category theory

# Contents

## Idea

A class $W$ of weak equivalences in a category $C$ is said to admit a calculus of fractions if it satisfies some axioms ensuring that its localization $C[W^{-1}]$ can be constructed in a particularly simple way using ‘one-step generalized morphisms.’ These axioms are a categorical analogue of the notion of a multiplicative system at which one can localize a ring.

Since composition in a category is generally non-commutative, we distinguish ‘left’ and ‘right’ calculi of fractions, just as for localization of non-commutative rings. In either case $C[W^{-1}]$ is referred to as a category of fractions, since its morphisms are two-step zigzags (either $\overset{f}{\to} \overset{w}{\leftarrow}$ or $\overset{w}{\leftarrow} \overset{f}{\to}$, depending on the handedness of the calculus) in which $w\in W$, which we can think of as ‘fractions’ $w^{-1} f$ or $f w^{-1}$. One sometimes also says that $(C,W)$ ‘admits a category of fractions.’

## Definition

A pair $(C,W)$ of a category $C$ and a class of morphisms $W$ is said to admit a calculus of right fractions if the following properties hold.

• $W$ is a wide subcategory of $C$ (that is, $W$ contains all identities and is closed under composition).
• (right Ore condition) Given an arrow $v:x\to z$ in $W$ and any arrow $f: y\to z$, there is an arrow $v':w \to y$ in $W$ and an arrow $f':w \to x$ in $C$ such that
$\begin{matrix} w& \stackrel{f'}{\to} & x \\ v' \downarrow&&\, \downarrow v\\ y &\underset{f}{\to} & z \end{matrix}$

commutes.

• (right cancellability) Given an arrow $v:y\to z$ in $W$ and a pair of parallel morphisms $f,g: x\to y$ such that $v\circ f = v \circ g$, there is an arrow $v':w\to x$ in $W$ such that $f\circ v' = g \circ v'$.

One may also say that $W$ is a right Ore system in $C$ (although this is potentially confusing since the Ore condition is only part of the definition), or that $(C,W)$ admits a category of right fractions. If $(C^{op}, W^{op})$ admits a calculus of right fractions, we say that $(C,W)$ admits a calculus of left fractions. Unfortunately there is no uniformity regarding the choice of ‘left’ versus ‘right;’ some authors use ‘left’ where we use ‘right’ and vice versa.

It is common to assume additional closure conditions on $W$ which make no difference to the localization. For example, one often assumes that $W$ contains all isomorphisms in $C$. One can also assume the 2-out-of-3 property (so that $(C,W)$ is a category with weak equivalences) or the stronger 2-out-of-6 property (so that $(C,W)$ is a homotopical category). Note that the 2-out-of-3 property includes closure under composition, and the 2-out-of-6 property together with containment of all identities implies containment of all isomorphisms.

In the presence of either sort of calculus of fractions, the 2-out-of-6 property is equivalent to saturation of $W$, i.e. that any morphism in $C$ which becomes an isomorphism in $C[W^{-1}]$ is already in $W$. Therefore, in this case we may equivalently call $(C,W)$ saturated. See 2-out-of-6 property for a proof, taken from 7.1.20 of Categories and Sheaves (where a pair $(C,W)$ admitting a calculus of left fractions is called a right multiplicative system).

## Construction of the localization

Suppose that $(C,W)$ admits a calculus of right fractions. Then the localization of $C$ at $W$ can be realized by taking the same objects as in $C$ and the hom-set $C[W^{-1}](a,b)$ to be the set of equivalence classes of spans whose left leg is in $W$, under the equivalence relation where $a\stackrel{v}\leftarrow a'\stackrel{f}\rightarrow b$ is equivalent to $a\stackrel{w}\leftarrow a''\stackrel{g}\rightarrow b$ iff there exists an object $\bar{a}$ and morphisms $s:\bar{a}\to a'$, $t:\bar{a}\to a''$ such that $f\circ s = g\circ t$, $v\circ s = w\circ t$, and $v\circ s = w\circ t$ is in $W$. We denote the equivalence class of $a\stackrel{v}\leftarrow a'\stackrel{f}\rightarrow b$ by $f\circ v^{-1}$.

