Theory and Applications of Categories, Vol. 11, No. 13, 2003, pp. 309–320.

PARTIAL TOPOSES

JEAN BÉNABOU AND THOMAS STREICHER

ABSTRACT. We introduce various notions of partial topos, i.e. “topos without terminal object”. The strongest one, called local topos, is motivated by the key examples of finite trees and sheaves with compact support. Local toposes satisfy all the usual exactness properties of toposes but are neither cartesian closed nor have a subobject classifier. Examples for the weaker notions are local homeomorphisms and discrete fibrations. Finally, for partial toposes with supports we show how they can be completed to toposes via an inverse limit construction.

1. Partial Toposes

For a category B with pullbacks its fundamental fibration P_{B} = ∂_{1} : B^{2} → B is well– powered [Bénabou, 1980] iff for every a : A → I in B/I there is an object p(a) : P (a) → I in B/I together with a subobject ∋_{a}: S_{a} ↣ P (a)×_{I}A such that for every b : B → I and subobject m of B×_{I}A there is a unique map χ : b → p(a) with m ≅ (χ×_{I}A)^{∗}∋_{a} as in the diagram

S  \overline{}S_{a}

[figure: B×_{I}A; m; χ×_{I}A; P (a)×_{I}A; ∋_{a}; A]

[figure: B; χ; P (a); p(a); I; a]

Thus B has pullbacks and its fundamental fibration P_{B} is well–powered if and only if every slice of B is an elementary topos. Accordingly, one may characterise elementary toposes as categories B with finite limits whose fundamental fibration P_{B} is well–powered.

General experience with the theory of fibred categories tells us that for bases of fibrations it is fairly irrelevant whether they have terminal objects, however, it is essential that they do have pullbacks. Thus, one may be inclined to define a partial topos as a category E all of whose slices are elementary toposes. This definition, however, is too general as it encompasses all groupoids as every slice of a groupoid is equivalent to the terminal category 1 and, therefore, is a trivial topos. Thus, there arises the question which conditions are missing in order to rule out such degenerate examples.

Received by the editors 2003-04-25 and, in revised form, 2003-06-23.

Transmitted by Anders Kock. Published on 2003-08-20.

2000 Mathematics Subject Classification: 18B25,18D30.

Key words and phrases: fibred categories, partial toposes.

\begin{matrix}c \\ ◯ \end{matrix} Jean Bénabou and Thomas Streicher, 2003. Permission to copy for private use granted.

First observe that a category B has pullbacks iff for every morphism α : J → I in B the functor Σ_{α} : B/J → B/I : β ↦ α∘β has a right adjoint α^{∗}, i.e., as shown in [Bénabou, 1980], that change of base for fibrations along Σ_{α} preserves smallness of fibrations. Thus, B has pullbacks iff for all morphisms α in B change of base along Σ_{α} preserves^{1} all good properties of fibrations as the Σ_{α} do preserve pullbacks anyway. From this point of view it appears as most natural to further require that for all objects I ∈ B the change of base along Σ_{I} = ∂_{0} : B/I → B (i.e. localisation to I) preserves all good properties of fibrations. As the Σ_{I} do preserve pullbacks anyway this requirement boils down to the condition that all Σ_{I} have a right adjoint I^{∗} : B → B/I, i.e. that B has binary products^{2}.

These considerations should motivate the relevance of the following notion.

1.1. Definition. A partial cartesian category is a category with pullbacks and binary products, i.e. a category with finite non–empty limits.

All partial cartesian groupoids are trivial for the following reason.

1.2. Lemma. Every groupoid G with binary products is trivial.

Proof. First of all notice that a groupoid G with binary products is strongly connected in the sense that for all X, Y ∈ G there is a morphism from X to Y as we have π_{1} : X × Y → X and π_{2} : X × Y → Y and, therefore, π_{2} ∘ π_{1}^{−1} : X → Y. Moreover, G is posetal as for morphisms f, g : X → Y we have ⟨f, g⟩ = δ_{Y} ∘ h = ⟨h, h⟩ for some h : X → Y because δ_{Y} = ⟨id_{Y} , id_{Y} ⟩ is an isomorphism.

