I've had trouble with starting a reply to Vaughan's most recent posting, the one headed, "Abelian-topos (AT) categories". It makes the wide gap that's come into being all too painful. Vaughan wrote, I don't see much discussion of closed structure for abelian categories. Why is this? Am I just reading the wrong stuff, or is the closed structure of abelian categories boring, or what? And that was after he had written, I only found out what abelian categories were a week ago. For a long time Category Theory existed (say, in the Mathematical Reviews) as a subset of Homological Algebra -- which is a way of saying that category theory was abelian category theory. (I can't remember a new result in the theory of abelian categories in the last quarter-century. I do remember, alas, a bunch of new announcements of such results.) I have trouble with Vaughan's phrase "the closed structure" on an abelian category. There can be many. The category of finite- dimensional complex representations of a compact group has a distinguished symmetric monoidal closed structure. If they are viewed just as categories then for any two compact groups with the same number of conjugacy classes the categories are isomorphic. If viewed just as monoidal closed categories then the necessary and sufficient condition that they be isomorphic is that the groups have isomorphic character tables. On the other hand, if viewed as a _symmetric_ monoidal closed categories, one can recover the group from the category. If you want a specific example consider the two non-abelian groups of order eight, the dihedral and the quaternian. In each case the plain category of representations is the 5-fold cartesian power of the category of finite-dimensional complex vector spaces. As monoidal closed categories they are isomorphic (but not isomorphic with the 5-fold cartesian power of the closed monoidal category of finite- dimensional complex vector spaces). As symmetric monoidal closed categories they are different. Anyway, there's a whole body of material. A lot of it is now viewed as standard in a number of (non-categorical) subjects and as for all successfull branches of category theory the theory of abelian categories is no longer considered to be a branch of category theory. Back to pratt cats: if one wants to axiomatize those categories that are products of abelian cats and topoi and not worry about the intervening families the axioms can be made quite simple. After saying that it's a regular category with a coterminator contained in its terminator, I'd start with the P-E-l-r-/\ structure as in my last post, define TX as the image of rX and prove it to be the correflection of X into the full subcategory of type-T objects. 0xX -> X is easily seen to be the correflection of X into the full subcategory of type-A objects. Then the axiom that these two correflections yield a coproduct decomposition for each object allows one to prove quickly that the category is the cartesian product of the two correfletive subcategories. The type-T objects clearly form a topos. All that's needed now is a couple of axioms to make the type-A objects abelian.