These equivalence classes compose as follows: take a representative $a\stackrel{v}\leftarrow a'\stackrel{f}\rightarrow b$ and a representative $b\stackrel{u}\leftarrow b'\stackrel{h}\rightarrow c$; then by the Ore condition there exist morphisms $z:d\to a'$ and $k:d\to b'$, where $z\in W$, such that $f\circ z= u\circ k$. The composition $(h\circ u^{-1})\circ (f\circ v^{-1})$ is the equivalence class of the span $a\stackrel{v\circ z}\leftarrow d\stackrel{h\circ k}\rightarrow c$. One proves that this definition does not depend on the choice of representatives, and that it is associative with units $1_a\circ 1^{-1}_a$. Obviously, the localization functor $C\to C[W^{-1}]$ sends a morphism $p : a\to b$ to $p\circ 1^{-1}_a$.

If instead $(C,W)$ admits a calculus of left fractions, the hom-sets of $C[W^{-1}]$ are equivalence classes of cospans (spans in opposite category). In fact, we can realize $C[W^{-1}]$ as $(C^{\mathrm{op}}[(W^{\mathrm{op}})^{-1}])^{\mathrm{op}}$. Note that two dualizations are involved, in order to get the cospans to be pointing in the correct direction.

An equivalent way to say this is that if $W$ admits a calculus of right fractions, then the hom-sets in $C[W^{-1}]$ are obtained as the colimit over maps in $C$ out of a $W$-replacement of the source object:

$Hom_{C[W^{-1}]}(X,Y) = \underset{X' \stackrel{p \in W}{\to}X}{colim} Hom_C(X',Y).$

Dually, if $W$ admits a calculus of left fractions, we instead take the colimit over maps into a $W$-replacement under the target object:

$Hom_{C[W^{-1}]}(X,Y) = \underset{Y \stackrel{i \in W}{\to} Y'}{colim} Hom_C(X,Y').$

If $W$ admits both a left and a right calculus of fractions, then both prescriptions coincide and are equivalent to taking a colimit over replacements of both objects:

\begin{aligned} Hom_{C[W^{-1}]}(X,Y) &= \underset{X' \stackrel{p \in W}{\to}X}{colim} Hom_C(X',Y) \\ &= \underset{Y \stackrel{i \in W}{\to} Y'}{colim} Hom_C(X,Y') \\ &= \underset{X' \stackrel{p \in W}{\to}X, Y \stackrel{i \in W}{\to} Y'}{colim} Hom_C(X',Y') \end{aligned} \,.

## 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, and therefore preserves all finite limits existing in $C$. Dually, if $W$ admits a calculus of left fractions, then $Q$ is right exact and preserves finite colimits.

Another important fact is that if

• $W$ admits a calculus of right fractions,
• $C$ admits all small filtered colimits, and
• for all $X \in C$ the category $W/X$, whose objects are morphisms $X'\to X$ in $W$ and whose morphisms are commutative triangles, is cofinally small,

then $C[W^{-1}]$ is locally small. In this case a left derived functor of any functor $F:C\to A$, where $A$ admits small filtered colimits, can be constructed as the colimit

$R_W F(X) = \underset{X \stackrel{s \in W}{\to} X'}{\colim} F(X').$

## Examples

• If $C$ is a category of fibrant objects and $\pi C$ its category of morphisms modulo homotopy, the collection of weak equivalences in $\pi C$ admits a calculus of right fractions. The corresponding localization is the homotopy category $\Ho(C)$ of $C$. Note that this example does not satisfy the 2-out-of-3 property.

• Given a null system $N$ in a triangulated category $C$, the collection of morphisms $f : X \to Y$ in $C$ such that there is a distinguished triangle $X \to Y \to Z$ where $Z \in N$ admits calculi of both left and right fractions.

• For $S$ a site, the collection of local epimorphisms in $C = [S^{op},Set]$ with respect to the given Grothendieck topology on $S$ admits a calculus of right fractions. In this case the localization is the category of sheaves on $S$.

## References

The above definition is due to Gabriel and Zisman in the book