Applied Categorical Structures 8: 7–15, 2000. G. Brümmer & C. Gilmour (eds), Papers in Honour of Bernhard Banaschewski. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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An ‘Unsitely’ Result on Atomic Morphisms Dedicated to Bernhard Banaschewski, who taught me the importance of always looking for the right proof PETER JOHNSTONE Department of Pure Mathematics, University of Cambridge, Cambridge CB2 1SB, U.K. (Received: 8 December 1996; accepted: 28 April 1997) Abstract. We give an ‘elementary’ proof, without mentioning sites, that any section of an atomic geometric morphism is open, and any section of a connected atomic morphism is an open surjection. Previously, these results were known only for bounded morphisms. As a by-product, we obtain a proof that any connected atomic morphism with a section is necessarily bounded. Mathematics Subject Classification (2000): 18B25. Key words: atomic morphism, topos theory.
Introduction A geometric morphism f : F → E between (elementary) toposes is said to be atomic if its inverse image f ∗ : E → F is a logical functor (i.e. preserves the elementary topos structure). Many important results (pullback-stability, descent theorems, etc.) are known about the class of atomic morphisms, but the known proofs of most of them rely on the characterization of atomicity for bounded morphisms (that is, for those f which have the property that F may be represented as the topos of sheaves on an internal site in E) in terms of ‘atomic sites’, as originally introduced (in the case when the base topos E is the topos of sets) by Barr and Diaconescu [1], and later extended to more general base toposes by Joyal and Tierney [6]. In particular, this means that the results are known only in the case of bounded morphisms. This latter restriction is not in practice a very irksome one, since very few examples of unbounded geometric morphisms are known. Also, since many of the results involve reference to pullbacks of geometric morphisms, and since the pullback of a pair of morphisms is known to exist only in case at least one of the pair is bounded, it is at least unclear whether some of them could be formulated at all for unbounded morphisms. Nevertheless, it remains of interest in at least some cases to see whether the results can be proved by purely ‘elementary’ means, without making reference to sites. In this paper, our aim is to provide a site-free proof of the result that any point of an atomic E-topos is open, and any point of a connected atomic E-topos is an
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open surjection. The only previous proof of this result (see [6], VII 4.1) makes use of atomic sites, and hence establishes the result only for bounded morphisms. (However, when E is the ‘classical’ topos Set of constant sets, both halves of the result are trivial: for if a topos F admits an atomic morphism to Set then it is Boolean (and so every morphism with codomain F is open), and if F → Set is also connected then F is two-valued, so that every morphism from a nondegenerate topos to F is surjective.) Our result is therefore a genuine extension of what was previously known; but it has to be admitted that it is unlikely to lead to any major new applications. Rather, the interest lies in the method of proof itself: the fact that such ‘elementary’ site-free arguments can be made to work in a context where they have not been employed before. The paper is organized in two sections: the first collects together some results from the literature (plus one result which, although we have not been able to find it in print, is surely well-known) which are relevant to the proof of our main result. Section 2 contains the proof of the main result itself, together with a few comments on it. 1. Preliminaries We recall first that, for any cartesian (= finite-limit-preserving) functor F : E → F between toposes, we have for each A ∈ ob E a comparison map φA : F ((E )A )
(F )FA ,
corresponding to the monomorphism F (∈A ) F (A × A) ∼ = F (A ) × FA. We say F is logical if this comparison map is an isomorphism for all A; as already indicated, we call a geometric morphism atomic if its inverse image is a logical functor. We recall also the following result from [4]: LEMMA 1.1. For a geometric morphism f : F → E , the following are equivalent: ∗ (i) For every object A of E, the comparison map φA : f ∗ ((E )A ) → (F )f A is monic. (ii) For every object A of E and every subobject B f ∗ A in F , the image of the composite geometric morphism F /B
F /f ∗ A
f/A
E/A
