2.1 Regular hyperdoctrines

Regular formulae are those with equality and closed under finitary conjunction and existential quantification; and regular logic is given by entailments between such regular formulae.

Aside The condition for equality is taken from Maietti and Rosolini (Reference Maietti and Rosolini2013a). They call a hyperdoctrine with conjunction and equality an elementary doctrine. They give a different but equivalent formulation in Maietti and Rosolini (Reference Maietti and Rosolini2013b). There they call a hyperdoctrine with conjunction and existential quantification an existential doctrine. So I think that in their terms, a regular hyperdoctrine is both elementary and existential. But these matters are very delicate and I am not sure that the details which I give are exactly right. (In particular, I have not checked exactly how much Beck–Chevalley should be assumed.) The test is that the evident interpretation of (many-sorted) regular logic in regular hyperdoctrines is sound and complete. That is what determines just how strong the various conditions should be. Maietti and Rosolini have developed a wealth of quite delicate theory around their weak notions and that has been extended by Trotta. My impression is that there is still much to explore.

2.2 The free regular category

The notion of a regular hyperdoctrine lies at the heart of indexed categorical logic because a regular hyperdoctrine $\mathcal{P}$ over $\mathbb{C}$ freely generates a regular category ${{\mathbb{C}}}({\mathcal{P}})$ .

The objects of ${{\mathbb{C}}}({\mathcal{P}})$ are of the form $R \in {\mathcal{P}}(X)$ with $X \in {{\mathbb{C}}}$ . Write such as $(X,R)$ and regard them as formal subobjects of the objects of $\mathbb{C}$ . Maps from $(X,R)$ to $(Y,S)$ are equivalence classes of $F \in {\mathcal{P}}(X \times Y)$ which in the internal logic (the logic of $\mathcal{P}$ ) are graphs of functions from $R$ to $S$ . Composition is effected using the existential quantifier to compose the graphs. Restricted equalities give the graphs of the identities. The category ${{\mathbb{C}}}({\mathcal{P}})$ comes equipped with a functor $\nabla : {{\mathbb{C}}} \to {{\mathbb{C}}}({\mathcal{P}})$ taking an object $X$ in ${\mathbb{C}}$ to $\top \in {\mathcal{P}}(X)$ and a map $u: X \to Y$ to $(u \times 1_Y)^*(\delta _Y)$ . Objects of this kind are called constant objects.

A category is said to be regular (in the sense of Barr) just when it is cartesian or left exact (i.e., has finite limits) and has stable factorisation of maps as surjections followed by monics. (There are many equivalent formulations – see Johnstone (Reference Johnstone2002).)

The proof should be routine: the regular structure is given in the internal logic. Of course each regular category has an associated subobject regular hyperdoctrine and the construction ${{\mathbb{C}}}({\mathcal{P}})$ has a universal property, but I leave that aside on this occasion. All the same I shall refer to ${{\mathbb{C}}}({\mathcal{P}})$ as the regular completion of $\mathcal{P}$ . Of course, regular completion refers to many other things, notably the regular completion of a left exact category. I hope that there will be no confusion.

2.3 The free exact category

In addition to the category ${{\mathbb{C}}}({\mathcal{P}})$ of formal subobjects, one can consider the category ${{\mathbb{C}}}[{\mathcal{P}}]$ of formal quotients of subobjects (subquotients of objects) of $\mathbb{C}$ . The objects of ${{\mathbb{C}}}[{\mathcal{P}}]$ are of the form $=_X \in {\mathcal{P}}(X \times X)$ where in the internal logic $=_X$ satisfies the statement that it is a partial equivalence relation (PER): in other words, it satisfies the standard conditions of symmetry and transitivity. Write such objects as $(X, =_X)$ . Maps from $(X, =_X)$ to $(Y,=_Y)$ are equivalence classes of $F \in {\mathcal{P}}(X \times Y)$ which satisfy in the internal logic the conditions determining a function between the subquotients. Again composition is defined by existential quantification, and now identities are given by the PERs themselves. There is a functor $\nabla : {{\mathbb{C}}} \to {{\mathbb{C}}}[{\mathcal{P}}]$ taking an $X \in {{\mathbb{C}}}$ to $\textrm {Id}_X \in {\mathcal{P}}(X \times X)$ and a map $u: X \to Y$ to $(u \times 1_Y)^*(\textrm {Id}_Y)$ . And again these are the constant objects.

In the logical setting, a category is said to be exact just when it is regular, it has quotients of equivalence relations and equivalence relations are effective (i.e., kernel pairs of the quotient). There are again many equivalent formulations; see Johnstone (Reference Johnstone2002).

Again the proof should be routine. One could formulate a universal property. Again I let that go but I still refer to ${{\mathbb{C}}}[{\mathcal{P}}]$ as the exact completion of $\mathcal{P}$ . Those new to the subject should be aware that there are other uses. At the end of this paper, I mention a different construction which is part of a wider picture. (A warning for the unwary is that particular care is needed with the free exact category over a regular one. The delicate issue is laid out in Carboni and Magno (Reference Carboni and Magno1982).)

Obviously, there is a functor from the regular category ${{\mathbb{C}}}({\mathcal{P}})$ to the exact category ${{\mathbb{C}}}[{\mathcal{P}}]$ . It takes an object $(X,R)$ of the first to an object $(X, =_R)$ of the second where the PER is defined in the internal logic by $x =_R y \leftrightarrow x \in R \land \delta (x,y)$ . As a result, we have functors

with the first preserving products and the second regular structure.

It is the exact category ${{\mathbb{C}}}[{\mathcal{P}}]$ which plays the central role in this paper. So for those new to the area, it may help to visualise the objects as follows. Given an object $(X, =_X)$ , setting $E_X = {\mathcal{P}}(\Delta )(=_X) \in {\mathcal{P}}(X)$ gives an induced existence predicate. Officially $(X, E_X)$ is an object of ${{\mathbb{E}}}({\mathcal{P}})$ , but it gives rise as just explained to an object of ${{\mathbb{E}}}[{\mathcal{P}}]$ which for simplicity I still write as $(X, E_X)$ . Then our original $(X,=_X)$ lies in a diagram

in ${{\mathbb{C}}}[{\mathcal{P}}]$ , where the horizontal arrow is a monic and the vertical one a surjection. Thus, the diagram exhibits $(X, =_{X})$ as a quotient of $(X,E_X)$ which is itself a subobject of the constant object $\nabla X$ .

2.4 Coherent hyperdoctrines

Coherent formulae are obtained by adding closure under finite disjunctions to the closure properties of regular formulae. Coherent logic is given by entailments between such formulae. There is an extensive treatment in Johnstone (Reference Johnstone2002). The corresponding hyperdoctrines do not really feature in this paper, but in view of the central role of coherent toposes in Topos Theory I give a quick sketch.

