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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