Theory and Applications of Categories, Vol. 8, No. 16, 2001, pp. 457–464.

PERFECT MAPS ARE EXPONENTIABLE - CATEGORICALLY

GÜNTHER RICHTER AND WALTER THOLEN

ABSTRACT. A categorical proof of the statement given by the title is provided, in generalization of a result for topological spaces proved recently by Clementino, Hofmann and Tholen.

1. Introduction

Using convergence structures, recently the authors of [CHT] established the fibred version of the fact that compact Hausdorff spaces are exponentiable in the category Top of topological spaces, by proving:

Theorem. Every perfect map of topological spaces is exponentiable in Top.

Here perfect means proper (=stably closed, [B]) and separated [J], while a map f : X → Y is exponentiable if it is an exponentiable object in the fibred category Top/Y of spaces over Y; equivalently, if the change-of-base functor

f^{∗} : Top/Y → Top/X

has a right adjoint [N]. The proof of the Theorem in [CHT] is based on a characterization of exponentiable maps in terms of ultrafilter convergence, which on the one hand gives an explicit description of the exponential structures involved (i.e., of the right adjoint of f^{∗}), but which on the other hand makes extensive use of the Axiom of Choice.

In this note we give an entirely constructive and general proof of the Theorem, in the sense that no Choice is used and that the argumentation is purely categorical, based on techniques developed in [CGT] and [T]. However, the result proved is also somewhat weaker than that of [CHT], in the sense that we show only preservation of quotient maps by the change-of-base functor, which then implies the existence of a right adjoint via Freyd’s Adjoint Functor Theorem. At the end of the paper we discuss applications of the Theorem to some categories other than Top. Acknowledgement. We thank Maria Manuel Clementino for valuable comments on the subject of this paper.

Partial financial assistance by NSERC is acknowledged.

Received by the editors 2001 January 8 and, in revised form, 2001 July 19.

Transmitted by Susan Niefield. Published on 2001 July 26.

2000 Mathematics Subject Classification: 18A20, 18B30, 54C10.

Key words and phrases: proper map, separated map, perfect map, quotient map, exponentiable map.

\begin{matrix}c \\ ◯ \end{matrix} Günther Richter and Walter Tholen, 2001. Permission to copy for private use granted.

2. The categorical theorem

2.1. We work in a finitely-complete category X with a proper, pullback-stable factorization system (E, M), as in [DG1], [DT], [CGT]. Referring to (isomorphism classes of) M-morphisms into an object X as subobjects of X, for every morphism f : X → Y we therefore have the image/inverse-image adjunction

f(−) ⊣ f^{−1}(−) : subY → subX,

with subX = (M/X)/ ≅. Stability means that in any pullback diagram

[figure: X; T; Y; S; p; q; h; g]  (1)

with g ∈ E also its pullback h along p must lie in E. Equivalently, the Beck-Chevalley Property (BCP) holds true:

g^{−1}(p(a)) = q(h^{−1}(a))

for all a ∈ subX.

2.2. In addition, we fix a pullback-stable subclass

F_{0} ⊆ M

which contains all isomorphisms and is closed under composition. Subobjects in F_{0} are referred to as closed subobjects. A morphism f : X → Y is closed if f(−) maps closed subobjects to closed subobjects. (Since F_{0} is closed under composition, closed subobjects are given by closed morphisms in M.) If every pullback of f is closed, then f is a proper morphism. The morphism f is separated if the diagonal map

δ_{f} = < 1_{X}, 1_{X} >: X → X ×_{Y} X

into its kernel pair is closed (as a morphism, or as a subobject). A perfect morphism is by definition both proper and separated.

An object X is compact (Hausdorff) if the morphism X → 1 to the terminal object of X is proper (separated, respectively); equivalently: if every projection X × Z → Z (the diagonal map δ_{X} : X → X × X, resp.) is closed.

A quotient map is a morphism f : X → Y in E with the property that a subobject b in Y is closed whenever f^{−1}(b) is closed.

