I was writing a very belated contribution to the earlier discussion about regular categories when I saw Peter Freyd's question about categories all of whose slices are topoi. Since the former bears on the latter here it is without further ado. The definition of Carboni and Street given in "Order ideals" is particularly nice because "E is regular" in their sense is exactly what one needs to define Rel(E) with no circumlocution. Recall that they require pullbacks, define strong epis in terms of (jointly) monic spans and require that every span factor, pullback stably, as a strong epi followed by a monic span. Then E is Barr regular iff E is C&S regular and has a terminal object. In joint (as yet unpublished) work with Carboni and Kelly we observed that the entire C&S definition can be encoded in a single adjunction. Let S be the category x<---r--->a, the "free-living" span. Let {S,E} denote the functor category, best thought of here as a power (=cotensor), and let (S,E) denote the full subcategory of it determined by the monic spans. Regard both categories and the inclusion, in:(S,E)--->{S,E} as data over ExE via the codomain(s) functors, c. Then E is a regular category iff "in" has a left adjoint "im" in the 2-category of fibrations over ExE. The point is that c:{S,E}--->ExE is a fibration iff E has pullbacks, whence c:(S,E)--->ExE is also a fibration; existence of im , merely as a functor over ExE, provides factorizations; and im being an arrow of fibrations is equivalent to stability under pullback. It is then fairly clear that one can define regular objects, E, in suitable 2-categories other than CAT, provided that one has also an interpretation for (S,E). This is not necessarily an idle generalization because such an approach dictates the reasonable definitions for both arrows and proarrows in the 2-categories of regular objects that result when regularity is studied in the presence of further properties or structure. (For a similar sort of situation consider studying (co)complete objects in the 2-category of finitely complete ordered sets where the result is locales, or the deeper idea of Kelly and Lawvere of studying finitely complete objects in the 2-category of Skolem categories.) But returning to regular categories, consider now Rel(E)(x,a) (where x and a are objects of E). It is reasonable to ask for representability in a , by an object P_a (but better S_a for see below). This was proposed by Street in the 1972 "Christmas card". C&S regular categories with this property, let us call one such an "opos" are rather good. In fact, E is a topos iff E is an opos and has a terminal object. (I believe that such categories have been considered by Benabou in work on trees and by Taylor in connection with laminations.) Every slice of an opos is a topos and if E is an opos then all Spn(E)(x,a) are again oposes. The present definition indicates that in addition to topoi, groupoids (and hence sets) are oposes. Another interesting example suggested by Pare is the category whose objects are topological spaces and whose arrows are local homeomorphisms. Still another is the category whose objects are sets and whose arrows are functions with finite fibres. The last example indicates that Cantor's theorem can fail in a non-degenerate opos, for there P_a is the set of finite subsets of a. It is probably better to write S_a for P_a and think along the lines of Joyal and Moerdijk as in their "Cumulative hierarchy of sets" paper. Although the diagonal argument has many forms, as Lawvere showed, one sees here that it is in a certain sense inevitable. On the geometric side one gains intuition about oposes by observing that a disjoint union of oposes is again an opos. The geometric intuition is further sharpened by recalling that functors between topoi that have pullback preserving left adjoints can be thought of as both partially defined geometric morphisms and families of geometric morphisms. These have been studied by Kennison, Pare, Thiebaud, Rosebrugh and the writer. The chief virtue of oposes seems to be that 2-categories of such are easier to work in than TOP, the 2-category of topoi and geometric morphisms.The 2-categories of oposes that I refer to are oposes together with fuctors that have pullback and monic span preserving left adjoints and natural transformations; oposes together with pullback and monic span preserving functors and natural transformations in the opposite direction. (The reason for dualizing at the level of 2-cells is so that the inclusion of the former in the latter is proarrow equipment as discussed in various papers by Rosebrugh and the writer. It enables a direct analogy with the pardigm CAT--->PRO, where the 1-arrows of PRO are profunctors.) It appears that various inverse limit constructions in TOP and related 2-categories have their complications concentrated in the passage from oposes to topoi but I have not yet thought this through. By the way, part of the reason that the "pro" version of a 2-category of oposes is so simple just resides in the trivial observation that naming an object, i, of an opos E by the functor i:1--->E is pullback preserving. While I am not entirely happy with the name "opos" and do not wish to tamper with the definition of topos there are several other indications that the role of the terminal object needs to be carefully understood. One concerns the real closed unit interval [0,1] regarded as a CCD lattice. The defining left adjoint to the supremum function for down-closed subsets preserves binary infima but not the top element. This suggests that [0,1), equivalently the non-negative reals, be studied in the spirit of the CCD programme but with arbitrary suprema replaced by suprema of bounded (down-closed) subsets. Another concerns the fact that the functor U:set--->CAT(set^op,set) is pullback preserving, where U-|V-|W-|X-|Yoneda . Rosebrugh and I have recently shown that if a locally small category B has such an adjoint string as above then B is equivalent to the category of sets. Regards, R.J. Wood ==============================================================================