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It might be a stupid question.

When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but morphism between two objects $M$ and $M'$ are isomorphism class of following diagrams:

$M'\leftarrow N\rightarrow M$

where the first morphism is admissible epimorphsim and second is admissble monomorphism.

I found the shape of this diagram is similar the definition of Gabriel-Zisman localization for category. Suppose we have categroy $C$, then objects in $\Sigma_{T}^{-1}C$ are the same as those in $C$ but morphism between two objects $M,M'$ is the equivalent class of following diagrams:

$M'\leftarrow N\rightarrow M$,where the first morphism $s\in \Sigma_{T}$.

The composition law is also similar to the above one.

Of course, the equivalence relations of these two constructions are different. But, are there any relationship between these two constructions? Maybe Quillen Q-constructions are inspired by the "localization constructions" ?

Any retags are welcome

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