Note that all notions defined in 2.2 depend on the parameter F_{0}. Whenever there is need to emphasize this dependency, we augment these notions by the prefix F_{0}. We also note that, denoting by F the class of closed morphisms in X, we have F_{0} = F ∩ M, and the notions defined here coincide with the ones given in [T] (except that there for proper morphisms the object term “compact” was used as well).

2.3. The fibres of a morphism f : X → Y are given by the pullback diagrams

[figure: X; Y; f^{−1}y; 1; f^{−1}(y); f; y]  (2)

Proper morphisms are, by definition, closed and have compact fibres (see [T], Cor.3.4(3)), while the converse proposition fails in general (see [CGT], Remark 5.14). We say that in X properness is fibre-determined if every closed morphism with compact fibres is proper. (Actually, we shall use this property only if the morphism in question is also in E and separated. Note that the fibres of a separated morphism are Hausdorff, since separatedness is, like properness, stable under pullback.)

Recall that the object 1 is said to be E-projective if for every morphism f : X → Y in E every point y : 1 → Y factors as f · x = y.

2.4. We list a few properties to be used in the proof of 2.5 below; proofs may be found in [T]:

(1) every monomorphism is separated, and separated morphisms are closed under composition and stable under pullback;

(2) if a composite g · f is separated, so is f;

(3) every morphism representing a closed subobject is proper, and proper morphisms are closed under composition and stable under pullback;

(4) if a composite g · f is proper with g separated, then also f is proper.

In the general setting of 2.1, 2.2 we are now able to prove:

2.5. Theorem. Let the terminal object of X be E-projective, and let properness in X be fibre-determined. Then the pullback of a quotient map along any perfect morphism is again a quotient map.

Proof. We consider the pullback diagram (1) with p perfect and g a quotient map. In order to show that h is a quotient map as well, we consider a ∈ subX with h^{−1}(a) ∈ subY closed and must show that a is closed.

Step 1: As a pullback of p the morphism q is closed, so that

q(h^{−1}(a)) = g^{−1}(p(a))

is closed, whence p(a) is closed by hypothesis on g. Hence, in the commutative diagram

[figure: A; p(A); h^{−1}(A); g^{−1}p(A); X; T; Y; S; h; h^{−1}(a); g^{−1}(p(a)); g′; p′; h′; q′; a; p(a); p; g; q]  (3)

all diagonal morphisms but a are known to be closed. By 2.4(3), (4) it now suffices to show that p′ is proper, since then the properness of (∗) p(a) · p′ = p · a and the separatedness of p give closedness of a. In order to show properness of p′ we use the hypothesis that this property is fibre-determined in X. (We note that p′ is in E and separated, which follows from (∗) and properties 2.4(1), (2).)

Step 2: In order to show that the morphism p′ is closed, we consider any b ∈ subA closed and form the commutative diagram

[figure: B; p(B); h^{−1}(B); g^{−1}(p(B)); A; p(A); h^{−1}(A); g^{−1}(p(A)); h′; (h′)^{−1}(b); (g′)^{−1}(p′(b)); g′′; p′′; h′′; q′′; b; p′(b); p′; g′; q′]  (4)

We note that q′ is closed, in fact proper, since

q · h^{−1}(a) = g^{−1}(p(a)) · q′

has this property (by 2.4(3),(4)). Furthermore, g′ is a quotient map, as a pullback of g along the closed subobject p(a); indeed, for any c ∈ subp(A) with (g′)^{−1}(c) closed also

g^{−1}(p(a)) · (g′)^{−1}(c) = g^{−1}(p(a) · c)

is closed, which implies closedness of p(a) · c and then of c.

Using BCP again, we conclude that

q′((h′)^{−1}(b)) = (g′)^{−1}(p′(b))

is closed, whence also p′(b) is closed.

