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\begin{document}

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\section*{calculus of fractions}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{additional_conditions}{Additional conditions}\dotfill \pageref*{additional_conditions} \linebreak
\noindent\hyperlink{construction_of_the_localization}{Construction of the localization}\dotfill \pageref*{construction_of_the_localization} \linebreak
\noindent\hyperlink{properties_of_the_localization}{Properties of the Localization}\dotfill \pageref*{properties_of_the_localization} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A class $W$ of [[weak equivalence]]s in a [[category]] $C$ is said to admit a \textbf{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.'

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A pair $(C,W)$ of a [[category]] $C$ and a class of [[morphism]]s $W$ is said to admit a \textbf{calculus of right fractions} if the following properties hold.

\begin{itemize}%
\item $W$ is a [[wide subcategory]] of $C$ (that is, $W$ contains all identities and is closed under composition).
\item (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{displaymath}
\begin{matrix}
w& \stackrel{f'}{\to} & x \\
v' \downarrow&&\, \downarrow v\\
y &\underset{f}{\to} & z
\end{matrix}
\end{displaymath}
commutes.


\item (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 $f\circ v' = g \circ v'$:\begin{displaymath}
w \xrightarrow{v'} x \underoverset{g}{f}{\rightrightarrows} y \xrightarrow{v} z
\end{displaymath}


\end{itemize}
One may also say that $W$ is a \textbf{right Ore system} in $C$ (although this is potentially confusing since the Ore condition is only part of the definition), or that $(C,W)$ \textbf{admits a category of right fractions}. If $(C^{op}, W^{op})$ admits a calculus of right fractions, we say that $(C,W)$ admits a \textbf{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.

\hypertarget{additional_conditions}{}\subsubsection*{{Additional conditions}}\label{additional_conditions}

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 \emph{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 $(C,W)$ \emph{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 \emph{right multiplicative system}).

\hypertarget{construction_of_the_localization}{}\subsection*{{Construction of the localization}}\label{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 [[span|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 [[cospan]]s (spans in opposite category). In fact, we can realize $C[W^{-1}]$ as $(C^{\mathrm{op}}[(W^{\mathrm{op}})^{-1}])^{\mathrm{op}}$. Note that \emph{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:

\begin{displaymath}
Hom_{C[W^{-1}]}(X,Y) = \underset{X' \stackrel{p \in W}{\to}X}{colim} Hom_C(X',Y).
\end{displaymath}
Dually, if $W$ admits a calculus of left fractions, we instead take the colimit over maps into a $W$-replacement under the target object:

\begin{displaymath}
Hom_{C[W^{-1}]}(X,Y) = \underset{Y \stackrel{i \in W}{\to} Y'}{colim} Hom_C(X,Y').
\end{displaymath}
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{displaymath}
\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}
  \,.
\end{displaymath}
\hypertarget{properties_of_the_localization}{}\subsection*{{Properties of the Localization}}\label{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 functor|exact]], and therefore preserves all finite [[limit]]s 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

\begin{itemize}%
\item $W$ admits a calculus of right fractions,
\item $C$ admits all small [[filtered colimits]], and
\item 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 category|cofinally small]],

\end{itemize}
then $C[W^{-1}]$ is [[locally small category|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

\begin{displaymath}
R_W F(X)
  =
  \underset{X \stackrel{s \in W}{\to} X'}{\colim} F(X').
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item 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.


\item 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.


\item For $S$ a [[site]], the collection of [[local epimorphism]]s 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 $S$.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[localization]]


\item [[colocalization]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The above definition is due to

\begin{itemize}%
\item [[Pierre Gabriel]], [[Michel Zisman]], \emph{[[Calculus of fractions and homotopy theory]]}, \emph{Ergebnisse der Mathematik und ihrer Grenzgebiete}, Band 35. Springer, New York (1967).

\end{itemize}
See also

\begin{itemize}%
\item [[Frank Adams]], part III, section 14 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Francis Borceux]], vol 1, chapter 5 of: \emph{[[Handbook of Categorical Algebra]], Cambridge University Press (1994)}



\end{itemize}
[[!redirects left calculus of fractions]] [[!redirects right calculus of fractions]] [[!redirects calculus of left fractions]] [[!redirects calculus of right fractions]]



\end{document}