The Fractions – icefog BLOG

Fractions are deceptively difficult. I have seen many parents agonize over how to explain concepts like 1/4 or 2/5 to an eight-year-old without simply resorting to rote formal operations. Paradoxically, I only fully realized the complexity of basic fractions by taking a detour into deep theory. While reading Weibel’s An Introduction to Homological Algebra, I found a fascinating exposition on the categorification of fractions—known as localization.

(A quick technical note: for the sake of rigor, we assume all categories discussed here belong to a Grothendieck universe and that the multiplicative system is locally small on the left. However, these set-theoretic details are not the focus of the article.)

Table of Contents

1. Multiplicative system

Definition. A multiplicative system $S$ in a category $C$ is a collection of morphisms that satisfies the following three self dual axioms:

1. $S$ is closed under composition and contains all identity morphisms for all objects in $C$ ;

2. (Ore condition) For each pair $g\in Mor(C),t\in S$ with $X\overset{g}{\to} Y \overset{t}{\leftarrow} Z$ , there is a weak pullback $f\in Mor(C),s\in S$ such that $gs=tf$ in $C$ . The dual statement is also valid (there exists a weak pushout with respect to a pair of morphisms in $C$ and $S$ );

3. (Cancellation) For each parallel morphisms $f,g:X\to Y$ in $C$ , the following two conditions are equivalent:
(3.a) there is a weak coequalizer of $f,g$ in $S$ , namely, $sf=sg$ for some $s\in S$ with $dom(S)=Y$ ;
(3.b) there is a weak equalizer of $f,g$ in $S$ , namely, $ft=gt$ for some $t\in S$ with $codom(S)=X$ ;

1.1. The Spirit of the Ore Condition: Right (Left) Permutability

Origin of the Name

The term comes from the Norwegian mathematician Øystein Ore. In 1931, he studied the problem of embedding a non-commutative ring into a division ring. He discovered that you cannot always form a field of fractions for a non-commutative ring. You can only do it if the ring satisfies the Ore Condition.

Intuition

In the localized category $C[S^{-1}]$ , we want to be able to compose a “fraction” $f s^{-1}$ with a morphism $g$ .

$(X \xleftarrow{s} X' \xrightarrow{f} Y) \circ (Y \xrightarrow{g} Z)$

We want the result to look like a fraction too. But structurally, this composition is $g \circ f \circ s^{-1}$ . The $s^{-1}$ is “trapped” on the right. This is fine.

However, consider composing two fractions:

$(f s^{-1}) \circ (g t^{-1})$

This looks like $f \circ s^{-1} \circ g \circ t^{-1}$ . To simplify this into a single (left) fraction (form $H K^{-1}$ ), we need to move the $s^{-1}$ past the $g$ . We need to “commute” them.

We want to find $g'$ and $s'$ such that $s^{-1} \circ g = g' \circ (s')^{-1}$ . Multiplying by $s$ on the left and $s'$ on the right (conceptually), we get $g \circ s' = s \circ g'$ .

This is exactly the Ore Condition (the weak pullback). It guarantees that any zigzag of morphisms can be reduced to a single roof (a span $X \xleftarrow{s} Z \xrightarrow{f} Y$ ).

1.2. The Spirit of Cancellation: Zero Divisor

Intuition

In ring theory, two fractions are equal if and only if

$\frac{a}{s} = \frac{b}{s} \iff t(a - b) = 0 \text{ for some } t \in S$

which implies $ta = tb$ .

In this concrete term, the property of an element in $S$ as a zero divisor (i.e. t(a-b)=0) is, in a categorical phrasing, well translated into the existence and the left cancellability of such an element of $S$ .

When we want to define an equivalence relation on fractions, we need to know when two parallel morphisms $f, g: X \to Y$ become equal in the localized category.

Further, the equivalence of conditions (3.a) and (3.b) asserts that it doesn’t matter which side you multiply on. Indeed, it imposes $S$ to behave as a collection of isomorphisms.