Aside A pretopos is a coherent category which is exact (here Johnstone says effective) and has disjoint finite coproducts (Johnstone calls this positive). Such categories are effectively invariants of coherent classifying toposes. The idea of conceptual completeness was formulated in these terms by Makkai and Lurie. Recent work – see in particular Hamad (Reference Hamad2025a) and Saadia (Reference Saadia2025) – extends the ideas to the general setting of classifying toposes for geometric theories. The geography of that setting, that is, of general Grothendieck toposes, is explored in detail in the recent Di Liberti and Ye (Reference Di Liberti and Ye2025).

2.5 First-order hyperdoctrines

The first treatments of the Kock–Mikkelsen factorisation concerned functors between toposes which preserved first-order intuitionistic logic. That does not play a big role in this paper, but it seems as well to have an account.

Aside There is some redundancy in the definition. Frobenius Reciprocity follows from preservation of Heyting implication under reindexing.

3.1 Two definitions

As explained in Pitts (Reference Pitts2002), the word tripos was coined for a general notion of an fibred logic or hyperdoctrine $\mathcal{P}$ over $\mathbb{C}$ for which the exact completion ${{\mathbb{C}}}[{\mathcal{P}}]$ is in fact a topos. The natural angle on this is to require that $\mathcal{P}$ interpret first-order logic and ask what more is needed to create the higher-order logic of a topos. Two distinct definitions have been considered. In both, we have a first-order hyperdoctrine $\mathcal{P}$ over $\mathbb{C}$ and equipped with generic predicates which is structure of the following shape. For each $A \in {{\mathbb{C}}}$ , there is $PA \in {{\mathbb{C}}}$ and a predicate $\textrm {In} \in {\mathcal{P}}(A \times PA)$ . The idea is that the predicate represents membership.

The original definition given in Hyland et al. (Reference Hyland, Johnstone and Pitts1980) and Pitts (Reference Pitts1981) was pragmatic; in that, it covered the leading examples and allowed the development of a substantial theory.

Of course, the label in the external sense did not appear in the original papers. I adopt it here because there is another possibly more compelling notion.

In Pitts (Reference Pitts2002), a paper written for a retrospective volume on realisability organised by Jaap van Oosten, Andy Pitts looked more closely at the question of what was needed to obtain a topos. He observed that there is a weaker but quite natural internal version of the condition that the generic predicate represents membership which suffices to ensure that the represented category ${{\mathbb{C}}}[{\mathcal{P}}]$ be a topos.

Definition 5. A first-order hyperdoctrine $\mathcal{P}$ indexed over $\mathbb{C}$ is a tripos in the internal sense just when it is equipped with generic predicates as above satisfying the following property. For every $R \in {\mathcal{P}}(A \times C)$ , the condition

\begin{equation*} \forall c \in C. \exists p \in PA . \forall a \in A. R(a,c) \iff \textrm {In}(a, p) \end{equation*}

holds in the logic of the hyperdoctrine.

Following Pitts (Reference Pitts2002), I shall call the condition the Comprehension Axiom, shortening that to CA.

Both the definitions in the external and the internal sense are due to Pitts. The merit of his second more relaxed definition is apparent in the following theorem from Pitts (Reference Pitts2002).

I shall call this the Pitts Caracterisation Theorem. Pitts (Reference Pitts2002) suggests that had he conducted the analysis earlier then he might have considered making what I am calling a tripos in the internal sense the official definition. That is tempting to a purist but as we shall see in this paper there is also something to be said in favour of the original definition.

To see what that is we need to understand what one gets from the external Skolemised definition of tripos. Obviously, it cannot be a property of the represented topos ${{\mathbb{C}}}[{\mathcal{P}}]$ . (After all every topos is represented over itself by its subobject tripos.) It has to involve its relationship with the base category ${\mathbb{C}}$ , that is, the functor $\nabla : {{\mathbb{C}}} \to {{\mathbb{C}}}[{\mathcal{P}}]$ . To understand that, it seems good to start with the notion of tripos in the internal sense and the Pitts Characterisation Theorem.

Aside Our two notions of tripos start with a first-order hyperdoctrine. One might wonder about other formulations. If the aim is to obtain a topos, one should consider that long ago Robinson and Rosolini (Reference Robinson and Rosolini1990) gave a construction of the effective topos via a delicate process of adding recursively indexed coproducts and then freely adding quotients as in Carboni and Magno (Reference Carboni and Magno1982). In a related direction of work, Carboni and Rosolini (Reference Carboni and Rosolini2000) considered what was required to ensure that an exact completion was locally cartesian closed; and in Menni (Reference Menni2003), Menni characterised the left exact categories whose exact completions are toposes. There are many unexplored directions of this kind.

3.2 Triposes in the internal sense

Let us try to understand the CA. In it, we are given some $R \in {\mathcal{P}}(A \times C)$ . That corresponds to a subobject $R$ of $\nabla A \times \nabla C$ in ${{\mathbb{C}}}[{\mathcal{P}}]$ . What then does the condition

\begin{equation*} \forall c \in C. \exists p \in PA . \forall a \in A. R(a,c) \iff \textrm {In}(a, p) \end{equation*}

give? Well our given structure $\textrm {In} \in {\mathcal{P}}(A \times PA)$ enables us to define an equivalence relation $\Leftrightarrow \, \in {\mathcal{P}}(PA \times PA)$ by the formula

\begin{equation*} \forall a \in A. \textrm {In}(a,p) \iff \textrm {In}(a,q) . \end{equation*}

That gives us a quotient $\widetilde {P}(\nabla A )= (PA , \Leftrightarrow\!)$ of $\nabla (PA)$ and the generic predicate induces a generic subobject of $\nabla A \times \widetilde {P}( \nabla A)$ which I continue to call $\textrm {In}$ . Now consider the relation defined in the internal logic by

\begin{equation*} \forall a \in A. R(a,c) \iff \textrm {In}(a, p) \end{equation*}

It is evidently a total functional relation from $C$ to $(PA, \Leftrightarrow\!)$ , so it represents a map $h : \nabla C \to \widetilde {P}\nabla A$ in ${{\mathbb{C}}}[{\mathcal{P}}]$ . It is easy to see that pulling back the subobject $\in$ along $\textrm {id} \times h$ gives the original $R$ .

The upshot is that the Pitts CA says exactly that the generic subobject $\textrm {In} \in \nabla A \times \widetilde {P}( \nabla A)$ satisfies the condition for being the power object of $\nabla A$ for subobjects indexed over objects themselves of the form $\nabla C$ . Thus, the force of the Characterisation Theorem is that this restricted condition implies that we have a full power object for all objects of the category. One can derive that fact in two steps. First using the internal logic extend from $\nabla C$ to an arbitrary indexing object $(C, =\!)$ so that $\textrm {In} \in \nabla A \times \widetilde {P}(\nabla A)$ is indeed the power object of $\nabla A$ . Then observe that each object $(A, =\!)$ of ${{\mathbb{C}}}[{\mathcal{P}}]$ is a quotient of a subobject of an object $\nabla A$ in the image of $\nabla$ as in the diagram

We obtain the power object of $(A,E)$ as a subobject of that of $\nabla A$ by direct image and the power object of $(A, =\!)$ as a subobject of that via inverse image. (That is easy: both the monics involved are split.)

