I have already mentioned topological functors here, only very quickly, where I mostly directed you to Sections 21 and 22 of . Topological functors are one way of describing categorically what makes the category Top of topological spaces so special. However, there are many other topological functors in nature. The situation is a bit similar as the notion of ring in algebra, which is one way of describing Z/nZ abstractly, although there are many other rings in nature.
Here is how I will proceed. I will explain what topological functors are by taking the example of Top. There are many other examples, such as preordered sets, prestreams, streams, or d-spaces. Before I come to them, I will list, and prove, a few useful properties of topological functors. I hope that this presentation will be easier to read than , which I find a bit demanding.
Topological functors: the canonical example
All right. The starting point is to consider not Top itself, rather the forgetful functor U : Top → Set that maps every topological space to its underlying set, and see what special properties U has.
Property 1: Faithfulness
One of those properties is that U is faithful. Given any two topological spaces X and Y, U maps every morphism f from X to Y in Top (a continuous map) to… simply f, seen as a mere function. Hence U is injective from the set Top(X,Y) of morphisms f in Top from X to Y, to Set(U(X),U(Y)). Top(X,Y) and Set(U(X),U(Y)) are examples of homsets, namely of sets of morphisms between two given objects. Hence U is injective on homsets, but the standard name for that property is to say that U is faithful.
Oh, U is certainly not surjective on homsets in general: that would mean that every function between topological spaces is continuous, which is simply wrong.
Property 2: every U-source has a U-initial lift (what gibberish! … but read on)
Let me call a family of topological spaces any collection (Yi)i ∈ I of topological spaces Yi indexed by some class I. In other words, a family is not a set, rather a map from I to the class of topological spaces. Perhaps more importantly, families may be indexed by a proper class, not just by sets I; and, in doing so, we are entitled to repeat several times the same space Yi.
Given a set E, any family of topological spaces (Yi)i ∈ I, and maps gi : E → Yi, one for each i in I, there is a coarsest topology on E that makes every gi continuous. This is the topology generated by the collection of sets of the form gi–1(V), where i ranges over I and V ranges over the open subsets of Yi. (Note that this collection of sets is itself a set, not a proper class, because they are all subsets of a common set, namely E.)
We model this property categorically as follows. Given any set E, the fiber above E (relative to the functor U) is the collection of topological spaces X such that U(X)=E. Any such topological space X is merely E itself, with some topology; and conversely, any topology on E gives you a topological space X in the fiber above E. Hence, instead of talking about topologies on E, we can (and will) talk about objects in the fiber above E. That is the same thing, but the latter only involves the functor U, not any externally defined notion such as a topology.
What does it mean to find a topology on E that would make every gi continuous? First, formally, saying that gi is a map from E to Yi is not quite right, categorically, since E and Yi live in different categories. Really, gi is a map from E to U(Yi). Finding a topology on E that would make every gi continuous then means finding an object X of Top in the fiber above E, first, and one such that gi lifts to a morphism ĝi from X to Yi, for each i in I. Formally, a lift of gi (with respect to U) is a morphism ĝi : X → Yi in Top whose image by U is gi. Since U is faithful, this lift is unique. Informally, that lift is simply the map gi itself, except that we know that it is continuous from X to Yi.
A family of morphisms gi : E → U(Yi), i ∈ I, with common domain E, is called a U-source. We say that a family of morphisms ĝi: X → Yi, i ∈ I, is a lift of that U-source if and if only X lies in the fiber above E and each ĝi is a lift of the corresponding gi: namely, U(X)=E and U(ĝi)=gi for every i in I. (The condition U(X)=E, namely that X is in the fiber above E, is redundant if I is non-empty.)
Good, but I have not just said that you could equip E with a topology that makes every gi continuous. If I had just said that, that would not be much: just equip E with the discrete topology. No, I have said that you can give E the coarsest topology that makes every gi continuous. One may think of modeling this categorically by requiring that the fiber above E be a complete lattice (topologies on a given set indeed form a complete lattice under inclusion, although a rather bizarre one—I may talk about it another day), but there is a sleeker and more comprehensive way.
