Why "right multiplicative system" for localization of category is called right instead of left?

Question

Let $C$ be a category. One says $S$, a family of morphisms of $C$ is a right multiplicative if it has the following property.

1) For all $X\in C$, $Id_X\in S$.

2) For all $f,g\in S$, if $g\circ f$ exists, then $g\circ f\in S$.

3) Given 2 morphisms, $f:X\to Y$ and $s:X\to X'$ where $s\in S$, there exists $t:Y\to Y',g:X'\to Y'$ with $t\in S$ and $g\circ s=t\circ f$.

4) Let $f,g:X\to Y$ be two parallel morphisms. If there exists $s:W\to X,s\in S$ s.t. $f\circ s=g\circ s$, then there exists $t\in S, t:Y\to Z$ s.t. $t\circ f=t\circ g$.

$\textbf{Q:}$ Why this is called "right" instead of "left"?

Ref. pg 90, Def 5.1.3 https://webusers.imj-prg.fr/~pierre.schapira/lectnotes/HomAl.pdf

Answer 1

In older literature, a family $\mathcal S$ which is right multiplicative in the sense of Shapira is said to admit a calculus of left fractions. See for example

Gabriel, Peter, and Michel Zisman. Calculus of fractions and homotopy theory. Vol. 35. Springer Science & Business Media, 2012.

The "old" definition is also used in ncatlab where the focus lies on systems admitting a calculus of right fractions which would be left multiplicative systems in the sense of Shapira.

In the above reference, the word "right" is used because of the right Ore condition and the right cancellability property. Of course, wording is a matter of taste. For example, Shapira's condition S4 says that if $f \circ s = g \circ s$ for $s \in \mathcal S$, then there exists $t \in \mathcal S$ such that $t \circ f = t \circ g$. Thus it seems in fact adequate to regard this as right cancellability of morphisms in $\mathcal S$. However, if you look at the diagram

then it seems to be more appropriate to regard it as left cancellability. As I said: It is a matter of taste.

Nevertheless, the old definition has the following "disadvantage" (quote from section 4 "Properties of the Localization"):

One important consequence of this construction is that when $W$ admits a calculus of right fractions, the localization functor $Q : C \to C[W^{−1}]$ is left exact.

Thus Shapira's definition has the benefit that a right multiplicative system yields a right exact localization functor.

Answer 2

It's really not a fixed convention.

There's a remark (7.1.8) in Schapira's book with Kashiwara where they state that other authors use "left" where they use "right", and vice versa.

They also say that they use right because under the localisation $Q:\mathcal{C}\to\mathcal{C}[\mathcal{S}^{-1}]$ one has $$\text{Hom}_{\mathcal{C}[S^{-1}]}(Q(X),Q(Y)) \simeq \varinjlim_{Y\to Y'}\text{Hom}_{\mathcal{C}}(X,Y')$$ with the direct limit being taken over 'right arrows'.