What I have just presented is a sketch proof of the Pitts Characterisation Theorem. With that explanation in place, I turn to the question of the force of the definition of tripos in the external sense.

Aside The definition of tripos in the original paper of Hyland et al. (Reference Hyland, Johnstone and Pitts1980) was unnecessarily demanding as regards the structure of the base category. The point that it suffices to consider $\mathcal{P}$ over categories with products is already made in Pitts (Reference Pitts1981). Note that van Oosten (Reference van Oosten2008) uses a seemingly different setting from Pitts (Reference Pitts2002). I do not believe there is really much difference between the two and certainly both cover the substantial examples.

3.3 Triposes in the external sense

In the discussion above, we got a generic subobject $\textrm {In}$ of $\nabla A \times \widetilde {P} A$ . Now given a subobject $\phi$ of $\nabla A \times \nabla C$ , the internal condition simply gave us a classifying map $\nabla C \to \widetilde {P}(A)$ in ${{\mathbb{C}}}[{\mathcal{P}}]$ . By contrast, the external condition gives us a classifying map $r : C \to PA$ in the base category ${\mathbb{C}}$ of the tripos. That gives a map $\nabla r : \nabla C \to \nabla PA$ in ${{\mathbb{C}}}[{\mathcal{P}}]$ and composing that with the quotient $\nabla PA \to (PA, \Leftrightarrow\!)$ gives the classifying map $\nabla C \to \widetilde {P}(A)$ . Thus, the additional force of the external condition amounts to this. Any map $\nabla C \to \widetilde {P}(A)$ factors through the quotient via a map of the form $\nabla r : \nabla C \to \nabla PA$ . As in our sketch of a proof of the Pitts Characterization Theorem, this condition extends from constant to all objects. This has an important consequence. I give a brief outline using some temporary homespun notation.

Any object in a topos is isomorphic to its object of singletons which is itself a subobject of its power object. Take an object $(A, =\!)$ of ${{\mathbb{E}}}[{\mathcal{P}}]$ . We observed above that it can be regarded as a quotient of a subobject of $\nabla A$ . But we can find another representation as a quotient of a subobject of a constant object as follows. In any topos, each object $X$ is a subobject of its power object $PX$ as the subobject $SX$ of singletons. But in ${{\mathbb{C}}}[{\mathcal{P}}]$ , the power object

\begin{equation*} \widetilde {P}(A,=\!) = (PA, \Leftrightarrow\!) \end{equation*}

is a quotient of a subobject of $\nabla PA$ . I write $(PA, E)$ for the subobject of $\nabla PA$ . Then I write $(SA, \Leftrightarrow\!)$ for the object of singletons and clumsily $(PA, S)$ for the result of pulling back that singleton subobject along the quotient map to $(PA, \Leftrightarrow\!)$ . That gives a diagram as follows.

which exhibits the object of singletons $(SA, \Leftrightarrow\!)$ and so the original object $(A,=\!)$ as the quotient of a subobject of $\nabla PA$ .

The value of the external definition of tripos seems to be that the presentation

enjoys a kind of lifting and extension property involving the functor $\nabla$ . Suppose that we have an object $(C,=\!)$ with some presentation

and suppose that in ${{\mathbb{C}}}[{\mathcal{P}}]$ we have a map from $f: (C,=\!) \to (SA, \Leftrightarrow\!)$ . Then there is a map $h: C \to PA$ in $\mathbb{C}$ with the following property. In ${{\mathbb{C}}}[{\mathcal{P}}]$ , the map $\nabla h : \nabla C \to \nabla (PA)$ first restricts (obviously uniquely) along the subobjects to give an intermediate map $(C,E) \to (PA, S)$ ; and that map then induces the given map $f: (C,=\!) \to (SA, \Leftrightarrow\!)$ between the quotients. Essentially this is what was called in Hyland et al. (Reference Hyland, Johnstone and Pitts1980) and Pitts (Reference Pitts1981) and then in van Oosten (Reference van Oosten2008) the property that $(SA, \Leftrightarrow\!)$ is weakly complete. The terminology comes from sheaf theory and does not seem brilliant, but I do not have a good alternative suggestion. (At Rosolini’s birthday meeting I just said that we have a good presentation which is no better.)

For the proof assume that $f: (C,=\!) \to (SA, \Leftrightarrow\!)$ is represented by a functional relation $F \in {\mathcal{P}}(C \times PA)$ and consider $H \in {\mathcal{P}}( C \times A)$ defined by

\begin{equation*} H(c,a) = \exists p . (F(c,p) \land \textrm {In}(a,p)) . \end{equation*}

By the tripos condition, we obtain from $H$ a map $h : C \to PA$ with the property that

\begin{equation*} \forall a .H(c,a) \iff \textrm {In}(a, h(c)) \end{equation*}

holds in the internal logic. It is then routine to check that $h$ satisfies the property explained above. The essential point – and this was the original formulation of weak completeness – is that

\begin{equation*} \exists p \in PA. F(c,p) \vdash F(c,h(c)) \end{equation*}

holds in ${\mathcal{P}}(C)$ .

3.4 Left exact maps of triposes

The property of weak completeness may seem arcane, but its use is essential for the theory of geometric morphisms of triposes. It is used in Hyland et al. (Reference Hyland, Johnstone and Pitts1980), Pitts (Reference Pitts1981) and then in van Oosten (Reference van Oosten2008) to show that these do indeed induce geometric morphisms at the level of the constructed toposes. More generally – as already remarked in passing in Hyland et al. (Reference Hyland, Johnstone and Pitts1980) – the idea applies to left exact or cartesian maps of triposes. We shall need this so I give a brief sketch.

Suppose that we have hyperdoctrines $\mathcal{P}$ over $\mathbb{C}$ and $\mathcal{Q}$ over $\mathbb{D}$ . A map of hyperdoctrines is given by a product-preserving $F: {{\mathbb{C}}} \to {{\mathbb{D}}}$ together with a map ${\Phi } : {\mathcal{P}} \to F^*{\mathcal{Q}}$ of fibrations over ${\mathbb{C}}$ . Explicitly the latter gives for each $A \in {{\mathbb{C}}}$ a map $\Phi _A : {\mathcal{P}}(A) \to {\mathcal{Q}}(FA)$ with the $\Phi _A$ commuting with reindexing up to isomorphism. Such a map is left exact just when $\Phi$ preserves finite conjunctions and equalities.

