deal of difficulty understanding them, and even occasionally make errors associated purely with transformation rules, apparently for lack of a conceptual framework to guide them. arguments about whether a certain tensor is or is not of type (r,s) or type (r+s,0), or whether it can be both, are of this nature, and rather depressing after a while, because it reveals a lack of interest in understanding what is going on.

so I have opened that can of worms again. but the discussion of it belongs here in a category discussion, since this is where the explanation is found.