Think of it this way. Given the coarsest topology on E that makes every gi continuous (and writing X for the topological space obtained by equipping E with that topology), how would you show that any given map h : Z → X from any given topological space Z is continuous? I claim that it is enough to show that gi o h : Z → Yi is continuous for every i in I. (Pause for a minute, and try to show it by yourself! If gi o h is continuous, then certainly the inverse image of gi–1(V) by h is open, for any i in I and for any open subset V of Yi, right?)
Categorically, this means that for every object Z in Top, for every map h : U(Z) → E (in Set), h lifts to a morphism ĥ from Z to X (in Top) if and only if every morphism gi o h : U(Z) → U(Yi) (in Set) lifts to a morphism from Z to Yi (in Top). In that case, ĝi o ĥ must be the lift of gi o h, since lifts are unique, owing to the fact that U is faithful, and we say that the family of morphisms ĝi: X → Yi, i ∈ I, is a U-initial lift.
Property 3: amnesticity (now we are speaking Greek)
There is a final, useful property that U has, and which is much simpler than its name: amnesticity. That is categorical wording for the fact that, given two topologies on the same set E, if one is both coarser and finer than the other one, then they are equal; in other words, that the inclusion preordering on the topologies on E is an ordering.
Given any faithful functor U from a category C to a category D, every object E in D has a fiber, which is the collection of objects X in C such that U(X)=E. (That collection may be a proper class.) This is preordered by:
X ≤ Y (in the fiber above E; read “X is finer than Y”) if and only if the identity morphism on E lifts to a morphism from X to Y in C.
In other words, and momentarily returning to the case C=Top and D=Set, X and Y stand for topologies on E, and X ≤ Y if and only if the identity map is continuous from X to Y, if and only if X stands for a finer topology than the topology that Y stands for.
The definition of topological functors, and basic properties
We obtain our definition:
Definition. A functor U : C → D is topological if and only if it is:
- faithful (injective on hom sets);
- amnestic, namely the “finer than” preordering (existence of a lift of the identity) on fibers is a partial ordering;
- and is such that every U-source has a U-initial lift.
Let me remind you that the latter means that every U-source gi : E → U(Yi), i ∈ I, has a lift ĝi : X → Yi, i ∈ I (i.e., U(X)=E and U(ĝi)=gi) that is U–initial, meaning that it satisfies the following universal property:
For every object Z in C, for every morphism h : U(Z) → E (in D), h lifts to a morphism ĥ from Z to X (in C) if and only if every morphism gi o h : U(Z) → U(Yi) (in D) lifts to a morphism from Z to Yi (in C).
In that case, ĝi o ĥ must be the lift of gi o h.
An alternate definition
Adámek, Herrlich and Strecker use another definition [1, Definition 21.1], which is more economical, but also more abstruse in my opinion. To them, a functor U is topological if and only if every U-source has a unique U-initial lift. Then, they immediately proceed to show that this implies that U is faithful [1, Theorem 21.3], using a clever but rather contorted argument.
The condition that U be faithful is actually redundant in the definition I use. The argument follows from Proposition 21.5 of , which in particular says that an amnestic functor U such that every U-source has a U-initial lift has the property that every U-source has a unique U-initial lift, and the latter implies that U is faithful, as I have just mentioned. The argument behind the latter result is tricky… so, no, I definitely prefer the presentation I gave above.
A useful consequence of the definition is that every topological functor U is uniquely transportable. In other words, every isomorphism f from U(Y) to E in D lifts to a unique isomorphism from Y to some (unique) object X. In the canonical example of the forgetful functor U : Top → Set, this translates to the following property, which one generally uses without thinking: if Y is a topological space and E is a set, and f is a bijection between Y and E, then there is a unique topology on E that makes f a homeomorphism.
How do you prove unique transportability? Let us do the exercise. This will train us and make us more comfortable with the definition. (It may even be useful for you to try to prove it by yourself without looking at the solution below.)
All right. Well, given an isomorphism f from U(Y) to E in D:
- Its inverse g ≝ f–1 defines a U-source with just one element.
- That U-source must have a U-initial lift, which must also consist of a single morphism ĝ : X → Y, with U(X)=E.
