Re: two categories of interest

[Richard Garner <richard.garner@mq.edu.au>]

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/ ]