The thrid construction, only applies to a small category $C$ but with no other assumptions. You look at the category $\widehat{C}$ of presheaves on $C$. Monomorphism in $C$ stay monomorphisms in $\widehat{C}$, so one can look at the category whose objects are objects of $C$ and whose arrow are partial maps in $\widehat{C}$. As $\widehat{C}$ as pullback this gives a solution to your problem.

The thrid construction, only applies to a small category $C$ but with no other assumptions. You look at the category $\widehat{C}$ of presheaves on $C$. Monomorphism in $C$ stay monomorphisms in $\widehat{C}$, so one can look at the category whose objects are objects of $C$ and whose arrow are partial maps in $\widehat{C}$. As $\widehat{C}$ has pullbacks this also gives a solution to your problem.

Here are two possibilities:

First you have the obvious 'universal solution':

You start from $C$ any category and $J$ class of maps, you can consider the category $C'$ freely generated from $C$ by adding a retract $r_i$ to each arrow $i \in J$.

this being said, I belive one can prove that the functor $C \rightarrow C'$ is faithfull if and only if all arrow in $J$ are monomorphisms. which might be interesting (I have an idea of how to prove that, but I haven't tried to write to complete proof of it yet, I would depending on the reaction to that comment).

A second construction, less general, but closer to the 'category of fractions' you mentioned.

Here are three possibilities:

First: you have the obvious 'universal solution':

You start from $C$ any category and $J$ a set of maps, you can consider the category $C'$ freely generated from $C$ by adding a retract $r_i$ to each arrow $i \in J$.

this being said, I belive one can prove that the functor $C \rightarrow C'$ is faithfull if and only if all arrow in $J$ are monomorphisms. which might be interesting (I will give a proof below in the case where $C$ is small, but it should be true in general either, eventually using a large cardinal axiom, or by just refining the construction a little).

Second construction, less general, but closer to the 'category of fractions' you mentioned.

The thrid construction, only applies to a small category $C$ but with no other assumptions. You look at the category $\widehat{C}$ of presheaves on $C$. Monomorphism in $C$ stay monomorphisms in $\widehat{C}$, so one can look at the category whose objects are objects of $C$ and whose arrow are partial maps in $\widehat{C}$. As $\widehat{C}$ as pullback this gives a solution to your problem.

This allows to proves the claim made in the first construction, at least for small category: for any small category $C$ and $J$ a set of monomorphisms in $C$, there exists a faithfull functor to a category $C'$ such that all arrow in $J$ have retraction in $C'$, hence the category obtained by universally adding the retraction is indeed a faithful extention of $C$.

Final remark:

None of the construction above, will produce a locally small category in full generality:

• In the first if you start form a proper class of arrow $J$ you will almost certainly end up with a non locally small category.
• In the second construction, the category of partial arrow will be locally small if and only if you started with a well-powered category.
• For the third one, you need a small category, although it might works for certain non-small category. Also you can adapt the idea and use another an embeddings into a finitely complete category. For example any accessible category can be embedded in $prsh(C)$ of $C$ small.