Almost all the proof is routine with one crucial exception which is this. Suppose that we have a map $r : (A, =_A) \to (B, =_B)$ in ${{\mathbb{C}}}[{\mathcal{P}}]$ represented by a functional relation $R \in {\mathcal{P}}(A \times B)$ , we would like to take that to a map $(FA, \Phi ( =_A)) \to (FB, \Phi (=_B))$ represented by $\Phi (R) \in {\mathcal{Q}}(FA \times FB)$ . Certainly, the objects $(FA, \Phi ( =_A))$ and $(FB, \Phi (=_B))$ lie in ${{\mathbb{D}}}[{\mathcal{Q}}]$ : being a PER is preserved. And $\Phi (R)$ is certainly a single-valued relation in the internal $\mathcal{Q}$ -logic. But if existential quantification is not preserved, it need not to be total.

That looks disastrous, but there is a way round the difficulty. Replace $(A, =_A)$ and $(B, =_B)$ by their isomorphic weakly complete versions $(SA, \Leftrightarrow\!)$ and $(SB, \Leftrightarrow\!)$ . So as not to overload notation, write again $r$ for the replacement map and $R \in {\mathcal{P}}(PA \times PB)$ for a representing functional relation. Now $R$ is total, in other words, $S(p) \vdash \exists q. R(p,q)$ holds in ${\mathcal{P}}(PA)$ . But by weak completeness of $(SB, \Leftrightarrow\!)$ , we have a map $h: PA \to PB$ in $\mathbb{C}$ with $\exists q. R(p,q) \vdash R(p, h(p))$ holding. So $S(p) \vdash R(p,h(p)$ holds in ${\mathcal{P}}(PA)$ . Applying the map of triposes, we see that $\Phi S(p) \vdash \Phi R(p, Fh(p))$ holds in ${\mathcal{Q}}(FPA)$ . But a fortiori that means that $\Phi S(p) \vdash \exists q. \Phi R(p,q)$ holds which gives the seemingly problematic totality of $\Phi R$ . The upshot is that we do get a map in the topos ${{\mathbb{D}}}[{\mathcal{Q}}]$ so long as we remember to replace objects by weakly complete versions of themselves.

Aside The problem which we handle above does not arise for maps of triposes which preserve regular structure. But without that condition, it simply is not the case that all total relations will be carried to total relations. The problem was recognised early in the development of Tripos Theory. The solution is adapted from the Theory of Sheaves and can now be regarded as standard. I sketch a fresh view on the topic at the end of this paper.

4.1 The basic hyperdoctrine

Given a left exact functor $L : {{\mathbb{E}}} \to {{\mathbb{F}}}$ between toposes, we can define a hyperdoctrine $\mathcal{P}$ indexed over $\mathbb{E}$ and a map of hyperdoctrines from the standard subobject hyperdoctrine $\textrm {Sub}$ over $\mathbb{E}$ to $\mathcal{P}$ . Every $A \in {{\mathbb{E}}}$ has a power object $PA$ , and we take ${\mathcal{P}}(A) = {{\mathbb{F}}}(1, L(PA))$ the points of the image of $PA$ in $\mathbb{F}$ . We have evident maps

\begin{equation*} L : \textrm {Sub}(A) \cong {{\mathbb{E}}}(1, PA) \to {{\mathbb{F}}}(1, L(PA)) = {\mathcal{P}}(A) \end{equation*}

and the hyperdoctrine structure is obtained from the internal structure in $\mathbb{E}$ by transferring along these maps. So to give reindexing along $u : A \to B$ in $\mathbb{E}$ , we have the internal pullback $u^* : PB \to PA$ in $\mathbb{E}$ and so we have $L(u^*) : L(PB) \to L(PA)$ in $\mathbb{F}$ . Composing with $L(u^*)$ gives the reindexing

\begin{equation*} {\mathcal{P}}(u) : {\mathcal{P}}(B) = {{\mathbb{F}}}(1, L(PB)) \to {{\mathbb{F}}}(1, L(PA)) = {\mathcal{P}}(A) \end{equation*}

Since $PA$ is an internal Heyting algebra in $\mathbb{E}$ , it is immediate that ${\mathcal{P}}(A)$ is Heyting algebra. Moreover, $u^*$ preserves the Heyting algebra structure in $\mathbb{E}$ ; so $L(u^*)$ does so in $\mathbb{F}$ ; and so ${\mathcal{P}}(u)$ does so externally. Finally, $u^*$ has left and right adjoints $PA \to PB$ in $\mathbb{E}$ ; these are mapped by $L$ to left and right adjoints in $\mathbb{F}$ and then as before we simply compose to get

\begin{equation*} \exists u : {\mathcal{P}}(A) \to {\mathcal{P}}(B) {\quad \textrm {and} \quad } \forall u: {\mathcal{P}}(A) \to {\mathcal{P}}(B) . \end{equation*}

The Beck–Chevalley conditions for pullbacks are themselves equational and so transfer without fuss. This makes $\mathcal{P}$ a first-order hyperdoctrine. Finally, we have the standard membership relation on $A \times PA$ in $\mathbb{E}$ is classified by a map $A \times PA \to \Omega$ so gives a point $1 \to P(A \times PA)$ . Applying $L$ , we obtain $\textrm {In} \in {\mathcal{P}} (A \times PA)$ . Pretty clearly $\mathcal{P}$ cannot in general be a tripos in the external sense, but we might hope to show that it is one in the internal sense. The natural thing to do is to use the fact that the topos $\mathbb{E}$ satisfies a fully internal form of the Comprehension Axiom:

\begin{equation*} \forall \phi \in P(A \times C). \forall c \in C. \exists p \in PA . \forall a \in A. \phi (a,c) \iff a \in p \end{equation*}

And indeed this does extend. The same holds in the logic of $\mathcal{P}$ (obviously with $\rm In$ replacing $\in$ ). The trouble, however, that there is no way in the hyperdoctrine to instantiate the variable $\phi$ at any given $R \in {\mathcal{P}}( C \times A)$ as is required for CA. One cannot see $R$ in the base $\mathbb{E}$ . To address that issue, I replace the hyperdoctrine by one over a richer category. I find the solution attractive and do not see any other way to proceed.

4.2 The extended base

The key idea is to replace the category $\mathbb{E}$ by one with maps more attuned to the hyperdoctrine structure. I first sketch some background.

It is an easy and familiar fact that any functor $L : {{\mathbb{E}}} \to {{\mathbb{F}}}$ between categories can be factored

as a bijective-on-objects functor $J: {{\mathbb{E}}} \to \widehat {{\mathbb{E}}}$ followed by a full and faithful functor $K: \widehat {{\mathbb{E}}} \to {{\mathbb{F}}}$ . This is a basic factorisation of category theory and unsurprisingly there are many versions of it. We need the following version sketched in passing many years ago by Kelly (Reference Kelly1969).

