Let $C$ be a category and assume either that $C$ has all binary pullbacks or that $C$ satisfies [right calculus of fractions](http://ncatlab.org/nlab/show/calculus+of+fractions). In both cases the localization of $C$ at every morphism (i.e. the groupoidification) can be represented by spans, i.e the objects are the same as in $C$ and morphisms are spans $A\leftarrow B\rightarrow C$ modulo some relation (see the link, section Construction of the localization).


My question is now: Does having binary pullbacks imply the right calculus of fractions? Or vice versa? Or is there no general relation between the two?



Relationship between pullbacks and the Ore condition
More generally: let $C$ have pullbacks and consider a class of morphisms $W$ containing the isos, pullback stable and satisfying 2 from 3. Benabou call this a pullback congruence. Lemma 1.2 of his paper "Some remarks on 2-categorical algebra" shows that such $W$ admits a calculus of right fractions.


Even better the category of fractions $C(W^{-1})$ admits pullbacks which are preserved by the quotient map $p:C \to C(W^{-1})$, and furthermore $p$ inverts precisely the $W$'s. This is all in the above paper.