Notice that we could have defined the notion of “partial cartesian category” equivalently as a category B with binary products all of whose slices are cartesian as the latter requirement is equivalent to the existence of all pullbacks. Notice that for all α : J → I in B the pullback functor α^{∗} : B/I → B/J preserves finite limits. Moreover, if B/I and B/J are cartesian closed or toposes then α^{∗} : B/I → B/J preserves exponentials and subobject classifiers, i.e. all structure under consideration.

This suggests the following notions of “partial locally cartesian closed category” and “partial topos”.

1.3. Definition. A partial locally cartesian closed category is a category with binary products all of whose slices are cartesian closed. A partial topos is a category with binary products all of whose slices are toposes.

1It has been shown in [Bénabou, 1980] that change of base along a functor F between categories with pullbacks preserves all good properties of fibrations if and only if F preserves pullbacks and has a right adjoint.

2For objects I and J in B their product cone is given by I^{∗}J : P → I and the counit ε_{J} : P → J of Σ_{I} ⊣ I^{∗} at J.

It is a straightforward exercise to show that B is partial cartesian closed if and only if B has binary products and P_{B} = ∂_{1} : B^{2} → B is locally small.

Partial toposes have conditional colimits in the following sense.

1.4. Lemma. In a partial topos every finite diagram has a colimit whenever it has a cocone.

Proof. Let E be a partial topos. A cocone for a finite diagram D : 𝔻 → E corresponds uniquely to a diagram D_{A} : 𝔻 → E/A which has a colimit in the topos E/A. Moreover, the functor Σ_{A} = ∂_{0} : E/A → E preserves all colimits as it has a right adjoint. Thus, applying Σ_{A} to a colimiting cocone for D_{A} gives rise to a colimiting cocone for D as required.

Thus, in particular, for partial toposes the situation is as follows.

1.5. Corollary. For a partial topos E we have that

(1) E has an initial object if and only if E is nonempty.

(2) The sum X + Y exists in E if and only if there are morphism X → Z and Y → Z in E for some object Z in E.

(3) A parallel pair of maps f, g : X → Y in E has a coequaliser if and only if h∘f = h∘g for some h : Y → Z in E.

Proof. Obvious from Lemma 1.4.

Next we discuss a few examples of properly partial toposes, i.e. partial toposes without a terminal object^{3}.

1.6. Example. For a topos E and a set I of subterminals, i.e. subobjects of 1_{E} let E_{/I} be the full subcategory of E on those objects X for which there is a morphism from X to some U ∈ I. One easily checks that E_{/I} is always a partial topos and that it is a topos iff I has a greatest element. The next three examples are instances of this quite general scheme.

1.7. Example. Let FinSet be the category of finite sets and arbitrary maps. The category of finitely branching trees is given by FinSet^{ω^{op}}, i.e. presheaves on ω with values in FinSet. It is a topos as ω/n is finite for all n ∈ ω. Let I be the set of all proper subobjects of the terminal object, i.e. all presheaves isomorphic to ∂_{0} : ω/n → ω for some n ∈ ω. Then T = FinSet_{/I}^{ω^{op}}, the category of finite trees, is a properly partial topos as I does not have a greatest element.

3Obviously, a partial topos is a topos iff it has a terminal object.

1.8. Example. The category of Set–valued presheaves over the large poset Ord of ordinals is a topos as Ord/α is a set for all α ∈ Ord. Let Wrk be the full subcategory on those presheaves A whose support is a set, i.e. for which there is an α ∈ Ord such that A(β) = ∅ for all β > α.