Suppose that $\mathbb{E}$ is a symmetric monoidal closed category, that ${\mathbb{F}}$ is a symmetric monoidal category and that $L: {{\mathbb{E}}} \to {{\mathbb{F}}}$ is monoidal in the usual lax sense. Then $L$ factorises as

where $\widehat {{\mathbb{E}}}$ is symmetric monoidal, the functor $J: {{\mathbb{E}}} \to \widehat {{\mathbb{E}}}$ is not only bijective on objects but also strong monoidal closed and the functor $K: \widehat {{\mathbb{E}}} \to {{\mathbb{F}}}$ is a normal monoidal functor. Here, normal means that the functor preserves points in the sense of monoidal category theory: that is, the evident maps $\widehat {{\mathbb{E}}}(I, A) \to {{\mathbb{F}}}(I. KA)$ are isomorphisms.

The construction of $\widehat {{\mathbb{E}}}$ is straightforward. It has the same objects as $\mathbb{E}$ , and its maps or hom-sets are given in terms of the closed structure $[A,B]$ in $\mathbb{E}$ by the formula

\begin{equation*} \widehat {{\mathbb{E}}}(A,B) = {{\mathbb{F}}}(I, L([A,B])) . \end{equation*}

Even in the late 1960s, Kelly did not feel the need to give further details and they can safely be left to the reader.

As a special case of the above factorisation, one can consider the situation where the closed structure on $\mathbb{E}$ is cartesian and ${\mathbb{F}}$ is a category with products. Suppose that the functor $L: {{\mathbb{E}}} \to {{\mathbb{F}}}$ preserves products. Then both functors in the factorisation

preserve products. In particular, $J: {{\mathbb{E}}} \to \widehat {{\mathbb{E}}}$ preserves the cartesian closed structure, while now $K$ preserves points.

We are going to extend the hyperdoctrine $\mathcal{P}$ over $\mathbb{E}$ to a hyperdoctrine $\widehat {\mathcal{P}}$ over $\widehat {{\mathbb{E}}}$ with the same fibres. Before doing that, let us review what we know about our intermediate category. First, $\widehat {{\mathbb{E}}}$ is cartesian closed and the bijective on objects functor $J: {{\mathbb{E}}} \to \widehat {{\mathbb{E}}}$ preserves that structure. It is also the case that $J$ preserves the pullbacks (finite limits) of $\mathbb{E}$ . On the other hand, $\widehat {{\mathbb{E}}}$ cannot be expected to have all pullbacks. Fortunately (see in particular Pitts (Reference Pitts2002)) to give a topos of the form $\widehat {{\mathbb{E}}}[{\mathcal{P}}]$ , we only need pullbacks arising from the product structure on $\widehat {{\mathbb{E}}}$ and these we do have.

4.3 The extended hyperdoctrine

The idea now is to consider the factorisation

of the previous section and to construct a tripos $\widehat {\mathcal{P}}$ in the external sense over the intermediate category $\widehat {{\mathbb{E}}}$ with the same fibres as $\mathcal{P}$ . Obviously, there will be maps of hyperdoctrines which we will be able to exploit.

When working with the hyperdoctrine $\widehat {\mathcal{P}}$ on $\widehat {{\mathbb{E}}}$ , we make constant use of the composite

of $L$ followed by the global sections functor $\Gamma$ on $\mathbb{F}$ . I introduce the notation $L_0 : {{\mathbb{E}}} \to \textbf {Sets}$ for this left exact composite.

We have already been making use of $L_0$ . We defined $\mathcal{P}$ on $\mathbb{E}$ by setting ${\mathcal{P}}(A) = L_0(PA)$ . We defined $\widehat {E}$ by $\widehat {E}(A,B) = L_0(B^A)$ . And now we define $\widehat {\mathcal{P}}$ on $\widehat {{\mathbb{E}}}$ by again setting $ \widehat {\mathcal{P}} (A) = L_0(PA)$ . So as promised $\widehat {\mathcal{P}}(A) = {\mathcal{P}}(A)$ . I now sketch the structure of a hyperdoctrine on $\widehat {\mathcal{P}}$ over $\widehat {{\mathbb{E}}}$ .

Reindexing Given $f \in \widehat {{\mathbb{E}}}(A,B) = L_0(B^A)$ , we want $\widehat {\mathcal{P}}(f) : \widehat {\mathcal{P}}(B) \to \widehat {\mathcal{P}}(A)$ . We give it uniformly in $f$ . There is an internal reindexing $PB \times B^A \to PA$ which is the evident internal composition $ \Omega ^A \times A^B \to \Omega ^B$ in $\mathbb{E}$ . Applying the product preserving $L_0$ gives the required reindexing

\begin{equation*} \widehat {\mathcal{P}}(A) \times \widehat {{\mathbb{E}}}(B,A) \to \widehat {\mathcal{P}}(B) . \end{equation*}

This simple procedure is a model for everything we do. All the tripos structure on $\widehat {\mathcal{P}}$ will be obtained in the same way from the internal topos structure in ${\mathbb{E}}$ by applying $L_0$ .

Propositional Logic We want to show that each $\widehat {\mathcal{P}}(A)$ has the structure of a Heyting algebra and the reindexing functors preserve that structure.

The first claim is clear. $\Omega$ and so each $PA$ is an internal Heyting algebra in $\mathbb{E}$ . As $L_0$ is left exact it preserves such structure. For the reindexing, we take $f: A \to B$ in $\widehat {{\mathbb{E}}}$ and want to show that reindexing $\widehat {\mathcal{P}}(f) : \widehat {\mathcal{P}}(B) \to \widehat {\mathcal{P}}(A)$ is a map of Heyting algebras. Again that goes through uniformly. To show that

\begin{equation*} \widehat {\mathcal{P}}(B) \times \widehat {{\mathbb{E}}}(A,B) \to \widehat {\mathcal{P}}(A) \end{equation*}

is a map of Heyting algebras indexed by $\widehat {{\mathbb{E}}}(A,B)$ we show that

\begin{equation*} \widehat {\mathcal{P}}(B) \to (\widehat {{\mathbb{E}}}(A,B) \Rightarrow \widehat {\mathcal{P}}(A) ) \end{equation*}

is a map of Heyting algebra, where on the right-hand side I use a temporary arrow notation for the function space. (Of course the function space is given the pointwise Heyting algebra structure.) To see this note that internally to ${\mathbb{E}}$ , the map from $P(B) \to (B^A \Rightarrow P(A)$ is a map of Heyting algebras. Applying $L_0$ we see that

\begin{equation*} L_0(P(B) ) \to L_0 ((B^A) \Rightarrow P(A)) \end{equation*}

is also a map of Heyting algebras. But the evident map

\begin{equation*} L_0 ((B^A) \Rightarrow P(A)) \to L_0 ((B^A) )\Rightarrow L_0(P(B)) \end{equation*}

also preserves the Heyting algebra structure. Composing gives the desired result.