If $s$ kills the difference on the left, there must be a $t$ that kills it on the right.

With these axioms, every morphism in $C[S^{-1}]$ is just a fraction $f s^{-1}$ .

2. Localization

Definition. Let $S$ be a collection of morphisms in a category $C$ . A localization $C[S^{-1}]$ of $C$ with respect to $S$ is a category together with a universal functor $q:C\to C[S^{-1}]$ in way that:

Any functor $F: C\to D$ such that $F(s)$ is an isomorphism for all $s\in S$ factors in a unique way through $q$ .

The definition ensures that $C[S^{-1}]$ is unique up to equivalence (of category) and $q(s)$ is an isomorphism in $C[S^{-1}]$ for all $s\in S$ .

Although localization with respect to a multiplicative system provides a formal definition, it leaves open the question of computation. Specifically, we lack a concrete representation for the morphisms in the localized category. This is known as the problem of constructability.

3. Construction of localization

Calculus of Fractions

While there are two primary methods for this construction¹, Weibel follows the later approach of defining fractions via equivalence classes.

Definition. Given a multiplicative system $S$ in a category $C$ , a (left) fraction with respect to $S$ is a chain in $C$ of the form

$fs^{-1}: X \overset{s}{\leftarrow} X_1 \overset{f}{\to} Y$

with $s\in S$ and $f\in Mor(C)$ .

By denoting the collection of all (left) fractions as $F(S)$ , we introduce an equivalence relation $\sim$ on $F(S)$ . This gives rise to the construction (and hence the existence) of the localization $C[S^{-1}] \cong F(S)/\sim$ . By the very definition of localization, this object is unique up to equivalence.

Definition. A fraction $fs^{-1}: X \overset{s}{\leftarrow} X_1 \overset{f}{\to} Y$ is called equivalent to $gt^{-1}: X \overset{t}{\leftarrow} X_2 \overset{g}{\to} Y$ , if there is a fraction $hu^{-1}: X \overset{u}{\leftarrow} X_3 \overset{h}{\to} Y$ fitting into a following commutative diagram in $C$ :

When I first saw this, my immediate reaction was: “Well, why does the equivalence of fractions take this form? And why must a third fraction, $hu^{-1}$ , be introduced?”

In the literature, this shape is called a Common Roof or Common Span. While the definition is complicated and might seem overwhelming to the untrained eye, I felt that this categorical interpretation effectively “factorizes”—or deconstructs—the essential complexity inherent in fractions.

In commutative ring, we say $\frac{a}{s} = \frac{b}{t}$ if and only if $at = bs$ . We can simply swap the denominators.

In a general category, we cannot do this because the domains of $f$ and $g$ ( $X_1$ and $X_2$ ) are different objects, and morphisms don’t commute. We cannot compare $f$ and $g$ directly — so we need a Common Denominator to compare them. The 3rd fraction (or the object $X_3$ ) acts as this common denominator.

It will be clear if we imagine when we check if the fractions $\frac{2}{3}$ and $\frac{4}{6}$ are equal, but we are not allowed to multiply numbers to cancel the denominators.

We then look for a common refinement—a third fraction that maps to both of them.

If we have two arbitrary fractions $\frac{a}{s}$ and $\frac{b}{t}$ , showing they are equal amounts to finding a third fraction $\frac{h}{u}$ (usually with a much larger denominator $u$ ) that simplifies down to $\frac{a}{s}$ and simplifies down to $\frac{b}{t}$ .

The properties of equivalent relation

If we defined equivalence as: there exists a map $k: X_1 \to X_2$ such that diagrams commute, this relation would not be symmetric. Furthermore, this allows the transitivity to be shown along with the Ore Condition, for which we are not giving a proof here.

The General Construction (“Zig-zags” or “Paths”) and The Calculus of Fractions (“Roofs”).