homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 above definition is due to
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 December 27, 2023 at 12:40:28. See the history of this page for a list of all contributions to it.