Definition 4.27.1. Let $\mathcal{C}$ be a category. A set of arrows $S$ of $\mathcal{C}$ is called a left multiplicative system if it has the following properties:

1. The identity of every object of $\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$.

2. Every solid diagram

$\xymatrix{ X \ar[d]_ t \ar[r]_ g & Y \ar@{..>}[d]^ s \\ Z \ar@{..>}[r]^ f & W }$

with $t \in S$ can be completed to a commutative dotted square with $s \in S$.

3. For every pair of morphisms $f, g : X \to Y$ and $t \in S$ with target $X$ such that $f \circ t = g \circ t$ there exists a $s \in S$ with source $Y$ such that $s \circ f = s \circ g$.

A set of arrows $S$ of $\mathcal{C}$ is called a right multiplicative system if it has the following properties:

1. The identity of every object of $\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$.

2. Every solid diagram

$\xymatrix{ X \ar@{..>}[d]_ t \ar@{..>}[r]_ g & Y \ar[d]^ s \\ Z \ar[r]^ f & W }$

with $s \in S$ can be completed to a commutative dotted square with $t \in S$.

3. For every pair of morphisms $f, g : X \to Y$ and $s \in S$ with source $Y$ such that $s \circ f = s \circ g$ there exists a $t \in S$ with target $X$ such that $f \circ t = g \circ t$.

A set of arrows $S$ of $\mathcal{C}$ is called a multiplicative system if it is both a left multiplicative system and a right multiplicative system. In other words, this means that MS1, MS2, MS3 hold, where MS1 $=$ LMS1 $+$ RMS1, MS2 $=$ LMS2 $+$ RMS2, and MS3 $=$ LMS3 $+$ RMS3. (That said, of course LMS1 $=$ RMS1 $=$ MS1.)

## Comments (2)

Comment #4336 by Manuel Hoff on

In LMS2, the notation doesn't make it entirely clear to me how $W$ is quantified. Maybe one wants to mention more explicitly that given $t$ and $g$ as in the diagram, there exist $W, f$ and $s$ as in the diagram (same applies to RMS2).

Comment #4486 by on

Since you parsed the condition correctly, I think others can too. Let's see if other people also feel we should clarify this. (There doesn't see to be a way to dotting the $W$ in the diagram.)

There are also:

• 12 comment(s) on Section 4.27: Localization in categories

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