Formally adding morphisms

Formally adding morphisms

[Joseph Collins]

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


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Re: Formally adding morphisms

[Ross Street]

Dear Joseph

In 2-categorical terminology, your A_f is the coinserter of the two functors X, Y : 1 --> A.

Ross

On 13 Nov 2019, at 5:04 AM, Joseph Collins <joseph.collins@strath.ac.uk<mailto:joseph.collins@strath.ac.uk>> wrote:

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).


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Re: Formally adding morphisms

[Fernando Lucatelli Nunes]

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
>

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