In the paper *Introduction to Extensive and Distributive Categories* by Carboni, Lack, and Walters, the authors write at the end of the introduction:



> 
> The Burnside rig of a distributive category is well known [6]. It has as elements isomorphism classes of objects of the category, and its addition and multiplication are given by sums and products in the category. Just as a distributive category can be thought of as a category with a rig-like structure, so should an extensive category be thought of as a category with an Abelian-group-like structure. As many results as possible will be proved using only this additive structure. In a later paper, the 2-category of extensive categories will be considered as analogous to the category $\mathsf{Ab}$ of Abelian groups, and, in particular, the tensor product defined.
> 
> 
> ... The isolation of extensivity, and the realization that it is
> an essentially additive notion was made over a period of time and is due to the
> authors, Lawvere and Schanuel. Lawvere has independently reported this discovery in [4].
> 
> 
> 


[4] F.W. Lawvere, Some thoughts on the future of category theory.


[6] S.H. Schanuel, Negative sets have Euler characteristic and dimension.




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I would like to understand in what sense "an extensive category should be thought of as a category with an Abelian-group-like structure". The word 'additive' does not seem to appear in [4] and I don't see where Lawvere reports the discovery of extensivity being an essentially additive notion (unless he means the characterization/definition via the coproduct functor from a product of slices being an equivalence). I am looking for an explanation and a reference.


I would also very much like to read this future paper, but I cannot find it (at least in the work of Lack; I don't know where to find the publications of Carboni).


So in what sense is extensivity an essentially additive property? Is it just a way to describe the characterization via the coproduct functor from a product of slices being an equivalence? What is the tensor product of extensive categories? What is the relation/analogy with abelian groups? Where can I read more about this?



References on extensivity as an essentially additive notion
This is not a complete answer, and in particular I don't know anything about a tensor product of extensive categories. But my understanding of the phrase in question is that it means that extensivity is a property of a category with coproducts only. One doesn't need to assume the category has *any* limits in order to state the property that the coproducts (which must be assumed to exist) are extensive, and there are extensive categories that don't have (for instance) finite products or even a terminal object. Since in a distributive category it is the coproducts that correspond to the addition in a ring, in general coproducts can be called the "additive" aspect of a category; I think that is what they mean by extensivity being an "additive property". 


The reason they emphasized this is that it *happens*, as a *consequence* of extensivity, that *certain* limits exist (certain squares involving coproduct injections are pullbacks), and so in the past (they say) people tended to define extensivity-like notions only in categories that *were* assumed to have (say) *all* pullbacks.


Formally, the only relation with abelian groups I know of is that the set of isomorphism classes of objects in an extensive category is an abelian monoid under coproduct, just as the set of isomorphism classes of objects in a distributive category is a semiring under product and coproduct. Although unlike the latter fact, the former doesn't require extensivity, merely the existence of coproducts.


