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September 10, 2012

Every Functor is a (Co)Reflection

Posted by Mike Shulman

Recall that a reflection is a left adjoint to the inclusion of a full subcategory, and a coreflection is a right adjoint to such an inclusion. Of course, it is not literally true that every functor is a (co)reflection! However, it is true that every functor can be made into a (co)relection by changing its domain. More precisely, we can prove:

Theorem: Let F:CDF:C\to D be any functor. Then there exists a category EE, which contains CC as a full subcategory and DD as a full reflective subcategory, such that when the reflector is restricted to CC it agrees with FF.

You might try thinking about how to prove this; I’ll give an answer below the fold.

Proof: Let EE be the collage of the corepresented profunctor D(F,1)D(F,1). That is, the objects of EE are the disjoint union of the objects of CC and DD, with E(x,y) =C(x,y) if x,yC E(x,y) =D(x,y) if x,yD E(x,y) =D(Fx,y) if ;xC and yD E(x,y) = if xD and yC. \array{ E(x,y) &= C(x,y) & \text{if }\; x,y\in C\\ E(x,y) &= D(x,y) & \text{if }\; x,y\in D\\ E(x,y) &= D(F x,y) & \text{if }; x\in C \;\text{ and }\; y\in D\\ E(x,y) &= \emptyset & \text{if }\; x\in D \;\text{ and }\; y\in C. } It’s easy to define the composition in EE, and to see that it satisfies the conclusion of the theorem. \Box

Of course, using the dual profunctor D(1,F)D(1,F) produces a dual version of the theorem where FF becomes a coreflection.

As usual with this sort of thing, the construction in the theorem is not very enlightening. The category EE is so cooked up that it can’t possibly give us any new insight about FF. But in the spirit of Lack’s coherence theorem (“naturally occurring bicategories are equivalent to naturally occurring 2-categories”), we can still use the idea of this theorem in a useful way. Namely, it may happen for a particular functor FF that there is some naturally occurring category EE which satisfies the conclusion of the theorem. If so, this can give us a new way to understand FF.

I have two examples of this in mind at the moment (but I’m hoping that people will be able to suggest others in the comments). The first is my exact completions paper. This is actually a categorified example, where CC is the 2-category of categories with weak finite limits, DD is the 2-category of exact categories, and FF is the free exact completion. When the domain of FF is restricted to categories with actual finite limits, then it becomes an adjoint to the forgetful functor from exact categories to finitely complete categories. However, this is not the case in general.

What was known before my paper is that there’s a profunctor HH, other than D(F,1)D(F,1), whose collage we can use in the above proof of the theorem. Of course, this implies that HH must be isomorphic to D(F,1)D(F,1), but the point is that HH is not defined to be D(F,1)D(F,1); in fact, its definition doesn’t refer to FF at all. Specifically, if XX has weak finite limits and YY is exact, then H(X,Y)H(X,Y) is a the full subcategory of Cat(X,Y)Cat(X,Y) whose objects are “left covering” functors. (The definition of left covering functor is irrelevant to this post, so I will omit it.)

This is interesting and somewhat useful — it means that FF does have some sort of universal property — but still somewhat mysterious. What I managed to do was find a naturally occurring 2-category EE, not the collage of any profunctor, for which the conclusion of the theorem holds. Namely, EE is the 2-category of unary sites. It so happens that if XX has weak finite limits and we give it the trivial topology, and YY is exact and we give it its regular topology, then E(X,Y)E(X,Y) is precisely the category of left covering functors from XX to YY. Thus we recover the previously known universal property, but we also learn a good deal more.

The reason I was inspired to write this post is that I recently learned of another such “naturally occurring” instance of this theorem. Let CC be the category of uniform spaces, DD the category of topological spaces, and FF the functor which associates to a uniformity its underlying topology. The functor FF has a left adjoint, but this doesn’t tell us much about FF beyond that it preseves limits; the only definition I know of for the adjoint is very adjoint-functor-theorem-y (given a topology τ\tau on XX, give XX the finest uniformity whose underlying topology is coarser than τ\tau).

However, there is a “naturally occurring” category EE which contains CC and DD as full subcategories, in which DD is coreflective, and such that the coreflector becomes FF when restricted to CC. This category EE consists of syntopogenous spaces. (That’s quite a mouthful!) The idea is:

  • A topogenous relation on a set XX is a relation of “nearness” between pairs of subsets of XX.
  • A syntopogenous structure on XX is a suitable collection of topogenous relations.

If XX is a topological space, then we define a topogenous relation on XX by saying that AA is near to BB if AA contains a point of the closure of BB. (Note that this is not a symmetric relation.) This relation clearly contains all the information of the topology of XX. The syntopogenous structure on XX generated by this single relation yields the embedding of topological spaces into syntopogenous ones.

On the other hand, if XX is a uniform space, then every entourage UU of XX yields a topogenous relation, where AA is near to BB if there are points xAx\in A and yBy\in B which are UU-close. The collection of these relations, as UU ranges over the entourages of XX, generates a syntopogenous structure on XX which yield the embedding of uniform spaces into syntopogenous ones.

One can characterize which syntopogenous structures arise from topologies, which arise from uniformities, and also which arise from proximities. It turns out that the topological ones are coreflective, and the coreflection computes the underlying topology of a uniformity, and also of a proximity. You can read more at the nLab page syntopogenous space, and in the reference cited therein (Ákos Császár, Foundations of General Topology).

I don’t claim that syntopogenous spaces are useful for anything. (Although I’d be amused to be proven wrong.) But it’s fun to discover new universal properties for familiar functors. Can anyone think of any other examples?

One thing that these two examples have in common, which I find interesting, is that in both cases we are used to thinking of the objects of DD as objects of CC with extra structure (or properties). However, inside of EE, the images of CC and DD are almost disjoint. An exact category with its regular topology, and with its trivial topology, are almost always different unary sites. And the only syntopogenous structures that are both topological and uniform are just sets equipped with equivalence relations. (And of course, in the collage of D(F,1)D(F,1), the images of CC and DD are literally disjoint.) There’s no other way the theorem could be true, of course, but I think it points out in a new way the relativity of the notion of “extra structure”.

Posted at September 10, 2012 5:46 AM UTC

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Re: Every Functor is a (Co)Reflection

I don’t have another example to offer, but I do have what might be the seed of an idea.

By definition, an adjunction is idempotent if its induced monad is idempotent, or equivalently its induced comonad is idempotent. (There are several other equivalent conditions.) For example, any reflection is idempotent, where here I’m using “reflection” to mean the whole adjunction. Dually, any coreflection is idempotent.

A little exercise, which I got from Peter Johnstone, is that any idempotent adjunction can be factored as a reflection followed by a coreflection. (Use the fact that any adjunction restricts canonically to an equivalence between full subcategories.)

Your theorem reminded me of this exercise. Maybe the two things can be coupled in some interesting way.

Posted by: Tom Leinster on September 10, 2012 7:04 AM | Permalink | Reply to this

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