TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The calculus of fractions constructs the localization

Statement

Under the smallness or supplied cofinal-denominator hypothesis of the multiplicative-system definition, roof classes with the preceding composition form a locally small localization $Q : C \to S^{- 1} C$ , with $Q (f) = [(1, f)]$ . For parallel $f, g : X \to Y$ , $Q (f) = Q (g)$ if and only if $f v = g v$ for some $v : W \to X$ in $S$ . Every arrow also has a right-roof presentation $Q (t)^{- 1} Q (h)$ , with $h : X \to V$ and $t : Y \to V$ in $S$ .

Facts & Assumptions

Given: Under the smallness or supplied cofinal-denominator hypothesis of the multiplicative-system definition, roof classes with the preceding composition form a locally small localization $Q : C \to S^{- 1} C$ , with $Q (f) = [(1, f)]$ . For parallel $f, g : X \to Y$ , $Q (f) = Q (g)$ if and only if $f v = g v$ for some $v : W \to X$ in $S$ . Every arrow also has a right-roof presentation $Q (t)^{- 1} Q (h)$ , with $h : X \to V$ and $t : Y \to V$ in $S$ .

[F1]

The roof composition is well defined, associative, and unital (Composition of roofs is well defined).

[F2]

A localization inverts $S$ and is universal for functors inverting $S$ , including descent of natural transformations (Localization of a category at a class of morphisms).

Proof

1.1

Composition, associativity and identities are supplied by the preceding lemma. Every roof into a fixed source $X$ refines to one with denominator in the supplied set $S_{X}$ ; its possible numerators lie in a set of Hom sets. Taking the quotient of this set by refinement gives a set of arrows from $X$ to $Y$ . In a small category the set of all roofs already suffices. This also constructs the empty localization when there are no objects.

F1given

1.2

Identity-denominator roofs show $Q (g f) = Q (g) Q (f)$ and preservation of identities. For $s : U \to X$ in $S$ , the roof $(s, 1_{U})$ is inverse to $Q (s)$ : the products are the identity at $U$ and the roof $(s, s)$ , which refines the identity at $X$ . Thus $Q$ inverts $S$ .

F1algebra

2.1

Let $F$ invert $S$ . Set $\overline{F} (s, f) = F (f) F (s)^{- 1}$ . For a refinement $s a = t b = r \in S$ , both $F (a) = F (s)^{- 1} F (r)$ and $F (b) = F (t)^{- 1} F (r)$ are invertible, so $f a = g b$ gives equal values. An Ore equality $f a = t b$ gives $F (t)^{- 1} F (f) = F (b) F (a)^{- 1}$ , proving preservation of composition. Every roof is $Q (f) Q (s)^{- 1}$ , forcing uniqueness. A natural transformation descends on the same objects: naturality for $s$ implies naturality for $Q (s)^{- 1}$ and hence for each roof. This proves the stated localization property.

F1F2step 1.2algebra

3.1

Equality of $(1, f)$ and $(1, g)$ gives refinement legs $a = b = v \in S$ with $f v = g v$ ; conversely such $v$ is a common refinement. For the dual presentation apply the other Ore axiom to $s : U \to X$ and $f : U \to Y$ , obtaining $t : Y \to V$ in $S$ and $h : X \to V$ with $t f = h s$ . Then $Q (f) Q (s)^{- 1} = Q (t)^{- 1} Q (h)$ . This describes right roofs in the same category and requires no second smallness assertion.

givenstep 1.2algebra∎

Depends on

Used by

Derived category of an abelian category Definition
fs-two-roofs-are-equal-whenever-their-right-hand-arrows-are-equal.md False statement
Addition of roofs makes an additive localization Lemma
Finite roof squares and composable pairs can be cleared Lemma
Morphisms from a homotopically projective complex need no roof Proposition

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

10.3.1–10.3.14, pp. 379–384 (standard reference, not scraped)