1.9. Example. Let X be a locally compact Hausdorff space and I the set of subterminals in Sh(X) consisting of the relatively compact open subsets of X, i.e. those open U ⊆ X whose closure \overline{U} is compact in X. Then Sh_{c}(X) = Sh(X)_{/I}, the category of sheaves on X “with compact support”, is a partial topos which is properly partial iff X is not compact.

1.10. Example. Let LH be the category of spaces and local homeomorphism. It can be shown that LH has binary products (see [Selinger, 1994]). Thus, LH is a partial topos as for every space X the slice of LH over X is equivalent to the topos Sh(X). It follows from Lemma 2.2 below that LH does not have a terminal object.

1.11. Example. Let dF be the category of small categories and discrete fibrations as morphisms. It can be shown that dF has binary products (the proof is a variation of Selinger’s proof in [Selinger, 1994] that LH has binary products). Thus, dF is a partial topos as for every small category ℂ the slice of dF over ℂ is equivalent to the presheaf topos ℂ̂ = Set^{C^{op}}. It follows from Lemma 2.2 below that dF does not have a terminal object.

We conclude this section by showing that partial cartesian categories cannot have exponentials or a subobject classifier unless they have already a terminal object, i.e. are properly cartesian.

1.12. Lemma. Let B be a partial cartesian category. If for some object A in B the exponential A^{A} exists in B then B has a terminal object, namely the equalizer of the identity on A^{A} and the morphism r : A^{A} → A^{A} defined as the exponential transpose of π_{2} : A^{A} × A → A.

Proof. Let e : T ↣ A^{A} be an equalizer of id_{A^{A}} and r. Observe that a map f : B → A^{A} equalizes id_{A^{A}} and r iff ev ∘ (f × A) = ev ∘ (r × A) ∘ (f × A), i.e. iff ev ∘ (f × A) = π_{2} ∘ (f × A) = π_{2}. Thus, for every object B in B there exists precisely one map from B to A^{A} equalizing id_{A^{A}} and r. Accordingly, for every object B in B there exists precisely one map from B to T, i.e. T is terminal in B as claimed.

1.13. Lemma. Let B be a partial cartesian category with a subobject classifier t : T ↣ Ω. Then T is terminal in B.

Proof. For every object A of B let ⊤_{A} : A → Ω be the classifier of id_{A}. Notice that for f : B → A we have ⊤_{A} ∘ f = ⊤_{B}. Let p : Ω → T be the unique map with t ∘ p = ⊤_{Ω}. We have t ∘ p ∘ t = ⊤_{Ω} ∘ t = ⊤_{T} = t from which it follows that p ∘ t = id_{T}. Thus, the monomorphism t is an equalizer for id_{Ω} and ⊤_{Ω}. Obviously, a map f : A → Ω equalizes id_{Ω} and ⊤_{Ω} iff f = ⊤_{Ω} ∘ f = ⊤_{A}. Thus, for every object A of B there exists precisely one map from A to T, i.e. T is terminal in B as claimed.

2. Partial Toposes with Supports

Now we will discuss some further property of partial toposes satisfied by all the examples above with the exception of Examples 1.10 and 1.11.

2.1. Definition. Let B be a category. An object U ∈ B is called subterminal if for all objects X ∈ B there exists at most one morphism in B from X to U. We write st(B) for the full subcategory of B on subterminal objects of B and say that B is a category with supports or has supports iff the inclusion i : st(B) ↪ B has a left adjoint supp.

Obviously, the category st(B) is posetal as for subterminals U and V there is at most one arrow from U to V. Accordingly, we write U ≤ V iff there is an arrow from U to V. Notice, moreover, that in a category with a terminal object the subterminals in the sense of the previous definition coincide with the subobjects of the terminal object.

Obviously, for U ∈ st(B) it leads to no confusion when identifying B/U with the full subcategory of B on those objects X for which there is a morphism X → U in B. Hence, in the sequel we employ this harmless identification whenever convenient.