- We need to show that ĝ is an isomorphism in C. To this end, we use the universal property of U-initial morphisms, applied to h ≝ f: h lifts to a morphism ĥ from Y to X in C if and only if g o h : U(Y) → U(Y) lifts to a morphism from Y to Y. And surely it does, since g o h = f–1 o f = idU(Y) lifts to the identity morphism on Y.
- Hence indeed h ≝ f lifts to a morphism ĥ from Y to X. Now U(ĝ o ĥ) = g o h = idU(Y), so by faithfulness ĝ o ĥ = idY; and similarly, ĥ o ĝ = idX. This shows that ĝ is an isomorphism in C, and that ĥ : Y → X is also an isomorphism (its inverse) that lifts f.
- Finally, we must show that X and ĥ are unique. Let us assume another object X’ and another isomorphism h’ : Y → X’ that lifts f. Then h’ o ĥ–1 = h’ o ĝ : X → X’ lifts f o f–1 = idE, so X is finer than X’ (in the fiber above E), by definition. We also have that ĥ–1 o h’ lifts f–1 o f = idU(Y) (use the fact that U, just like any functor, maps inverses to inverses), so X’ is also finer than X. Since U is amnestic, X=X’. Then both ĥ and h’ are morphisms from Y to X and both lift f; since U is faithful, this implies that ĥ=h’.
The duality of topological functors
In Top, it is not only true that given any U-source gi : E → Yi, i ∈ I, you can always equip E with the coarsest topology that makes every gi continuous, but the dual property holds: given any U–sink hi : Xi → E, i ∈ I (any family of maps with the same codomain), there is a finest topology on E that makes every hi continuous.
That dual property is a consequence of the first one, as we will now see. This is Theorem 21.9 in .
In general, given a functor U from C to D, a U-sink is a family of morphisms hi : U(Xi) → E in D, i ∈ I, with the same codomain. A lift of that U-sink is a family of morphisms ĥi : Xi → Y in C such that U(ĥi)=hi, one for each i ∈ I, where Y is in the fiber over E. Such a lift is U-final if and only if it satisfies the following universal property:
For every object Z in C, for every morphism g : E → U(Z) (in D), g lifts to a morphism ĝ from Y to Z (in C) if and only if every morphism g o hi : U(Xi) → U(Z) (in D) lifts to a morphism from Xi to Z (in C).
Theorem A. Let U : C → D be a functor such that every U-source has a U-unitial lift. Then every U-sink has a U-final lift.
Proof. The proof is similar to the following famous observation in lattice theory: every poset with all infima has all suprema. The proof is simple: given any family A of elements in that poset, we build the set A↑ of all upper bounds of A (the elements that are above all the elements of A); A↑ has a greatest lower bound x (an infimum), by assumption; it is easy to check that x is above every element in A, hence is in A↑; it follows that x is the least element of A↑, namely the least upper bound of A (a.k.a. its supremum).
Let’s get to the proof. We consider a U-sink hi : U(Xi) → E in D, i ∈ I. We form the class of all the morphisms g with domain E in D such that g o hi has a lift for every i ∈ I. (This is the analogue of A↑, considering that A is the analogue of the U-sink we started with.) We enumerate them as gj : E → U(Yj), j ∈ J.
This forms a U-source gj : E → U(Yj), j ∈ J. By assumption, that U-source has a U-initial lift ĝj : Y → Yj, j ∈ J (i.e., U(Y)=E and U(ĝj)=gj; this plays the role of the element x).
- For each i ∈ I, we check that hi has a lift ĥi : Xi → Y. It suffices to verify that gj o hi has a lift for every i ∈ I: we are using the definition of U-initiality here; and gj o hi indeed has a lift, because this is how we defined the morphisms gj, literally.