is an open subtopos of E/A. (Here f/A is the geometric morphism whose inverse image is obtained by applying f ∗ to morphisms with codomain A.) A geometric morphism with the equivalent properties of 1.1 is said to be open. The following weak version of the descent theorem for atomic morphisms (cf. [7],
AN ‘UNSITELY’ RESULT ON ATOMIC MORPHISMS
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1.10), though not required for the proof of our main result, turns out to be quite closely related to it. LEMMA 1.2. Let h
G
F f
g
E be a commutative triangle of geometric morphisms, such that g is atomic and h is an open surjection. Then f is atomic. Proof. For any object A of E, we have a commutative triangle g ∗ ((E )A )
h∗ (φA )
∗
h∗ ((F )f A ) χf ∗ A
ψA
(G )g
∗A
where φ, ψ, χ are the comparison maps for the functors f ∗ , g ∗ , h∗ respectively. Since ψA is an isomorphism, it follows that χf ∗ A is (split) epic; but χf ∗ A is monic in any case since h is open, and hence it is an isomorphism. So h∗ (φA ) is an isomorphism; but h∗ reflects isomorphisms since h is surjective, so φA is an isomorphism. 2 A logical functor between toposes has a left adjoint if and only if it has a right adjoint (see [2]); thus a sufficient condition for a logical functor to be the inverse image of an atomic morphism is that it should have either a left or a right adjoint. The following result on left adjoints of logical functors is not new, but we have not been able to find a reference for it. LEMMA 1.3. Let f ∗ : E → F be a logical functor between toposes having a left adjoint f! . Then f! preserves monomorphisms; and in fact, for every object A of F , the assignment m 7→ f! m is a bijection from (isomorphism classes of) subobjects of A in F to subobjects of f! A in E. Proof. We clearly have a bijection between subobjects A0 A classified by morphisms χ : A → F ∼ = f ∗ E and subobjects of f! A classified by morphisms χ : f! A → E ; the problem is to show that it is induced by the action of f! on monomorphisms. For the moment, let us write λA0 f! A for the subobject corresponding to m : A0 A. Since the diagram f! A0
f! 1 f! >
f! m
f! A
1
f! χ
f ! F
1 >
E
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is readily seen to commute (where is the counit of (f! a f ∗ )), we have a canonical morphism f! A0 → λA0 ; to show that this is an isomorphism, we shall demonstrate that λA0 has the same universal property as f! A0 . For any object B of E, morphisms A0 → f ∗ B can be regarded as partial maps ∗ B satisfying δg = χ, A + f ∗ B, and so correspond to morphisms g : A → fg ∗ ∗ B. B → F classifies the canonical monomorphism if ∗ B : f ∗ B fg where δ : fg ∗ But since f is logical, it commutes with the construction of partial-map rep∗B ∼ ˜ which identifies δ with resenters; so we have an isomorphism fg = f ∗ (B), ∗ 0 0 ˜ f (δ ) where δ : B → E classifies iB . Thus morphisms A0 → f ∗ B correspond to morphisms g : f! A → B˜ satisfying δ 0 g = χ, and these in turn correspond to 2 morphisms λA0 → B, as required. A geometric morphism f : F → E is called a local homeomorphism if there is an object A of E and an equivalence F ' E/A which identifies f ∗ and f∗ with the pullback functor A∗ : E → E/A and its right adjoint 5A . Local homeomorphisms are atomic; conversely, an atomic morphism f is a local homeomorphism provided the left adjoint f! of f ∗ preserves equalizers ([3], 1.47), or equivalently provided f! is faithful. We note in passing that a local homeomorphism E/A → E is surjective iff A → 1 is epic in E ([3], 4.12(iii)). We recall that a geometric morphism f : F → E is said to be surjective if f ∗ is faithful, and connected if f ∗ is full and faithful. f is called hyperconnected if it is connected and the image of f ∗ is closed under subobjects in F . A connected atomic morphism is hyperconnected: this follows easily from 1.3, since for any object A of E the functor f! induces a bijection from subobjects of f ∗ A to subobjects of f! f ∗ A ∼ = A, whose inverse must be induced by f ∗ . Corresponding to 1.3, we have: LEMMA 1.4. For a geometric morphism f : F → E , the following are equivalent: (i) f is hyperconnected. (ii) f∗ preserves ; i.e. the comparison map φ1 : f∗ (F ) → E is an isomorphism. (iii) For any object A of E, f ∗ induces a bijection from the subobject lattice of A in E to that of f ∗ A in F . Proof. See [5], 1.5. 2 We recall also from [5] that every geometric morphism factors, uniquely up to unique isomorphism, as a hyperconnected morphism followed by one which is localic. For atomic morphisms, this factorization has a particularly simple form, resulting from the fact that a localic atomic morphism is necessarily a local homeomorphism. LEMMA 1.5. Let f : F → E be an atomic geometric morphism. Then (i) f is connected iff f! preserves 1.