For partial cartesian categories B with supports Lemmas 1.12 and 1.13 can be shown even more easily as follows. If A^{A} exists in B then for all objects B in B there is a map from B to A^{A} (the exponential transpose of π_{2} : B × A → A) and accordingly supp(B) ≤ supp(A^{A}) from which it follows that supp(A^{A}) is terminal in B. If there is a subobject classifier t : T ↣ Ω in B then for all objects B in B there is a map from B to Ω (the classifier of id_{B}) and accordingly supp(B) ≤ supp(Ω) from which it follows that supp(Ω) is terminal in B.

The next lemma shows that not every partial topos has supports.

2.2. Lemma. Both in LH and dF there are objects X such that there is no map from X to a subterminal object U.

Proof. First notice that up to isomorphism the monos in LH are the open subspace inclusions. Let ∇(2) be the codiscrete space with 2 elements. The space ∇(2) cannot be subterminal as it admits endomaps different from id_{∇(2)}. As every local homeomorphism h : ∇(2) → X must have discrete fibres it follows that h is one–to–one, i.e. monic and, therefore, isomorphic to a subspace inclusion. Thus, if there were a map h : ∇(2) → U with U subterminal then ∇(2) were subterminal, too. Thus, there is no map from ∇(2) to a subterminal object.

First notice that up to isomorphism the monos in dF are inclusions of sieves. Let ∇(2) be the codiscrete category with 2 elements. The category ∇(2) cannot be subterminal as it admits endomaps different from id_{∇(2)}. As every discrete fibration P : ∇(2) → X must have discrete fibres it follows that P is isomorphic to a sieve inclusion and, therefore, a monomorphism. Now if P : ∇(2) → 𝔹 were a discrete fibration with 𝔹 subterminal then ∇(2) were subterminal, too. Thus, there is no map from ∇(2) to a subterminal object.

Obviously, from Lemma 2.2 it follows that LH and dF have neither supports nor a terminal object.

Notice that already for the space ℝ of real numbers there cannot exist a reflection map η_{ℝ} from ℝ to a subterminal U. The reason is that for every c ∈ ℝ the map f(x) = c + x is an automorphism of ℝ from which it follows that η_{ℝ} : ℝ → U has to be constant and, therefore, cannot be a local homeomorphism as it has a non-discrete fibre, namely ℝ.

There is also another argument showing that dF is not a category with supports. First observe that for a subterminal object 𝔹 in dF it holds that objects I, J ∈ 𝔹 are isomorphic whenever 𝔹/I ≅ 𝔹/J. Let ℂ be the category

[figure: A; α; T; β; B]

where the objects A, B and T are all different. We show that for every discrete fibration P : ℂ → 𝔹 the category 𝔹 cannot be subterminal. The objects P (A) and P (B) are not isomorphic as otherwise there were a map from A to B in ℂ. But on the other hand we have

𝔹/P (A) ≅ ℂ/A ≅ 1 ≅ ℂ/B ≅ 𝔹/P (B)

from which it follows by the above observation that P (A) and P (B) are isomorphic in 𝔹. Thus, the category 𝔹 cannot be subterminal in dF and, accordingly, in dF there is no map from ℂ to a subterminal object. From this it follows that dF has neither supports nor a terminal object.

Next we show how under the assumption of having supports partial cartesian categories can be characterised more simply.

2.3. Lemma. Let B be a category with supports and pullbacks. Then B has binary products if and only if st(B) has binary infima.

Proof. Assume that B has supports and pullbacks.

If B has binary products then st(B) has binary infima as subterminals are closed under binary products.

On the other hand if st(B) has binary infima then the binary product of A and B is constructed as follows

[figure: A × B; p_{2}; B′; m_{2}; B]

[figure: A′; p_{1}; U ∩ V; V]

[figure: A; m_{1}; U]

where U = supp(A), V = supp(B) and U ∩ V is the infimum of U and V in st(B). The projections π_{i} are given by m_{i} ∘ p_{i} as indicated in the diagram.

