Re: Formally adding morphisms

[Fernando Lucatelli Nunes <lucatellinunes@uc.pt>]

Dear Joseph,

The objects of a category A is in bijective relation with the functors 1\to
A, in which 1 is the terminal category.

The free addition of a morphism in a category is actually a very particular
case of a Cat-weighted colimit (see Limits indexed by category-valued
2-functors, Street), called coinserter (see pag. 307 of Elementary
observations on 2-categorical limits, Kelly), in Cat.
Given a pair of objects x: 1\to A, y: 1\to A, the coinserter of the pair x:
1\to A and y: 1\to A is the category you are looking for (and, as any
Cat-enriched colimit of them Cat-category Cat, it can be constructed out of
the coequalizers, coproducts and products).




Fernando

On Thu, Nov 14, 2019 at 2:56 PM Joseph Collins <joseph.collins@strath.ac.uk>
wrote:

> Hey all
>
> Suppose that we have a category A. If we want to formally add a single
> morphism, say f:X -> Y, where X,Y are in A, but f is not in A, we can do
> the following: we look at the discrete category containing  only X and Y -
> let us  denote that as (X   Y) - and the category with two objects and only
> a single morphism between them. Let's call this one (X -> Y).
>
> There are natural embeddings (X   Y) -> A and (X    Y) -> (X -> Y). We
> take  the pushout of these functors, and as one might expect, we get the
> union of A and (X -> Y). This is basically A, but with an extra morphism
> formally added in. Let's call this new morphism f and the new category A_f.
> This category is not particularly interesting, but I can then quotient it
> by some equations involving f and it becomes more interesting.
>
> I don't think that I am doing anything particularly modern, and I expect
> that someone else will have done something similar in the past, but my
> search  has not been very fruitful. Does anyone have any references that
> they can throw my way?
>
> Thanks
> Joe
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]