Dear Paul,
Have either of the following categories been studied before?
1. A "set with loners" is a set A with a subset U, whose elements are called "loners". A "loner-respecting function" (A,U) --> (B,V) is a function A --> B such that for any x in U, f(x) is in V and its only f-preimage is x. Let SWL be the category of sets with loners and loner-respecting functions, and Inj the category of sets and injections. Both Set and Inj are isomorphic to full subcategories of SWL.
I don't know if this category has been studied, but it looks like you can also describe it as follows. Take the category Inj x Set. On here there is a monad defined by T(A,B) = (A,A+B). The Kleisli category of this monad appears to be SWL. That presentation seems to make it look dual to Dialectica-type stuff.
2. For sets A and B, a "sum preorder" from A to B is a preorder on A+B. Example: A is the set of men, B is the set of women, take the preorder "younger than or the same age as". An equivalence relation on A+B is called a "corelation" from A to B. Given sum preorders R : A --> B and S : B --> C, obtain the composite by taking the least preorder on A+B+C that contains R and S, and then restricting to A+C. Let SumPreord be the category of sets and sum preorders, Rel the category of sets and relations, and Corel the category of sets and corelations. Both Rel and Corel are isomorphic to lluf subcategories of SumPreord.
Yes: Dosen, Petric, "Syntax for split preorders", Annals of Pure and Applied Logic 164 (2013) 443???481 I think Danos and Regnier might also talk about related things somewhere, but I can't exactly tell you where (or if I am remembering correctly). Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]