We have already observed that most examples of partial toposes satisfy the additional property of having supports. In the Examples 1.7–1.9 it holds moreover that the subcategory of subterminals is filtered in the sense that for all subterminals U and V there is a subterminal W with U, V ≤ W.

Next we show that categories B with supports and pullbacks have binary products whenever st(B) is filtered.

2.4. Lemma. Let B be a category with pullbacks and supports where st(B) is filtered, i.e. for all U, V ∈ st(B) there is a W ∈ st(B) with U, V ≤ W. Then B has binary products.

Proof. By Lemma 2.3 it suffices to show that st(B) has binary infima. Suppose U and V are in st(B). As st(B) is filtered there is a W in st(B) with U, V ≤ W. Then the infimum of U and V is given by the pullback

[figure: U ∩ V; V; U; W]

The object U ∩ V is subterminal as for every object X there is at most one arrow from X to U ∩ V (because there is at most one arrow from X to U and at most one arrow from X to V).

In the light of the previously considered examples the following more restrictive notion of partial topos seems to be also the more useful one.

2.5. Definition. A category E is called a local topos iff the following three conditions hold

(1) every slice of E is a topos (2) the inclusion i : st(E) → E has a left adjoint supp

(3) the category st(E) is filtered, i.e. for all U, V ∈ st(B) there is a W ∈ st(B) with U, V ≤ W.

Notice that for partial toposes E_{/I} as in Example 1.6 their categories of subterminals are filtered if and only if I is filtered. This, however, need not be the case as e.g. if E = Set^{I} for some infinite set I and I is given by all subsets of I having at most n elements (for a finite cardinal n fixed in advance).

We conclude this section by observing that most of the exactness properties of toposes extend to local toposes.

From Lemma 1.4 it follows that finite colimits exists (as filtered supports guarantee that every finite diagram has a cocone). Moreover, finite colimits are preserved by pullbacks as every slice is a topos and in toposes colimits are preserved by pullbacks. For the same reason a morphism is epic iff it is a coequaliser (of its kernel pair) and monos are preserved by pushouts along arbitrary morphism and such pushout squares are also pullback squares. Furthermore, all local toposes are extensive categories with pullbacks because they have finite sums and all its slices are extensive as they are toposes. Moreover, all properties of internal categories, functors and distributors (as studied in [Johnstone, 1977]) are valid also for local toposes. Finally, notice that all partial toposes are locally cartesian closed, i.e. all pullback functors f^{∗} have right adjoints Π_{f} satisfying the (Beck-) Chevalley condition. Thus, fibred over itself every partial topos has internal products (see [Bénabou, 1980, Streicher, 2003]).

3. Topos Completion of Partial Toposes with Supports

Now we will show how to complete a partial topos E with supports to a topos E_{∞}. For the rest of this section all partial toposes are assumed to have supports.

Notice that for every map i : U → V between subterminals for the induced essential geometric morphism

Σ_{i} ⊣ i^{∗} ⊣ Π_{i} : E/U → E/V

it holds that i^{∗} is logical and its adjoints are full and faithful.

We write E/(−) for the contravariant (pseudo-)functor from st(E) to the (2-)category Log of toposes and logical morphisms sending U to E/U and i : U → V to the logical functor i^{∗} : E/V → E/U.

3.1. Theorem. Let E be a partial topos with supports and

E_{∞} = Lim_{U∈st(E)} E/U

the inverse limit of E/(−) : st(E)^{op} → Log called topos completion of E.

Then for every U ∈ st(E) the projection π_{U} : E_{∞} → E/U is equivalent to U^{∗} : E_{∞} → E/U and has full and faithful left and right adjoints Σ_{U} and Π_{U}, respectively.

Moreover, st(E) can be embedded into st(E_{∞}) by sending U ∈ st(E) to ι(U) = (U ∩ V)_{V ∈st(E)}. Then E can be recovered up to equivalence from E_{∞} as E_{∞}_{/st(E)} where st(E) is considered as included into st(E_{∞}) via ι.

