Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ and the following universal property. For all $\varphi\in S$ the morphism $F(\varphi)$ is an isomorphism. Now let $G: \mathcal{C} \longrightarrow \mathcal{D}$ be another functor such that $G(\varphi)$ is an isomorphism for all $\varphi \in S$. Then $G$ factors via $HF$ for another functor $H: \mathcal{C}[S^{-1}] \longrightarrow \mathcal{D}$. One can explicitly construct $\mathcal{C}[S^{-1}]$ via the calculus of fractions under some conditions on $S$.

Now my question is the following: In the text above replace "isomorphism(s)" by "morphism(s) with right inverse". Are there similiar conditions on $S$ such that we can do the same construction via something like calculus of fractions?

Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ and the following universal property. For all $\varphi\in S$ the morphism $F(\varphi)$ is an isomorphism. Now let $G: \mathcal{C} \longrightarrow \mathcal{D}$ be another functor such that $G(\varphi)$ is an isomorphism for all $\varphi \in S$. Then $G$ factors via $HF$ for another functor $H: \mathcal{C}[S^{-1}] \longrightarrow \mathcal{D}$. One can explicitly construct $\mathcal{C}[S^{-1}]$ via the calculus of fractions under some conditions on $S$.

Now my question is the following: In the text above replace "isomorphism(s)" by "morphism(s) with right inverse". Are there similar conditions on $S$ such that we can do the same construction via something like calculus of fractions?

Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ and the following universal property. For all $\varphi\in S$ the morphism $F(\varphi)$ is an isomorphism. Now let $G: \mathcal{C} \longrightarrow \mathcal{D}$ be another functor such that $G(\varphi)$ is an isomorphism for all $\varphi \in S$. Then $G$ factors via $HF$ for another functor $H: \mathcal{C}[S^{-1}] \longrightarrow \mathcal{D}$. One can explicitly construct $\mathcal{C}[S^-1]$ via calculus of fractions under some conditions on $S$.

Now my question is the following: In the text above replace "isomorphism(s)" by "morphism(s) with right inverse". Are there similiar conditions on $S$ such that we can do the same construction via something like calculus of fractions?

Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ and the following universal property. For all $\varphi\in S$ the morphism $F(\varphi)$ is an isomorphism. Now let $G: \mathcal{C} \longrightarrow \mathcal{D}$ be another functor such that $G(\varphi)$ is an isomorphism for all $\varphi \in S$. Then $G$ factors via $HF$ for another functor $H: \mathcal{C}[S^{-1}] \longrightarrow \mathcal{D}$. One can explicitly construct $\mathcal{C}[S^{-1}]$ via the calculus of fractions under some conditions on $S$.

Now my question is the following: In the text above replace "isomorphism(s)" by "morphism(s) with right inverse". Are there similiar conditions on $S$ such that we can do the same construction via something like calculus of fractions?