calculus of fractions


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Basic facts




A class WW of weak equivalences in a category CC is said to admit a calculus of fractions if it satisfies some axioms ensuring that its localization C[W 1]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]C[W^{-1}] is referred to as a category of fractions, since its morphisms are two-step zigzags (either fw\overset{f}{\to} \overset{w}{\leftarrow} or wf \overset{w}{\leftarrow} \overset{f}{\to}, depending on the handedness of the calculus) in which wWw\in W, which we can think of as ‘fractions’ w 1fw^{-1} f or fw 1f w^{-1}. One sometimes also says that (C,W)(C,W) ‘admits a category of fractions.’


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

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


  • (right cancellability) Given an arrow v:yzv:y\to z in WW and a pair of parallel morphisms f,g:xyf,g: x\to y such that vf=vgv\circ f = v \circ g, there is an arrow v:wxv':w\to x in WW such that fv=gvf\circ v' = g \circ v':
    wvxgfyvz w \xrightarrow{v'} x \underoverset{g}{f}{\rightrightarrows} y \xrightarrow{v} z

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

Additional conditions

It is common to assume additional closure conditions on WW which make no difference to the localization. For example, one often assumes that WW contains all isomorphisms in CC. One can also assume the 2-out-of-3 property (so that (C,W)(C,W) is a category with weak equivalences) or the stronger 2-out-of-6 property (so that (C,W)(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 WW, i.e. that any morphism in CC which becomes an isomorphism in C[W 1]C[W^{-1}] is already in WW. Therefore, in this case we may equivalently call (C,W)(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)(C,W) admitting a calculus of left fractions is called a right multiplicative system).

Construction of the localization

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

These equivalence classes compose as follows: take a representative avafba\stackrel{v}\leftarrow a'\stackrel{f}\rightarrow b and a representative bubhcb\stackrel{u}\leftarrow b'\stackrel{h}\rightarrow c; then by the Ore condition there exist morphisms z:daz:d\to a' and k:dbk:d\to b', where zWz\in W, such that fz=ukf\circ z= u\circ k. The composition (hu 1)(fv 1)(h\circ u^{-1})\circ (f\circ v^{-1}) is the equivalence class of the span avzdhkca\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 a1 a 11_a\circ 1^{-1}_a. Obviously, the localization functor CC[W 1]C\to C[W^{-1}] sends a morphism p:abp : a\to b to p1 a 1p\circ 1^{-1}_a.

If instead (C,W)(C,W) admits a calculus of left fractions, the hom-sets of C[W 1]C[W^{-1}] are equivalence classes of cospans (spans in opposite category). In fact, we can realize C[W 1]C[W^{-1}] as (C op[(W op) 1]) op(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 WW admits a calculus of right fractions, then the hom-sets in C[W 1]C[W^{-1}] are obtained as the colimit over maps in CC out of a WW-replacement of the source object:

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

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

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

If WW 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:

Hom C[W 1](X,Y) =colimXpWXHom C(X,Y) =colimYiWYHom C(X,Y) =colimXpWX,YiWYHom C(X,Y). \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 WW admits a calculus of right fractions, the localization functor Q:CC[W 1]Q:C\to C[W^{-1}] is left exact, and therefore preserves all finite limits existing in CC. Dually, if WW admits a calculus of left fractions, then QQ is right exact and preserves finite colimits.

Another important fact is that if

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

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

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


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

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

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


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

See also

Revised on June 23, 2017 08:21:20 by Mike Shulman (