Factorization of Gabriel-Zisman localization construction?

Question

My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$.

The localization of a category $\mathcal{C}$ by a collection of maps $S$ is understood to mean a new category $S^{-1}\mathcal{C}$ and a functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ with the property that

• For any functor $f : \mathcal{C} \to \mathcal{D}$ such that $f(s)$ is an isomorphism for all $s\in S$, there is a functor $\bar{f} : S^{-1}\mathcal{C}\to \mathcal{D}$ which satisfies $f = \bar{f} \circ Q$.

Ignoring set theoretic issues, this category always exists but is difficult to control for sets of maps $S$ which aren't particularly well-behaved. When $S$ satisfies a number of reasonable properties there is a Gabriel-Zisman model for $S^{-1}\mathcal{C}$. The objects of $S^{-1}\mathcal{C}$ are the same as those of $\mathcal{C}$ but the morphisms are now roofs: $$X \xleftarrow{s} Y \xrightarrow{f} Z \quad\quad{ where } \quad\quad s\in S,\, f\in \mathcal{C}$$

This construction is desirable because it gives us some control over $S^{-1}\mathcal{C}$.

Now the condition that any given map $f(s)\in \mathcal{D}$ is an isomorphism can be split into two parts:

• there is a map $a$ such that $a f(s) = 1$
• there is a map $b$ such that $f(s) b = 1$

For any $S\subset \mathcal{C}$, again ignoring set theoretic concerns, there are categories $S^l \mathcal{C}$ and $S^r \mathcal{C}$ each corresponding to the conditions above in the sense that there are functors $Q^l : \mathcal{C} \to S^l \mathcal{C}$ and $Q^r : \mathcal{C} \to S^r\mathcal{C}$. Each of which satisfy the analogue of the universal property for $Q$ in the second paragraph above. Since this construction is just obtained by factoring the quotient defining $S^{-1}\mathcal{C}$ into parts I'd expect $$S^r S^l\mathcal{C} \cong S^{-1}\mathcal{C} \cong S^l S^r \mathcal{C}$$

My question is whether the Gabriel-Zisman construction can be factored to produce the $S^r$ and $S^l$ constructions above. Are there roofs with broken symmetry which give only $S^r$ or $S^l$ appearing somewhere in the literature?

Answer 1

I agree with Tyler that it's not clear what the universal property of, say $S^l C$ is supposed to be. Does a functor $G: S^l C \to D$ correspond to (i) a functor $F: C \to D$ with the property that $F(s)$ is a left inverse for every $s \in S$ or (ii) a functor $F: C \to D$ along with, for every $s \in S$, a choice of right inverse $\rho_s$ to $F(s)$, or (iii) something like (ii), but with $\rho_s$ required to be functorial in $s$, or (iv) something else?

I don't believe that an object with the universal property (i) exists outside of degenerate cases, maybe.

A category with the universal property (iii) should be obtainable in a straightforward way by deleting some of the identifications in the description of Gabriel-Zisman localization.