Entailment It is good to be clear about the preorder $\vdash$ on the hyperdoctrine $\mathcal{P}$ . The internal order on the subobject classifier $\Omega$ of ${\mathbb{E}}$ is given by the subobject $\Omega _1 \to \Omega ^2$ which is classified by the Heyting implication $\Rightarrow$ . So we have a pullback

and since exponentiation preserve pullbacks and $\Omega ^A \cong PA$ , we have for every $A \in {{\mathbb{E}}}$ a pullback diagram

where $\Rightarrow _A$ is the Heyting implication on $PA$ and $PA_1$ is the internal order relation on $PA$ .

Also in ${\mathbb{E}}$ , we have for every $A$ an infimum map $\bigwedge _A :PA \to \Omega$ which classifies the top element of $PA$ as in the pullback

Composing this with the previous pullback gives a big pullback

showing that the internal order $PA_1$ on $PA$ is classified by the top composite. That demonstrates that for $R, S \in {\mathcal{P}}(A) = \widehat {{\mathbb{E}}}(A, \Omega )$ we have

\begin{equation*} R \vdash S \quad \textrm {if and only if} \quad \bigwedge _A (R \Rightarrow S) = \top \end{equation*}

in the internal logic of $\mathbb{E}$ .

Adjoints Reindexing $\widehat {\mathcal{P}}(B) \times \widehat {{\mathbb{E}}}(A,B) \to \widehat {\mathcal{P}}(A)$ was obtained by applying the functor $L_0$ to the internal reindexing map $PA \times A^B \to PB$ of ${\mathbb{E}}$ . We give the left and right adjoints in the same way. Within ${\mathbb{E}}$ we have existential quantification and universal quantification maps

\begin{equation*} \exists : B^A \times PA \to PB {\quad \textrm {and} \quad } \forall : B^A \times PA \to PB , \end{equation*}

and applying $L_0$ gives maps

\begin{equation*} \exists : \widehat {{\mathbb{E}}}(A,B) \times {\mathcal{P}}A \to {\mathcal{P}} B {\quad \textrm {and} \quad } \forall : \widehat {{\mathbb{E}}}(A,B) \times {\mathcal{P}}A \to {\mathcal{P}}B . \end{equation*}

So for $f \in \widehat {{\mathbb{E}}}(A,B)$ , we have $\exists f : {\mathcal{P}}(A) \to {\mathcal{P}}(B)$ and $\forall f : {\mathcal{P}}A \to {\mathcal{P}}B$ . That these give adjoints is again uniform in $f$ . Those are expressed in $\mathbb{E}$ by commuting diagrams. For $\exists$ it is of shape

and similarly for $\forall$ . (In the diagram, $\textrm {ev}$ is the evaluation map giving reindexing.) Applying $L_0$ gives a commuting diagram of shape

Input $R \in \widehat {\mathcal{P}}A$ , $f \in \widehat {{\mathbb{E}}}(A,B)$ and $S \in \widehat {\mathcal{P}}B$ on the left-hand side. The top composite takes them to $\top$ just when $R \vdash \widehat {\mathcal{P}}f S$ ; and the bottom composite takes them to $\top$ just when $\exists _f R \vdash S$ . So the two conditions are equivalent. Of course mutatis mutandis the same argument goes through for universal quantification. We have both left and right adjoints to reindexing.

Aside Weirdly these general adjoints are not of much interest or use to us. That is because good adjoints should satisfy some kind of Beck–Chevalley condition. We cannot have this in general because we have no handle on pullbacks in $\widehat {{\mathbb{E}}}$ . Fortunately for a tripos, we only need quantification along product projections and Beck–Chevalley for pullbacks of them.

Beck–Chevalley The Beck–Chevalley condition refers to a commutation of substitution and quantification. For a tripos, we need this only for quantification along product projections, that is, for quantification in the usual restricted sense of standard logic. For $g \in \widehat {{\mathbb{E}}}(C,D)$ , we have an evident $g \times A \in \widehat {{\mathbb{E}}}(C \times A, D \times A)$ and a pullback diagram of the form

For existential quantification, there are two induced maps $\exists _{\pi } . \widehat {\mathcal{P}}(g \times A)$ and $\widehat {\mathcal{P}}g . \exists _{\pi }$ from $\widehat {\mathcal{P}}(D \times A)$ to $\widehat {\mathcal{P}}(C)$ . Adjunction induces a natural comparison from the first to the second and the Beck–Chevalley condition states that this is an isomorphism. More concretely for $R \in \widehat {\mathcal{P}}(D \times A)$ , we have an isomorphism

Again we find an internal version of this condition in ${\mathbb{E}}$ . The diagram

commutes. Applying $L_0$ gives a commuting diagram of the same shape and as was the case for the adjoints we can read off from it the required Beck–Chevalley condition.

Equality I put off saying anything about equality because we can now deal with it quickly. We want $\delta _A \in \widehat {\mathcal{P}}A$ satisfying the axioms for equality. We get exactly what we had for $\mathcal{P}$ . We simply apply $L_0$ to the map $1 \to P(A \times A)$ picking out the internal equality on $A \in {{\mathbb{E}}}$ . With the structure, we now have it is sufficient to show that for $R \in \widehat {\mathcal{P}}(A \times A)$ we have

\begin{equation*} {\delta }_A \vdash R \quad {\textrm {if and only if} }\quad \top _A \vdash \widehat {\mathcal{P}}(\Delta _A)(R) . \end{equation*}

That is easy as we have essentially set ${\delta }_A = \exists _{\Delta _A} (\top _A)$ .

Generic predicates At this point, we know that we have a first-order hyperdoctrine $\widehat {\mathcal{P}}$ over our category $\widehat {{\mathbb{E}}}$ and it remains to show that it has generic predicates. In a sense that is the easiest property. $\widehat {{\mathbb{E}}}$ is exactly constructed to have maps making $\widehat {\mathcal{P}}$ a tripos in the external sense. We just transfer structure from $\mathbb{E}$ .

In $\mathbb{E}$ , we have the membership predicate as a subobject of each $A \times PA$ . It is classified by an evaluation map $A \times PA \to \Omega$ and that corresponds by transpose to a point $P(A \times PA)$ . Applying $L$ gives a point of $LP(A \times PA)$ and that gives the membership relation $\textrm {In} \in {\mathcal{P}}(A \times PA)$ .

Suppose that we are given $R \in \widehat {\mathcal{P}}(A \times C)$ . That is a point of $LP(A \times C) \cong L(PA^C)$ and that gives a map $r : C \to PA$ in $\widehat {{\mathbb{E}}}$ . That has to do the trick.

Why? Well in $\mathbb{E}$ we have $P(A \times C) \cong PA^C$ and there is an evident map $PA^C \to (A \times PA)^{(A \times C)}$ which is taking the product with the identity on $A$ . So we have $P(A \times C) \to (A \times PA)^{(A \times C)}$ . Take the product of that with the point of $P(A \times PA)$ above to give

\begin{equation*} P(A \times C) \to (A \times PA)^{(A \times C)} \times P(A \times PA) \end{equation*}

Composing with the internal compositon (which we use to define reindexing)

\begin{equation*} (A \times PA)^{(A \times C)} \times P(A \times PA) \to P(A \times C) \end{equation*}

returns the identity on $P(A \times C)$ . Applying $L_0$ to that unravels to show that for each $R \in \widehat {\mathcal{P}}(A \times C)$ with corresponding $r : C \to PA$ in $\widehat {{\mathbb{E}}}$ as above, the reindexing $\widehat {\mathcal{P}}(A \times r)(\textrm {In})$ is $R$ as required.

