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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.’
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 identity morphisms and is closed under composition).
(right Ore condition) Given a morphism $v \colon x\to z$ in $W$ and any morphism $f \colon y\to z$, there is a morphism $v' \colon w \to y$ in $W$ and a morphism $f' \colon w \to x$ in $C$ such that the following diagram commutes:
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).
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:
Dually, if $W$ admits a calculus of left fractions, we instead take the colimit over maps into a $W$-replacement under the target object:
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:
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
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
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 $S$.
See also references at category of fractions.
The original monograph:
See also
Frank Adams, part III, section 14 of Stable homotopy and generalised homology, 1974
Francis Borceux, vol 1, chapter 5 of: Handbook of Categorical Algebra, Cambridge University Press (1994)
Tobias Fritz, Categories of Fractions Revisited, Morfismos 15 2 (2011) 19-38 [arXiv:0803.2587, morfismos:vol15-n2-3]
Generalization from categories to quasi-categories (“$(\infty,1)$-calculus of fractions”, cf. at localization of an $(\infty,1)$-category):
Exposition:
Chris Kapulkin, Calculus of Fractions for Quasicategories (Part I), talk at CQTS (18 Oct 2023) [video:YT]
Daniel Carranza, Calculus of Fractions for Quasicategories (Part II), talk at CQTS (25 Oct 2023) [video:YT]
Last revised on May 30, 2024 at 16:28:29. See the history of this page for a list of all contributions to it.