Peter's response to my question about categories having all the common properties of toposes and abelian categories was a very pleasant surprise. First I was not at all expecting such a definite answer. Second I did not expect a representation theorem, especially not such a nice one as "every such category is a product of an abelian category and a topos". This is the converse of the automatic (by Horn) fact that such a product is such a category, where (for this converse at least) the language can be taken to be as large as the full Horn theory common to abelian categories and toposes. Third I did not expect at all that the class would be finitely axiomatizable (but on the other hand neither did I have any reason to suppose not). Fourth and very nice, unless I'm overlooking something, Peter's representation theorem implies that my original question (does the common *elementary* theory allow anything other than abelian categories and toposes?) can now be answered in the negative. The elementary sentence "either all the objects are A-type or all the objects are T-type" is obviously true in both abelian categories and (pre)toposes, and rules out hybrids like Set x Ab. I'd been wondering what elementary sentence might fail for that example, and Peter's A-type and T-type constructs lead straight to it. Minor remarks: The trick of multiplying by 0 to tease out both the A- and T-components at once is beautiful, not just intrinsically but also because it reveals a certain duality between A and T that is not at all apparent (to me anyway) from their separate definitions. The actual multiplication projects out the T part without disturbing the A part, and then the pushout AX (where AX = 0xX) / \ 0 X \ / TX neatly quotients out that surviving A part by forcing the left side through the zero object while on the right AX->X communicates only the A part which is therefore the only part that the pushout quotients out from 0 + X (if that's not too clumsy a way of putting it). The upshot is then that the AX's form (as an abelian category) a coreflective subcategory of the whole while the TX's form (as a pretopos) a reflective subcategory. That's really beautiful! The following is presumably the key to proving completeness of Peter's axiomatization...
VAE) For every X there's a map TX -> X such that
AX TX \ / X
...by permitting any model to be constructed directly as a product of an abelian category and a topos (not even "subcategory" seems to be needed here, so this is an even stronger representation theorem than the representation of a Boolean algebra as a subalgebra of a power set).
And note that the type-A objects can not be reflective: if 1 has a map to any type-A object the entire category collapses.
Whoops, a bug here if I'm understanding things. Only the T part collapses, if the category were abelian to begin with then there'd be no collapse.
Define TX -> X as the image of rX.
This was lovely (getting rid of the E in VAE above). Some questions. 1. Can this representation theorem be extended to the full Horn theory, i.e. any number of alternations of quantifiers? (Note that my disjunction---every object is A-type or every object is T-type---is not a Horn sentence.) (Now that I look again, I guess the answer must be yes, since products preserve *all* Horn sentences.) 2. In a text that treated both toposes and abelian categories, would your axiomatization be a good starting point prior to doing either? You could then quickly define an abelian category to be a category of this general type with the additional property of being strongly connected. I don't see right off how to define a pretopos as slickly. 3. What about bringing in the closed structure? Might that permit some simplification of your axiomatization? For example it looks as though you could define a T-type object to be one having a unique map to the tensor unit. Then you could answer the last sentence of the previous paragraph simply with "the tensor unit is terminal." Vaughan Pratt