AN ‘UNSITELY’ RESULT ON ATOMIC MORPHISMS
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(ii) f can be factored, uniquely up to unique isomorphism, as a connected atomic morphism followed by a local homeomorphism. Proof. (i) If f is connected, then the counit map f! f ∗ → 1E is an isomorphism, so f! 1 ∼ = f! f ∗ 1 ∼ = 1. Conversely, from the fact that f ∗ preserves exponentials we deduce the ‘Frobenius reciprocity’ condition that the natural map f! (B × f ∗ A) → f! (B) × A is an isomorphism for all A and B; putting B = 1, we deduce that the counit f! f ∗ → 1 is an isomorphism, i.e. that f is connected. (ii) Given f , we set A = f! 1. Then we have a canonical factorization of f through the local homeomorphism E/A → E, whose inverse image is given by applying f ∗ to objects over f! 1 and then pulling back along the unit 1 → f ∗ f! 1. It is easy to see that this factorization is atomic, and (using (i)) that it is connected. This establishes the existence of the factorization; the uniqueness follows from [5] and the fact that connected atomic morphisms are hyperconnected. 2
2. Proof of the Theorem We are now ready to state our main result. THEOREM 2.1. Let f : F → E be an atomic morphism, let A be an object of E and let h : E/A → F be a geometric morphism over E. Then (i) h is open. (ii) If f is connected and A → 1 is epic, h is surjective. This is a slight generalization of the result as usually stated for bounded f (see, for example, [6], Proposition VII 4.1), which corresponds to the special case A = 1. However, the general case stated here follows easily from the special one plus known stability properties of atomic morphisms under pullback along local homeomorphisms. Our reason for stating the result in the more general form is that we shall need to allow the object A to vary in the course of our proof. As a first step in the proof, we show: LEMMA 2.2. Let f : F → E be an atomic geometric morphism, and let η denote the unit of the adjunction (f! a f ∗ ). Then (i) For every monomorphism B 0 B in F , the naturality square B0
ηB 0
ηB
f ∗ f! B 0
f ∗ f! B B is a pullback. (ii) If f is connected, ηB is epic for all B.
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Proof. (i) Let χ : B → F ∼ = f ∗ (E ) be the classifying map of B 0 B. By the proof of 1.3 we have a pullback square f! B 0
1 >
f! B
χ
E
in E, where χ is the transpose of χ; but f ∗ preserves this pullback, and the composite f ∗ (χ).ηB equals χ, so the result follows. (ii) Since both f ∗ and f∗ preserve , their left adjoints f! and f ∗ both induce isomorphisms of subobject lattices, by 1.3 and 1.4; and by part (i) the inverse of the induced bijection Sub(B) → Sub(f ∗ f! B) is given by pullback along ηB . But the image of ηB and the top subobject of f ∗ f! B both pull back to the top subobject of B; so they must be equal. 2 We may now deduce the second assertion of 2.1: Proof. For any m : B 0 B in F , we have a pullback square f ∗ f! B 0
B0 m
f ∗ f! m
f ∗ f! B
B
by 2.2(ii). Applying h∗ to this (and recalling that h∗ f ∗ ∼ = A∗ ), we obtain a pullback square h∗ B 0 h∗ m
h∗ B
A∗ f! B 0 A∗ f! m
A∗ f! B
in E/A. So h∗ m is an isomorphism ⇒ A∗ f! m is an isomorphism ⇒ f! m is an isomorphism, since A∗ is faithful ⇒ m is an isomorphism, by 1.3. In other words, h∗ ‘preserves properness of subobjects’; but since it also (unlike f! ) preserves equalizers, it follows that it is faithful. 2 We note that the above proof still works if we replace the topos E/A by any topos G admitting a surjective geometric morphism to E. Similarly, the proof of 2.1(i) which follows can be made to work when E/A is replaced by a topos G admitting a bounded open morphism to E.