Proof. We first give a more detailed description of the inverse limit E_{∞} = Lim_{U∈st(E)} E/U. Its objects are cartesian sections of the fibration corresponding to the pseudo-functor E/(−) : st(E)^{op} → Log. More explicitly, an object of E_{∞} is a function A picking an object A_{U} ∈ E/U for every U ∈ st(E) and a (mono)morphism A_{V,U} : A_{V} → A_{U} with

[figure: A_{V}; A_{V,U}; A_{U}; V; U]

for every pair V ≤ U in st(E) in a functorial way, i.e. satisfying A_{U,U} = id_{A_{U}} and A_{V,U} ∘ A_{W,V} = A_{W,U} whenever W ≤ V ≤ U. For A, B ∈ E_{∞} a morphism from A to B is given by a family (f_{U} : A_{U} → B_{U})_{U∈st(E)} such that

A_{V}[figure: A_{V,U}] A_{U}

[figure: B_{V}; f_{V}; B_{V,U}; B_{U}; f_{U}]

[figure: V; U]

whenever V ≤ U.

It is evident that E_{∞} has finite limits which are constructed componentwise. A (canonical) terminal object 1 in E_{∞} for example is given by 1_{U} = U and 1_{V,U} the unique morphism from V to U in E.

For A, B ∈ E_{∞} their exponential B^{A} is constructed by putting (B^{A})_{U} = B_{U}^{A_{U}} (exponential in E/U) and for V ≤ U the map (B^{A})_{V,U} : B_{V} ^{A_{V}} → B_{U}^{A_{U}} is given as the transpose of B_{V,U} ∘ ẽv where ẽv : B_{V} ^{A_{V}} × A_{U} → B_{V} is the unique map with ẽv ∘ (B_{V} ^{A_{V}} × A_{V,U}) = ev : B_{V} ^{A_{V}} × A_{V} → B_{V} as in the diagram

[figure: B_{V} ^{A_{V}} × A_{V}; ev; B_{V}; B_{V,U}; B_{U}]
[figure: ẽv]
B_{V} ^{A_{V}} × A_{U} B_{V} ^{A_{V}} × A_{V,U} ≅_{}

Thus E_{∞} is cartesian closed.

A subobject classifier Ω in E_{∞} can be constructed by choosing for every U ∈ st(E) a generic mono t_{U} : 1_{U} → Ω_{U} in E/U and then defining for V ≤ U in st(E) the map Ω_{V,U} : Ω_{V} → Ω_{U} as the unique morphism in E/U satisfying

[figure: 1_{V}; 1_{U}; Ω_{V}; t_{V}; Ω_{V,U}; Ω_{U}; t_{U}]

from which definition it is clear that Ω_{U,U} = id_{Ω_{U}} and Ω_{W,U} = Ω_{V,U} ∘ Ω_{V,U} as

[figure: 1_{W}; 1_{V}; 1_{U}]

[figure: Ω_{W}; t_{W}; Ω_{W,V}; Ω_{V}; t_{V}; Ω_{V,U}; Ω_{U}; t_{U}]

whenever W ≤ V ≤ U in st(E). That Ω_{V,U} fits into a pullback diagram

[figure: Ω_{V}; Ω_{V,U}; Ω_{U}; V; i = 1_{V,U}; U]

can be seen as follows. As i^{∗} : E/U → E/V is a logical functor it in particular preserves subobject classifiers. Thus, we have

[figure: 1_{V}; ≅; i^{∗}1_{U}; 1_{U}]

[figure: Ω_{V}; t_{V}; χ ; ≅; i^{∗}Ω_{U}; i^{∗}t_{U}; m; Ω_{U}; t_{U}]

[figure]

[figure: V; i; U]

The map χ is an isomorphism because i^{∗}t_{U} is a subobject classifier in E/V as i^{∗} is logical. Thus, we have Ω_{V,U} = m ∘ χ form which the claim follows.