In the discussion above, we have sketched the proof of our main technical theorem.

It is evident from the definitions that we can identify the categories $\widehat {{\mathbb{E}}}[\widehat {\mathcal{P}}]$ and ${{\mathbb{E}}}[{\mathcal{P}}]$ . In line with the notation of Kock and Mikkelsen (Reference Kock, Mikkelsen, Hurd and Loeb1974) I write ${{\mathbb{E}}}_*$ for either version. Now $\widehat {\mathcal{P}}$ is a tripos in the external sense so we can deduce the following.

By the Pitts Characterisation Theorem, we can now deduce that $\mathcal{P}$ is a tripos over $\mathbb{E}$ in the internal sense.

4.4 The logical functor

We have a map of triposes from $\rm Sub$ over $\mathbb{E}$ to $\mathcal{P}$ over $\mathbb{E}$ . It is the identity on $\mathbb{E}$ . On fibres it is the following map $\textrm {Sub}(A) \to {\mathcal{P}}(A) = L_0(PA)$ . A subobject $R$ of $A$ is classified by a map $A \to \Omega$ and taking the transpose we get a point $r: 1 \to PA$ ; then $R$ is taken to $L(r) \in L_0(PA)$ . It is clear from the definitions that this is what one could call a logical map of triposes. In fact, the structure here is precisely preserved. So we can deduce at once the following.

4.5 The left exact functor

We have a map of hyperdoctrines from $\widehat {\mathcal{P}}$ over $\widehat {{\mathbb{E}}}$ to the subobject hyperdoctrine $\textrm {Sub}$ over $\mathbb{F}$ . The details need a little care.

At the category level, we take an object $A$ in $\widehat {{\mathbb{E}}}$ , equivalently in $\mathbb{E}$ to the object $L(A)$ of $\mathbb{F}$ . But we also need to give the action on maps. Suppose we have $f \in \widehat {{\mathbb{E}}}(A, B) = L_0(B^A)$ . Since $L$ preserves products, there is an evident standard comparison $L(B^A) \to (LB)^{LA}$ and composing with it gives a point $1 \to (LB)^{(LA)}$ . The exponential transpose is then the required map $LA \to LB$ in $\mathbb{F}$ .

At the level of the fibres, we have $\widehat {\mathcal{P}}A = L_0(PA)$ . Again as $L$ preserves products we have the comparison $L(PA) = L(\Omega ^A) \to (L\Omega )^{LA}$ . And as the functor $L$ is left exact, we get a classifier $L\Omega \to \Omega$ of the subobject $L\top \to L\Omega$ . So we have $L(PA) \to \Omega ^{LA} = P(LA)$ . Given $R : 1 \to L(PA)$ in $\widehat {\mathcal{P}}A$ we compose to get a point $1 \to P(LA)$ and hence a subobject of $LA$ as required.

While the structure of this map of hyperdoctrines is natural enough, it is not quite straightforward to see that it induces even a functor between the generated toposes as generally the map will preserve left exact structure but not regular structure. Fortunately, we dealt with that problem in a general setting in Proposition 6. We can apply that here and deduce the following.

4.6 The factorisation

Given that the definition of the functor $\Lambda$ involves replacing objects by an isomorphic weakly complete one, it is a bit fiddly to check properties of it. All the same it is fairly clear that it does preserve elements or points, the condition identified by Kock and Mikkelsen (Reference Kock, Mikkelsen, Hurd and Loeb1974). Moreover with a little more attention to detail one sees that the composite

is isomorphic to the original $L : {{\mathbb{E}}} \to {{\mathbb{F}}}$ .

4.7 The regular completion

We have paid a good deal of attention to the power objects $\widetilde {P} (\nabla A)$ of constant objects $\nabla A$ of our toposes. For a general tripos $\mathcal{P}$ over $\mathbb{C}$ , the power object is given by $(PA , \Leftrightarrow\!)$ where $p \Leftrightarrow q$ is defined in the internal logic by the formula $\forall a. \textrm {In}(a,p) \Leftrightarrow \textrm {In}(a,q)$ . Evidently $\Leftrightarrow$ is an equivalence relation and the power object is the quotient of $\nabla PA$ which it determines.

That is the typical situation but when we consider the triposes $\mathcal{P}$ over $\mathbb{E}$ or $\widehat {\mathcal{P}}$ over $\widehat {{\mathbb{E}}}$ arising from a left exact $L : {{\mathbb{E}}} \to {{\mathbb{F}}}$ something unusual happens. For in $\mathbb{E}$ the relation $\Leftrightarrow$ on $PA$ is just equality. It follows that the quotient $\nabla (PA) \to \widetilde {P}(\nabla A)$ is an isomorphism. (In one reading of the situation it is literally the identity.) But any object of our toposes is (isomorphic to) a subobject of the power object $\widetilde {P}(\nabla ) A)$ . Hence in the case of the (equivalent) toposes ${{\mathbb{E}}}[{\mathcal{P}}]$ and $\widehat {{\mathbb{E}}}[\widehat {\mathcal{P}}]$ every object is isomorphic to a subobject of a constant object. It follows that the standard embeddings ${{\mathbb{E}}}({\mathcal{P}}) \to {{\mathbb{E}}}[{\mathcal{P}}]$ and $\widehat {{\mathbb{E}}}(\widehat {\mathcal{P}}) \to \widehat {{\mathbb{E}}}[\widehat {\mathcal{P}}]$ of the regular completion in the exact completion are in fact equivalences. That explains the fact that no quotients are to be found in the old treatments of Kock and Mikkelsen (Reference Kock, Mikkelsen, Hurd and Loeb1974) and Volger (Reference Volger, Lawvere, Maurer and Wraith1975). In a sense that reflects very well on those involved. The natural perspective is surely via the Tripos Theory of Pitts and avoiding the quotients along the way made the issues they faced just a bit harder.

4.8 First-order functors

In Kock and Mikkelsen (Reference Kock, Mikkelsen, Hurd and Loeb1974), the assumption that their functor $L : {{\mathbb{E}}} \to {{\mathbb{F}}}$ is first-order logic preserving. At one point they appear to say that this means that the subobject classifier is preserved. While that is true in their leading example, it is stronger than is needed and the paper suggests that they intended simply that it preserves first-order logic in what is now the standard sense mentioned in section 1.2. In other words, they assume that $L$ is first-order. Now that implies that the various comparisons

\begin{equation*} L\Omega \to \Omega , \quad L(B^A) \to (LB)^{(LA)}, \quad L(PA) \to P(LA) , \end{equation*}

which we have seen above are all monics. (It seems that considerations of this kind were first studied by Mikkelsen (Reference Mikkelsen2022) in his thesis. Readers are likely to be familiar with the condition for inverse images of geometric morphisms. A geometric morphism of toposes is open (see Johnstone (Reference Johnstone1980) for example) just when the inverse image functor is first-order.)