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For the latter proof, we first observe that we may reduce to the case A = 1 (i.e. to the case G = E): for we have a pullback square of toposes and geometric morphisms F /f ∗ A
F
f/A
E/A
f
E
where f/A is atomic, and thus h induces a morphism E/A → F /f ∗ A over E/A, whose openness implies that of h itself. Having reduced to the case A = 1, we note that we have a natural transformation h∗ → f! , obtained by applying h∗ to the unit 1F → f ∗ f! . We now show that h satisfies condition (ii) of 1.1. Proof. Let B be an object of F , and suppose we are given a subobject C h B. By 1.3, there is a unique subobject B 0 B such that f! B 0 f! B is the image of the composite C h∗ B → f! B. By 2.2(i) and the fact that h∗ preserves pullbacks, we have a pullback ∗
h∗ B 0
f! B 0
h∗ B
f! B
;
so B 0 is the unique smallest subobject of B such that C 6 h∗ B 0 . Now, if we apply the factorization of 1.5(ii) to the composite F /B 0 → F → E, we obtain F /B 0 → E/f! B 0 → E; so the first factor of the latter is connected and atomic. Also, the composite E/C → E/ h∗ B 0 → F /B 0 is a morphism over E/f! B 0 ; and since C → f! B 0 is epic by the definition of B 0 , it follows from 2.1(ii) that this latter composite is a surjection. So the image of E/C → E/ h∗ B → F /B (in the sense of the surjection–inclusion factorization) is the open inclusion 2 F /B 0 → F /B. Combining 2.1(i) with 1.2, we immediately obtain: COROLLARY 2.3. Let f : F → E be a geometric morphism, and suppose there exists an object A of E and a surjection h : E/A → F over E (that is, ‘F has a sufficient set of E-valued points’). Then f is atomic iff h is open. As we remarked in the Introduction, the extension of the result of 2.1 from bounded morphisms to arbitrary ones does not seem likely to lead to any important new applications. There are two main applications of (the bounded version of) 2.1 in the existing literature: the first is the characterization of atomic morphisms as those open morphisms whose diagonals are open ([6], Theorem VII 4.1), and the
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second uses 2.1(ii), in conjunction with the descent theorem for open surjections, to characterize connected atomic E-toposes with points (for a given base topos E) as the categories of continuous actions of localic groups in E ([6], Theorem VIII 3.1). The first of these clearly requires the existence of a pullback of f along itself before we can even state it for a morphism f ; and so, in the absence of any known method of constructing pullbacks of unbounded morphisms, we cannot hope to extend it to unbounded atomic morphisms. However, the second application does not require the assumption of boundedness for the morphism f from which we start, even though the descent theorem is only valid for bounded open surjections. The point is that, if we are given f : F → E equipped with a morphism h : E/A → F over E (for some object A of E), then h is necessarily bounded (indeed, localic), since the composite f h is localic (cf. [3], 4.44(ii)). Thus we can form the kernel-pair of h (and prove that it is localic over E) in any case; and, provided h is an open surjection, we may then apply the descent theorem to obtain an equivalence between F and the topos of continuous G-actions in E, where G is a localic groupoid in E whose object of objects is the discrete locale A (in particular, if A = 1, then G is a localic group in E). However, a topos of this particular form is necessarily bounded over E (cf. [7]). We therefore obtain the following somewhat unexpected result: COROLLARY 2.4. Let f : F → E be a connected atomic geometric morphism, and suppose there exists a morphism h : E/A → F over E where A → 1 is epic in E. Then f is bounded. Similarly, we may deduce from 2.1(i) that any atomic E-topos with a sufficient set of E-valued points is bounded over E. The unexpectedness of 2.4 derives from the fact that the two best-known examples of unbounded geometric morphisms F → Set (those obtained by taking F to be the topos of set-actions of a ‘large’ group which is an inverse limit of small ones, or by taking it to be the topos of ‘uniformly continuous’ actions of a topological group with no smallest open subgroup – cf. [3], 4.49) are both connected and atomic. However, neither of them has a section: although, in both cases, the forgetful functor F → Set ‘looks as if it should be an inverse image functor’, in that it preserves finite limits and all colimits which exist in F , it fails to have a right adjoint in either case. (In the first case, the solution-set condition required for the Adjoint Functor Theorem is not satisfied, and in the second the category F is not cocomplete.) Thus, although 2.4 itself is a new result, it does not seem likely to lead us to any better understanding of unbounded geometric morphisms in general, or even of connected atomic ones in particular. References 1. Barr, M. and Diaconescu, R.: Atomic toposes, J. Pure Appl. Algebra 17 (1980), 1–24. 2. Johnstone, P. T.: Adjoint lifting theorems for categories of algebras, Bull. London Math. Soc. 7 (1975), 294–297.
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3. Johnstone, P. T.: Topos Theory, London Mathematical Society Monographs No. 10, Academic Press, 1977. 4. Johnstone, P. T.: Open maps of toposes, Manuscripta Math. 31 (1980), 217–247. 5. Johnstone, P. T.: Factorization theorems for geometric morphisms, I, Cahiers Top. Géom. Diff. 22 (1981), 3–17. 6. Joyal, A. and Tierney, M.: An Extension of the Galois Theory of Grothendieck, Mem. Amer. Math. Soc., 1984, p. 309. 7. Moerdijk, I.: The classifying topos of a continuous groupoid, I, Trans. Amer. Math. Soc. 310 (1988), 629–668.