- Next, we verify that the family of morphisms ĥi : Xi → Y, i ∈ I, is a U-final lift of the U-sink hi : U(Xi) → E, i ∈ I. This means showing that, given any morphism g : E → U(Z) in D, g lifts to a morphism ĝ from Y to Z if and only if every morphism g o hi : U(Xi) → U(Z) lifts to a morphism from Xi to Z. If g has a lift, then so does every morphism g o hi, and we only have to show the converse implication. Hence we assume that every morphism g o hi : U(Xi) → U(Z) lifts to a morphism from Xi to Z. By definition, this means that g is equal to some gj and that Z=Yj. Now we know that every gj has a lift ĝj from Y to Yj, and this is the desired lift from Y to Z. ☐
This theorem can be stated in the following more elegant, but perhaps less immediately operative fashion. We consider the opposite categories Cop and Dop: going to opposite categories turns U-sources into U-sinks and conversely, and U-initial lifts into U-final lifts and conversely. Therefore:
Corollary (duality theorem for topological functors). Every topological functor U : C → D induces a topological functor U from Cop to Dop.
Taking opposites once again, we obtain that if U is topological from Cop to Dop, then it is also topological from C to D. Alternatively, reversing arrows in the statement of Theorem A: every functor U such that every U-sink has a U-final lift is also such that every U-source has a U-unitial lift.
Topological functors are left and right adjoint
Given any set E, there is a finest topology on E at all. This is the discrete topology on E, whose open sets are all the subsets of E. You can do the same with topological functors U : C → D: given any object E of D, the empty U-sink with codomain E has a U-final lift; that lift simply consists of one object X of C such that U(X)=E, and no incoming morphism; and the universal property reads: “for every object Z in C, for every morphism h : E → U(Z) in D, h lifts to a morphism ĥ from X to Z (period)”. (In the case of Top, that means that every map from the set E to the topological space Z is already continuous, provided that E is given the discrete topology.)
Let me write D(E) for that object X, the discrete object over E. We have U(D(E))=E, meaning that D(E) is in the fiber above E. If we specialize the universal property to objects E in the fiber above E, and if we take h≝idE, we obtain that h lifts to a morphism from D(E) to Z. In other words, D(E) is finer than Z. Since Z is arbitrary, D(E) is the finest possible object in the fiber above E (~ the finest topology, in the case of Top).
Also, there is a morphism η : E → U(D(E)) —that is simply the identity morphism— and the universal property can be rephrased as follows: for every object Z in C, for every morphism h : E → U(Z) in D, there is a unique morphism ĥ from X to Z such that U(ĥ) o η = h. The latter simply means that ĥ lifts h, since η is the identity morphism. The uniqueness of ĥ follows from the fact that U is faithful. Anyway, formulated this way, you should recognize a familiar property; this is the situation of Diagram (5.2) in the book (Section 5.5.2).
In other words, D(E) is the free object of C over E; alternatively, D defines a functor from D to C that is left adjoint to U.
Entirely symmetrically, there is a coarsest possible object C(E) in the fiber over E, which I will call the coarse object over E, and C defines a functor from D to C that is right adjoint to U. In the case of Top, and as you may have guessed, C(E) is just E with the indiscrete topology, whose only open sets are the empty set and E.
Hence we have the following remarkable string of adjunctions:
D ⊣ U ⊣ C
Remember that right adjoints preserve all existing limits, and left adjoints preserve all existing colimits? (In the book, have a look at Section 5.5.2.)
Since a topological functor is both left and right adjoint, it automatically preserves all (existing) limits and colimits. As a sanity check, check that, indeed, all limits and colimits taken in Top are computed by first taking limits and colimits in Set, and equipping them with appropriate topologies.
Given a topological functor U : C → D, it is more interesting to show how one can build limits and colimits of diagrams in C from limits and colimits in D rather than the other way around, and that works, too.
Let us have a brief look at limits. We consider a diagram F : I → C. Then U o F is a diagram in the category D. Let us assume that U o F has a limit in D.
- That limit of U o F is given by a (universal) cone, whose apex is an object of D, let us call it E.
- We also have cone morphisms gi : E → U(F(i)), one for each object i of I. That they are cone morphisms means that for every morphism f : i → j in I, U(F(f)) o gi = gj.
- The family of all cone morphisms gi : E → U(F(i)) is a U-source. Hence we can form its U-initial lift, which is composed of morphisms ĝi : X → F(i), one for each object i of I.