Step 3: We must still show that p′ has compact fibres. But any point z : 1 → p(A) factors as z = g′ · w, with w : 1 → f^{−1}(p(A)), by E-projectivity of 1, and we can form the commutative diagram

[figure: (p′)^{−1}z; 1; (q′)^{−1}w; 1; A; p(A); h^{−1}(A); g^{−1}(p(A)); h′; h^{∗}; z; w; p′; g′; q′]  (5)

Its front face is a pullback diagram since the back-, top- and bottom-faces are pullback diagrams. Hence h^{∗} is an isomorphism, so that with the compact fibre (q′)^{−1}w of the proper morphism q′ also (p′)^{−1}z is compact.

2.6. If, in the setting of 2.1, we are given a closure operator c (in the sense of [DG1], [DT], [CGT]), then we may choose for F_{0} the class of c-closed subobjects, which is pullbackstable, and also closed under composition if c is weakly hereditary. Furthermore, if c is idempotent, “c-preserving” of [CGT] becomes “closed” in this paper’s terminology, while “proper”, “separated”, “perfect” read as “c-compact”, “c-Hausdorff”, “c-perfect” in [CGT], respectively. Writing “c-quotient” instead of “(F_{0}-)quotient”, we deduce from 2.5:

2.7. Corollary. For an idempotent and weakly hereditary closure operator c, let cproperness be fibre-determined in X, and assume the terminal object to be E-projective. Then the pullback of a c-quotient map along any c-perfect morphism is again a c-quotient map.

3. Examples

3.1. (Topological spaces). An application of 2.5 to the category Top of topological spaces with its (surjective, embedding)-factorization structure and the usual notion of closure gives a choice-free proof of:

Pullbacks of quotient maps along perfect maps are quotient maps.

From this fact one derives with Freyd’s Adjoint Functor Theorem (using the Axiom of Choice) Theorem A of [CHT]:

Perfect maps are exponentiable in Top.

If instead of F_{0} = {closed embeddings} we choose F_{0} = {open embeddings}, 2.5 still yields an interesting but much easier and well-known statement:

Local homeomorphisms are exponentiable in Top.

3.2. (Partially ordered sets). Call a subset A of a poset X (up-)closed if x ≥ a ∈ A always implies x ∈ A. Then, in the category PoSet of posets and monotone maps with its (surjective,embedding)-factorization structure, closed maps f : X → Y are characterized by the property that for all y_{2} ≥ f(x_{1}) one finds x_{2} ≥ x_{1} with f(x_{2}) = y_{2}. Such maps are already proper, so that every poset is compact, and trivially properness is fibre-determined. Perfect maps have the additional property that f(x_{2}) = f(x′_{2}) with x_{2}, x′_{2} ≥ x_{1} always implies x_{2} = x′_{2}. Theorem 2.5 confirms that pullbacks of quotient maps along perfect maps are quotient maps. It is worth noting that in PoSet, like in Top, our notion of quotient map assumes the usual meaning as a regular epimorphism of the category. Indeed, for f : X → Y and y, y′ ∈ Y, let us write

y ≤_{f} y′ :⟺ ∃x, x′ ∈ X : x ≤ x′, f(x) = y, f(x′) = y′;

then f is a regular epimorphism precisely if the order of Y is the transitive hull of ≤_{f}. Such maps clearly have the property that B ⊆ Y is closed whenever f^{−1}(B) ⊆ X is; conversely, for y ≤ y′ in Y, one lets B be the up-closure of {y} with respect to the transitive hull of ≤_{f} and obtains that B must be closed. We note that generally quotient maps are not stable under pullback along arbitrary morphisms. Those f : X → Y that are stable have the property that ≤_{f} is the order of Y; they are also known as biquotient maps (see [CH]). We note furthermore that we may also order-dualize the result obtained from 2.5: pullbacks of quotient maps along local homeomorphisms are quotient maps; as in [CH], here we call f : X → Y a local homeomorphism if for all y_{1} ≤ f(x_{2}) there is exactly one x_{1} ≤ x_{2} with f(x_{1}) = y_{1}.

