Lemma 4.27.19 (04VL)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.27: Localization in categories
Lemma 4.27.19 (cite)

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Lemma 4.27.19. Let $\mathcal{C}$ be a category and let $S$ be a multiplicative system. The category of left fractions and the category of right fractions $S^{-1}\mathcal{C}$ are canonically isomorphic.

Proof. Denote $\mathcal{C}_{left}$, $\mathcal{C}_{right}$ the two categories of fractions. By the universal properties of Lemmas 4.27.8 and 4.27.16 we obtain functors $\mathcal{C}_{left} \to \mathcal{C}_{right}$ and $\mathcal{C}_{right} \to \mathcal{C}_{left}$. By the uniqueness statement in the universal properties, these functors are each other's inverse. $\square$

Comments (2)

Comment #1710 by Keenan Kidwell on November 30, 2015 at 19:50

In the last sentence of the proof, there is an instance of "they" which shouldn't be there.

Comment #1754 by Johan on December 15, 2015 at 19:03

Thanks, fixed here.

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