- For every morphism f : i → j in I, the morphisms F(f) o ĝi and ĝj have the same image through U, namely U(F(f)) o gi = gj. But U is faithful, so F(f) o ĝi = ĝj. Therefore X and the morphisms ĝi form a cone for the diagram F.
- Let us consider any other cone for F, with apex X’ and cone morphisms fi : X’ → F(i), one for each object i of I. Hence, for every morphism f : i → j in I, F(f) o fi = fj. We apply U to that equation, and we obtain that the object U(X’) and the morphisms U(fi) form a cone for U o F. By the universal property of the limit, there is a unique morphism h : U(X’) → E such that gi o h = U(fi) for every i. We wish to lift that morphism h to a (unique) morphism ĥ : X’ → X such that ĝi o ĥ = fi for every i. Remember that the family of morphisms ĝi is a U-initial lift: h : U(X’) → E lifts to a morphism from X’ to X if and only if gi o h lifts to a morphism from X’ to F(i) for every i; and it does! by definition of h, gi o h = U(fi).
- Hence the U-initial lift (ĝi : X → F(i)) of the U-source gi : E → U(F(i)), where i ranges over the objects of I, is a limit of F in C, where X is the apex and the morphisms gi are the cone morphisms of the limit of U o F in D.
Let us recapitulate. In order to form the limit of a diagram F : I → C in C, it is necessary and sufficient that the limit of U o F exists in D. If that limit exists in D, then it is a universal cone, which happens to be a U-source, and we obtain the limit of F in C by taking a U-initial lift of that U-source.
The case of colimits is completely symmetrical. Given a diagram F : I → C, F has a colimit in C if and only if the colimit of U o F exists in D. If so, then that colimit, seen as a universal cocone, is a U-sink, and we obtain the colimit of F as a U-final lift of that U-sink.
Proposition. Let U : C → D be a topological functor. Then C is complete if and only if D is complete, and C is cocomplete if and only if D is cocomplete.
In the case of topological spaces, we obtain that Top is both complete and cocomplete… since Set is.
Further examples, non-examples, and applications
There are many other examples of topological functors that the forgetful functor from Top to Set.
Preordered sets (over Set)
A related topological functor is the forgetful functor from Pre, the category of preordered sets and monotonic maps, to Set. In that setting, a U-source is a family of maps gi from a given set E to U(Yi), where each Yi is preordered and U(Yi) is the underlying set. Its lift is obtained by equipping E with the coarsest preordering ≤ that makes every gi monotonic; namely, a≤b if and only if gi(a) is below gi(b) for every i.
Ordered sets (over Set)
As a non-example, the forgetful functor from Ord, the category of ordered, not just preordered, sets to Set, is not topological. The intuitive reason is that the preordering we have just defined is not an ordering. If you quotient E by the equivalence relation associated with that preordering, you will not be able to stay in the fiber above E. A formal argument runs as follows. If the forgetful functor from Ord to Set were topological, then every set E would have a coarsest ordering ≤, namely an ordering ≤ such that, if a is below b with respect to any ordering on E, then we would have a≤b. But no set of cardinality at least 2 has any such coarsest ordering: picking any two distinct elements a and b from that set, there are orderings that set a below b, some other that set b below a, so we would have a≤b and b≤a, hence a=b (because ≤ is antisymmetric; that is the key); and that contradicts our assumption that a and b are distinct.
T0 topological spaces (over Set)
Similarly, the forgetful functor from the category of T0 topological spaces and continuous maps to Set is not topological.
Prestreams (over Top)
However, the forgetful functor from PreStr, the category of prestreams, to Top, is topological [2, Proposition 5]. That forgetful functor U maps every prestream (X, (⊑U)U ∈ OX) to the underlying topological space X, and every prestream morphism to itself, seen as a mere continuous map. I will skip over the easy verification that U is faithful and amnestic.
