Lex total categories and Grothendieck toposes

This is the first of a series of connected posts. The next post is here.

In the very valuable volume Sketches of an Elephant, A Topos Theory Compendium, Volume 1, Peter Johnstone mentions a lecture of Andre Joyal at the 1981 Cambridge Summer Meeting where he listed seven different descriptions of 'what a topos is like'. His fifth one was

(v) 'A topos is a totally cocomplete object in the meta-2-category of  finitely complete categories'.



Peter also writes that in the 20 years that have passed since that lecture, the category-theory community (and computer scientists) have added a few more descriptions to the list, and he mentions another six including

 (ix ) 'A topos is the category of maps of a power allegory'

Recently a young topos theorist asked me about (v), which was a view of toposes first presented in [2]. I think that the description by Peter Johnstone makes mysterious what I believe is the simplest conceptual description of the notion of topos. (I haven't found any expanded explanation in the Elephant of (v).)

The idea begins as follows:
 The open sets of a topological space form a partially ordered set with arbitrary sups and finite intersections, with the property that intersection distributes over arbitrary sups: what is called abstractly a locale. We can state this in a slightly more sophisticated way.  

Instead of considering the sup of a subset we may look at sups of downsets (subsets V such that if x < = y and y is in V then also x is in V).
It is clear that every subset U generates a downset ((U)) which has a sup iff U does, and supU=sup((U)).
Let PX be the poset of downsets of X. PX may also be seen as the set of ordered functions X^opp -> 2 (2={0 < = 1}); just look at the inverse image of 1 to find the corresponding downset.
Then there is an order preserving function yoneda: X -> PX which takes x to ((x)), the principal downset generated by x.

FACT: yoneda:X->PX has a left adjoint iff X has arbitrary sups, and the left adoint takes U to supU.
proof. The adjunction condition is that supU < = x iff U < = ((x)), which clearly characterizes sup.

FACT: if X has finite infs and arbitrary sups then sup : PX -> X preserves finite infs iff finite infs distribute over arbitrary sups in X; ie iff X is a locale.
proof. It is easy to see that U inf V = ((u inf v; u in U, v in V)). The preservation of finite infs means that
sup{u inf v; u in U, v in V} = sup((u inf v ; u in U, v in V)) = sup(U inf V) = supU inf supV.
A special case of this which however implies the general case is that (reading backwards)
u inf supV = sup(u inf v; v in V), the distribution of finite limits over arbitary sups.

NEW DEFINITION: So we have a new definition of poset X being a LOCALE, namely that it is finitely complete and that  yoneda has a left exact left adjoint. This definition has a straightforward extension to categories.
A category Y is LEX TOTAL  [1] iff it is finitely complete and yoneda : X -> PX has a left exact left adjoint colimit : PX -> X where PX is the category of presheaves on X, or alternatively the category of discrete fibrations with small fibre over X.  We will see that it is in the second sense that the left adjoint may be called colimit.

Let me make some connection with the ideas I described above about locales.

FACT: A category X has the property that yoneda : X -> PX has a left adjoint iff every discrete fibration with small fibres over X has a colimit. Further the adjoint to yoneda is colimit (regarding the object of  PX as discrete fibrations).
proof. Yoneda takes object x in X to the fibration ((x))->X where ((x)) has objects arrows (y->x)
projecting to x, and arrows (z->y) : (z->y->x)->(y->x) projecting to z->y.
Hence the adjunction condition is that there is a natural bijection between arrows colimitF -> x and
functors over X,  F -> ((x)). But it is not difficult to check that a functor (over X) from F to ((x)) is the same as a cocone from F to x, and hence the result.

What about the fact that when we considered posets we said that in considering sups it was sufficient
to consider sups of downsets? The corrsponding fact here for categories [2] is that any functor Y->X where Y is small has a factorization through a discrete fibration over X with small fibres with the same colimit (if the colimit exists). However there is something new.

SOMETHING NEW Not all discrete fibrations with small fibres come from functors with small domain.
So we have a new notion of cocompleteness - TOTAL COCOPLETENESS which means cocomplete for all discrete fibrations with small fibre. This is a much stronger condition (and in our opinion a sadly neglected one) than small completeness, and has a much stronger adjoint functor theorem.
It holds for most commonly used small complete categories, the first being sheaves on a small category where the left adjoint to yoneda of PX is P of yoneda of X.

I think I will continue in another post. Suffice to say here that lex total categories, the direct generalization of locales, have all the completeness and exactness properties of Grothendieck toposes (lacking only possible a small set of generators). They also have, via the adjoint functor theorem, the constructions of elementary toposes, subject again to some smallness conditions.

We believe that the notion of lex total category is the simplest conceptual description of Grothendieck
toposes, and deserves wider attention and use.

NOTE: I should point out that there has been appreciation and development of these ideas by Wood and Rosebrugh, Tholen, Kelly, Street and Freyd, and others, beyond their introduction in the papers below by Street and Walters. I will give some references in the next post.

I almost forgot to make a remark about Peter's aspect (ix) of toposes. I think it is a pity that he did not use the characterization of elementary toposes in terms of bicategories of relations in [3].

[1] RH  Street, RFC Walters, The comprehensive factorization of a functor, Bull. Amer. Math. Soc. 79, 1973
[2] RH Street,  RFC Walters, Yoneda structures on 2-categories, J. Algebra, 50, 1978.
[3] A. Carboni, RFC Walters, Cartesian bicategories I, JPAA, 1987.