In essence, for both topological spaces and partially ordered sets, the chosen class F_{0} is able to identify important classes of exponentiable maps since it “contains” essential information about the structures themselves, as is demonstrated negatively also by the following example.

3.3. (Pretopological spaces). A pretopological space may, like a topological space, be defined as a set with a closure operation, except that this operation is not required to be idempotent. Like in Top, there are therefore natural notions of closure and convergence in the category PrTop. In contrast to Top though, a pretopological space in which all ultrafilters converge is (F_{0}-) compact (with F_{0} = {closed embeddings}), but not conversely: see [DG2]. Exponentiable objects in PrTop were completely characterized in [LS] as the finitely-generated pretopological spaces, a result which was generalized to all quotient-reflective subcategories of PrTop in [R]. In particular, we see that compact and separated objects in these categories are not exponentiable, in general.

We finally give an example showing that the crucial hypothesis of Theorem 2.5 (that properness be fibre-determined) fails to be a necessary condition.

3.4. (Continuous maps). We consider the category Top^{2} (with 2 the two-element chain), whose objects are continuous maps of topological spaces, and whose morphisms (p, q) : h → g are given by commutative squares as in diagram (1) in Top. It inherits the “pointwise” factorization structure from Top, and we may declare F_{0} to consist of the pointwise closed embeddings, i.e., of those morphisms (p, q) for which both p and q are closed embeddings. Then an arbitrary morphism (p, q) is closed (proper, separated, a quotient map) if both p and q have the respective property in Top. Consequently, pullbacks of quotient maps along perfect maps in Top^{2} are quotient maps. However, properness is not fibre-determined in Top^{2}, simply because the points of an object h in Top^{2}, given by the map h : Y → X in Top, are completely determined by the points of Y. Hence, they will not describe all points of X, unless h is surjective. For example, for every non-empty space X, the Top^{2}-map from (∅ → X) into the terminal object is closed and has compact fibres (since there are none), but fails to be proper.

References

[B] N. Bourbaki, Topologie Générale (Hermann, Paris 1961).

[CGT] M.M. Clementino, E. Giuli, and W. Tholen, Topology in a category: compactness, Port. Math. 53 (1996) 397–433.

[CH] M.M. Clementino and D. Hofmann, Triquotient maps via ultrafilter convergence, preprint (Coimbra 2000).

[CHT] M.M. Clementino, D. Hofmann, and W. Tholen, The convergence approach to exponentiable maps, preprint (Coimbra 2000).

[DG1] D. Dikranjan and E. Giuli, Closure Operators I, Topology Appl. 27 (1987) 129–143.

[DG2] D. Dikranjan and E. Giuli, Compactness, minimality and closedness with respect to a closure operator, in: Proc. Int. Conf. on Categorical Topology, Prague 1988 (World Scientific, Singapore 1989), pp.284–296.

[DT] D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators (Kluwer, Dordrecht 1995).

[J] I.M. James, Fibrewise Topology (Cambridge University Press, Cambridge 1989).

[LS] E. Lowen-Colebunders and G. Sonck, Exponential objects and Cartesian closedness in the construct Prtop, Appl. Categorical Structures 1 (1993) 345–360.

[N] S. Niefield, Cartesianness: topological spaces, uniform spaces and affine schemes, J. Pure Appl. Algebra 23 (1982) 147–167.

[R] G. Richter, More on exponential objects in categories of pretopological spaces, Appl. Categorical Structures 5 (1997) 309–319.

[T] W. Tholen, A categorical guide to separation, compactness and perfectness, Homology, Homotopy and Applications 1 (1999) 147–161.

Fakultät für Mathematik, Universität Bielefeld, Universitätsstr., 33615 Bielefeld, Germany and Dept. of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ont., Canada M3J 1P3

Email: heidrich@mathematik.uni-bielefeld.de and tholen@mathstat.yorku.ca

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