Given any U-source gi : E → (Yi, (⊑iV)V ∈ OYi), i ∈ I, where E is a topological space now, its U-initial lift is the family of prestream morphisms ĝi : (E, (⊑U)U ∈ OE) → (Yi, (⊑iV)V ∈ OYi), i ∈ I, where:
- (⊑U)U ∈ OE is the coarsest precirculation on E that makes every gi locally monotonic (i.e., such that for every i in I, for every open subset V of Yi, for all points x, y in U ≝ gi–1(V) such that x ⊑U y, gi(x) ⊑iV gi(y)); it is easy to see that, for every open subset U of E, ⊑U is defined by x ⊑U y if and only if for every i in I, for every open subset V of Yi such that U ⊆ gi–1(V), gi(x) ⊑iV gi(y);
- ĝi is simply gi, which is now both continuous and locally monotonic, hence a prestream morphism.
Hence PreStr is automatically complete and cocomplete. We even know how to compute limits and colimits: taken the relevant limits and colimits in Top, and then compute the appropriate precirculation by taking U-initial lifts, as described above, for limits; or U-final lifts for colimits.
There is a more complete set of concrete descriptions of limits and colimits in PreStr in Section 4 of .
Streams (over Top)
There is a cosheafication functor, from PreStr to the category Str of streams. Using a pretty general lemma (Lemma 15 in , which I may talk about some day), one can show that this entails that the forgetful functor from Str to Top is topological. Given any U-source gi : E → (Yi, (⊑iV)V ∈ OYi), i ∈ I, where E is a topological space and each (Yi, (⊑iV)V ∈ OYi) is a stream, not just a prestream (and U is the forgetful functor from Str to Top), its U-initial lift is the family of prestream morphisms ĝi : (E, (⊑’U)U ∈ OE) → (Yi, (⊑iV)V ∈ OYi), i ∈ I, where (E, (⊑’U)U ∈ OE) is the cosheafification of the prestream (E, (⊑U)U ∈ OE) we have built in the similar case of prestreams.
As a consequence, Str also has all limits and colimits. This was first shown by Sanjeevi Krishnan , and that is the paper that originally induced me into trying and understanding topological functors in the first place. As usual, one can find more concrete descriptions of limits and colimits in Str in [1, Section 4].
d-Spaces (over Top)
There is a forgetful functor U that maps every d-space (X, dX) to its underlying topological space X. Given any U-source gi : E → (Yi, dYi), i ∈ I, where E is a topological space, its U-initial lift is the family of d-space morphisms ĝi : (E, dE) → (Yi, dYi), i ∈ I, where:
- dE is the collection of paths γ : [0,1] → E such that for every i ∈ I, the path gi o γ is in dYi. It is easy to see that this collection contains all the constant paths, and is closed under concatenation and under monotonic reparametrization.
- ĝi is just gi, which is not just continuous but also a d-space morphism.
Hence, once again, we obtain that the category dTop of d-spaces is complete and cocomplete, and I guess you see how one can compute limits and colimits in dTop, at least in principle.
I have entitled this post “Topological functors I”. Yes, I plan to write a “Topological functors II” post, and perhaps more. This is an very rich notion.
Here is a teaser. In the book (Section 5.6), there is a construction of so-called C-generated spaces, which form Cartesian-closed subcategories of Top, generalizing the so-called compactly generated spaces. This is an incredibly nifty construction due to Escardó, Lawson, and Simpson . And, surprise, surprise… this construction works for any well-fibered topological functor [2, Theorem 1, Theorem 2]! Oh, I didn’t say what well-fibered means. I’ll say that another time; all the examples I have given here are well-fibered.
- Jǐrí Adámek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories. The Joy of Cats. John Wiley and Sons, 1990. Online edition, 2004.
- Jean Goubault-Larrecq. Exponentiable streams and prestreams. Applied Categorical Structures 22, pages 515–549, 2014. The published version, available from Springer Link, contains two mistakes, which are repaired in the HAL report.
- Sanjeevi Krishnan. A convenient category of locally preordered spaces. Applied Categorical Structures, 17(5):445–466, 2009.
- Martín Escardó, Jimmie Lawson, and Alex Simpson. Comparing Cartesian Closed Categories of (Core) Compactly Generated Spaces. Topology and Its Applications, 143(1–3):105–146, 2004.
— Jean Goubault-Larrecq (September 22nd, 2021)