two categories of interest

two categories of interest

[Paul B Levy]

Dear all,

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.

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.

Paul



-- 
Paul Blain Levy
School of Computer Science, University of Birmingham
http://www.cs.bham.ac.uk/~pbl


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Re: two categories of interest

[Richard Garner]

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


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