A class of weak equivalences in a category is said to admit a calculus of fractions if it satisfies some axioms ensuring that its localization 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 is referred to as a category of fractions, since its morphisms are two-step zigzags (either or , depending on the handedness of the calculus) in which , which we can think of as ‘fractions’ or . One sometimes also says that ‘admits a category of fractions.’
One may also say that is a right Ore system in (although this is potentially confusing since the Ore condition is only part of the definition), or that admits a category of right fractions. If admits a calculus of right fractions, we say that 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 which make no difference to the localization. For example, one often assumes that contains all isomorphisms in . One can also assume the 2-out-of-3 property (so that is a category with weak equivalences) or the stronger 2-out-of-6 property (so that 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 , i.e. that any morphism in which becomes an isomorphism in is already in . Therefore, in this case we may equivalently call saturated. See 2-out-of-6 property for a proof, taken from 7.1.20 of Categories and Sheaves (where a pair admitting a calculus of left fractions is called a right multiplicative system).
Suppose that admits a calculus of right fractions. Then the localization of at can be realized by taking the same objects as in and the hom-set to be the set of equivalence classes of spans whose left leg is in , under the equivalence relation where is equivalent to iff there exists an object and morphisms , such that , , and is in . We denote the equivalence class of by .
These equivalence classes compose as follows: take a representative and a representative ; then by the Ore condition there exist morphisms and , where , such that . The composition is the equivalence class of the span . One proves that this definition does not depend on the choice of representatives, and that it is associative with units . Obviously, the localization functor sends a morphism to .
If instead admits a calculus of left fractions, the hom-sets of are equivalence classes of cospans (spans in opposite category). In fact, we can realize as . 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 admits a calculus of right fractions, then the hom-sets in are obtained as the colimit over maps in out of a -replacement of the source object:
Dually, if admits a calculus of left fractions, we instead take the colimit over maps into a -replacement under the target object:
If 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 admits a calculus of right fractions, the localization functor is left exact, and therefore preserves all finite limits existing in . Dually, if admits a calculus of left fractions, then is right exact and preserves finite colimits.
Another important fact is that if
If is a category of fibrant objects and its category of morphisms modulo homotopy, the collection of weak equivalences in admits a calculus of right fractions. The corresponding localization is the homotopy category of . Note that this example does not satisfy the 2-out-of-3 property.
For a site, the collection of local epimorphisms in with respect to the given Grothendieck topology on admits a calculus of right fractions. In this case the localization is the category of sheaves on .
The above definition is due to Gabriel and Zisman in the book