I'm studying the localization functor that makes morphism in $\Sigma$ invertible, where $\Sigma$ is class of morphism in a category (the Gabriel-Zisman localization).
It is known that if $\Sigma$ admits a right calculus fraction, then the localization functor preserves finite limits (see, for example, Proposition 5.2.5 of Borceux - Handbook of Categorical Algebra - Vol. 1).
I'm trying to find a study on properties of the localization when $\Sigma$ does not admit a right calculus fraction. Would you have any references? I'm in a situation where $\Sigma$ may admit some kind of weaker right calculus fraction, and I hope this will give that the localization preserves some finite limits, but not all. If you could share an example of localization that behaves like this, I would appreciate it.