Suppose that $L:{{\mathbb{E}}} \to {{\mathbb{F}}}$ is first order and return to the description of the map of triposes from $\widehat {\mathcal{P}}$ over $\widehat {{\mathbb{E}}}$ to $\textrm {Sub}$ over $\mathbb{F}$ . Since $L$ is first-order the comparison $L(B^A) \to (LB)^{(LA)}$ is monic and that implies that the map of triposes is faithful at the level of categories. But as $L(PA) \to P(LA)$ is monic, it is also faithful at the level of fibres. Since in particular $L$ preserves existential quantification, we have no need to consider replacement by weakly complete objects so we immediately deduce the following.

This proposition explains why Kock and Mikkelsen were able to carve out ${{\mathbb{E}}}_*$ as a subcategory of the ambient category $\mathbb{F}$ .

5.1 Artin Gluing

Clearly, the Kock–Mikkelsen factorisation of a left exact functor between toposes into a logical one followed by a left exact one preserving points belongs in the part of Topos Theory concerned with logical functors. What should the place of the factorisation be in the Topos Theory story? The referee suggested saying more about one aspect.

Artin glueing takes a finite-limit-preserving functor $L:{{\mathbb{E}}}\to {{\mathbb{F}}}$ between toposes and gives the glued topos ${\mathbb{G}}$ with an open inclusion $u: {{\mathbb{E}}} \to {{\mathbb{G}}}$ and corresponding closed inclusion $v: {{\mathbb{F}}} \to {{\mathbb{G}}}$ . The functor $L$ is then the composite of the direct image $u_* : {{\mathbb{E}}} \to {{\mathbb{G}}}$ and the inverse image $v^* {{\mathbb{G}}} \to {{\mathbb{F}}}$ . There is a detailed discussion in Johnstone (Reference Johnstone2002).

It is a perhaps confusing feature of Artin gluing that the inverse image $u^* : {{\mathbb{G}}} \to {{\mathbb{E}}}$ is logical. By contrast, this paper takes a finite-limit-preserving functor $L: {{\mathbb{E}}} \to {{\mathbb{F}}}$ and provides a logical functor ${{\mathbb{E}}} \to {{\mathbb{E}}}_*$ . It feels as if there should be something to say here, but I have not thought it through. Quite specifically, the referee notes that Artin gluing can be thought of as a tabulation and asks whether there is a notion of cotabulation for toposes. That is a good question. I have no idea about it, but that certainly seems worth pursuing.

5.2 Methodology

In a different direction, one might wonder about the methodology of this paper, specifically the use of the tripos $\widehat {\mathcal{P}}$ on the intermediate category $\widehat {{\mathbb{E}}}$ . Is all that really necessary? As to that I would remark first that both Kock and Mikkelsen (Reference Kock, Mikkelsen, Hurd and Loeb1974) and Volger (Reference Volger, Lawvere, Maurer and Wraith1975) are led in their treatments to consider a category corresponding to $\widehat {{\mathbb{E}}}$ . Indeed Kock and Mikkelsen explicitly draw attention to it in their leading example as a category of standard objects and non-standard maps. More than that however I feel – and regret not having conveyed better – that the theory of the tripos $\widehat {\mathcal{P}}$ over $\widehat {{\mathbb{E}}}$ is incredibly smooth. It exploits systematically the way in which the structure of an elementary topos is internalised within the topos itself and carries that over to the new tripos using the functor $L_0$ as explained above. That leads me to think that the construction may be a fundamental one.

5.3 Weak completeness

In the first version of this paper, I observed that the need for the seemingly technical notion of weak completeness in the development of the theory told against the ideas of this paper. I hoped to clarify that to make it all seem completely natural.

Since writing that I have had the great pleasure, at her invitation, to spend a week in June 2025 with Milly Maietti in Padova. For my visit, she arranged what was an ideal mini-conference: it had just a few talks and ample time for discussion and reflection. In the course of it, I learnt much of current research in the general area of doctrines. The meeting began with a wonderful survey of the field by Rosolini and prompted by that I came to see that the story of weak completeness has a natural home within the theory of doctrines. Here, I give just a glimpse of what is involved.

First from the Maietti–Rosolini perspective, it is natural to consider for suitable doctrines $\mathcal{P}$ over $\mathbb{C}$ not only the categories ${{\mathbb{C}}}({\mathcal{P}})$ of section 2.2 and ${{\mathbb{C}}}[{\mathcal{P}}]$ of section 2.3 but also versions of these where one restricts to maps which are tracked by maps in ${\mathbb{C}}$ . Put otherwise, we have in each case the same objects, but maps are equivalence classes of maps in ${\mathbb{C}}$ satisfying evident properties in the internal logic. Just for now I shall refer to these as the tracked maps in the larger categories. The simple but crucial point exploited in section 3.3 is that a left exact map of doctrines automatically acts on the (categories of) tracked maps.

The notion of weak completeness makes sense in this context. An object $B$ is weakly complete just when for all objects $A$ all maps $A \to B$ are tracked. So evidently left exact maps of doctrines act on the subcategory of weakly complete objects.

The story is completed by showing that for a tripos $\mathcal{P}$ over $\mathbb{C}$ in the external sense all objects in ${{\mathbb{C}}}[{\mathcal{P}}]$ are isomorphic to weakly complete ones. I discussed that in section 3.3, but perhaps made too much fuss. The genericity condition pretty much says that power objects of constant objects are weakly complete. That easily extends to all power objects. But in a topos, every object is isomorphic to the subobject of singletons in the power object and the usual presentation makes these weakly complete.

Of course the proper context for the story just sketched would be an extensive development of the theory of doctrines. As I learnt during my recent visit to Padova that is a very lively area. When a definitive account is written, one might hope that better understanding will suggest a better term than ‘weakly complete’.

5.4 Thanks

Given what I have just said I obviously owe an intellectual debt as regards this paper to Pino Rosolini for whom it is written. Thanks are also due to Milly Maietti for her determined promotion of research in the area of doctrines and more generally for her insistence on precision in research on categorical logic. Alongside that I would like to thank all who came to the meeting in June. I learnt a great deal from the presentations and from the discussions. The area of doctrines has proved to be (perhaps surprisingly) subtle and there remains much still worth pursuing.

Finally, any reader of this paper will recognise that it has been substantially enriched by suggestions made by a sympathetic, thoughtful and well-informed referee. It is probably as well that such contributions to research remain anonymous. However, I would like the referee and the editor who chose that referee to know that I appreciate their efforts. As ever, it remains the case that the inevitable errors and infelicities in the paper are my responsibility.