This finishes the proof that E_{∞} is a topos.

One easily sees that the projection π_{U} : E_{∞} → E/U is equivalent to the logical functor U^{∗} : E_{∞} → E_{∞}/U ≃ E/U. One can embed E into E_{∞} via a functor I sending A ∈ E to I(A) ∈ E_{∞} where I(A)_{U} = A × U and I(A)_{V,U} = A × i_{V,U} as illustrated in the diagram

[figure: A × V; A × i_{V,U}; A × U; V; π_{2}; i_{V,U}; U; π_{2}]

and f : A → B in E to I(f) : I(A) → I(B) with I(f)_{U} = f × U. The functor U^{∗} has a full and faithful left adjoint Σ_{U} equivalent to the restriction of I to E/U. As U^{∗} is logical and has a full and faithful left adjoint it follows that U^{∗} has also a right adjoint Π_{U} which is also full and faithful.

Obviously, the the restriction of I to st(E) factors through st(E_{∞}) and is equivalent to ι : E → E_{∞} : U ↦ (U∩V)_{V ∈st(E)} as U × V ≅ U ∩ V for subterminals U, V. Thus, we have E/U ≃ E_{∞}/ι(U) for every U ∈ st(E) from which it follows that E ≃ E_{∞}_{/st(E)} as desired.

Notice that for defining the embedding I : E → E_{∞} we have used essentially that E has supports and binary products.

We observe that applying the topos completion procedure of Theorem 3.1 to the partial toposes of the Examples 1.7–1.9 gives rise to the toposes from which they are were constructed, i.e. T_{∞} ≃ FinSet^{ω^{op}}, Wrk_{∞} ≃ Set^{Ord^{op}} and Sh_{c}(X)_{∞} ≃ Sh(X) for locally compact Hausdorff spaces X.

Finally we want to remark that the topos completion procedure of Theorem 3.1 extends to NNOs (Natural Numbers Objects) in the following way. If E is a partial topos such that for every U ∈ st(E) there is a NNO

[figure: U; z_{U}; N_{U}; s_{U}; N_{U}]

in E/U then these can be organised into a single NNO in E_{∞} choosing for V ≤ U the map N_{V,U} : N_{V} → N_{U} as the unique arrow making the diagram

[figure: V; U]

[figure: N_{V}; z_{V}; N_{V,U}; N_{U}; z_{U}]

[figure: N_{V}; s_{V}; N_{V,U}; N_{U}; s_{U}]

commute. Existence and uniqueness of N_{V,U} follows from the fact that the pullback functor V^{∗} : E/U → E/V preserves NNOs.

Acknowledgements. We acknowledge the use of Paul Taylor’s diagram macros used for writing this paper.

References

J. Bénabou (1980) Des Catégories Fibrées lecture notes by J.-R. Roisin of a course at Univ. Louvain-la-Neuve.

P. T. Johnstone (1977) Topos Theory Academic Press.

P. Selinger (1994) LH has nonempty products unpublished note, electronically available

    at http://quasar.mathstat.uottawa.ca/˜ selinger/papers/ lh.ps.gz

T. Streicher (2003) Fibred Categories à la J. Bénabou extended version of lecture notes for a course given in Munich in April 1999, electronically available at

    http://www.mathematik.tu-darmstadt.de/˜ streicher/FIBR/FibLec.ps.gz

20 rue Lacépède, F-75005, Paris, France

FB4 Mathematik, TU Darmstadt, Schloßgartenstr.7, D-64289, Germany

Email: jean.benabou@wanadoo.fr
      streicher@mathematik.tu-darmstadt.de

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/13/11-13.{dvi,ps}

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Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

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Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi, Postscript and PDF. Details will be e-mailed to new subscribers. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh,

rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at http://www.tac.mta.ca/tac/. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of Wales Bangor: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Palo Alto Research Center: paiva@parc.xerox.com Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca