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# category of fractions

Recall that given a commutative ring $R$ and a subset $S$ that is multiplicatively closed and does not contain any zero divisors, we can then form the ring of fractions $S^{{-1}}R$ by *formally* inverting elements of $S$. $S^{{-1}}R$ has the following universal property: there is a ring homomorphism $\phi:R\to S^{{-1}}R$ with the property that

$\phi(s)$ is a unit in $S^{{-1}}R$ for every $s\in S$;

and if $\sigma:R\to T$ is another ring homomorphism with the above property, then we get a unique ring homomorphism $\delta:S^{{-1}}R\to T$ such that $\delta\circ\phi=\sigma$. The category of fractions is the generalization of this concept to category theory.

Definition. Let $\mathcal{C}$ be a category, and $\Sigma$ a class of morphisms in $\mathcal{C}$. A *category of fractions* of $\mathcal{C}$ over $\Sigma$ is a pair $(\mathcal{D},F)$, where

1. $\mathcal{D}$ is a category and $F:\mathcal{C}\to\mathcal{D}$ is a functor, such that

$F(f)$ is an isomorphism for every $f$ in $\Sigma$,

2. if $(\mathcal{E},G)$ is another such a pair satisfying condition 1 above, then there is a unique functor $H:\mathcal{D}\to\mathcal{E}$ with $H\circ F=G$.

Equivalently, consider the (large) category $\mathcal{Q}$ with objects all pairs $(\mathcal{D},F)$ satisfying condition 1 above, and a morphism from $(\mathcal{D}_{1},F_{1})$ to $(\mathcal{D}_{2},F_{2})$ is a functor $G:\mathcal{D}_{1}\to\mathcal{D}_{2}$ where

$\xymatrix@+=1.5cm{&\mathcal{C}\ar[dr]^{{F_{1}}}\ar[dl]_{{F_{2}}}&\\ \mathcal{D}_{1}\ar[rr]_{{G}}&&\mathcal{D}_{2}}$ |

is a commutative diagram. An initial object in $\mathcal{Q}$ is called a *category of fractions* (of $\mathcal{C}$ over $\Sigma$).

It is clear that a category of fractions is unique up to natural isomorphism, so we call $(\mathcal{D},F)$ *the* category of fractions of $\mathcal{C}$ over $\Sigma$, and we denote the category $\mathcal{D}$ by $\mathcal{C}\Sigma^{{-1}}$.

For example, let $\mathcal{C}$ is the category with objects $A,B$ and morphisms $1_{A},1_{B}$ and $f:A\to B$, and $\Sigma=\{f\}$. Consider the category $\mathcal{D}$ with the same two objects $A,B$ and morphisms $1_{A},1_{B},g:A\to B$ and its inverse $g^{{-1}}:B\to A$, and the functor $F:\mathcal{C}\to\mathcal{D}$ given by $F(A)=A,F(B)=B$ and $F(1_{A})=1_{A},F(1_{B})=1_{B}$ and $F(f)=g$. Then $(\mathcal{D},F)$ is the category of fractions of $\mathcal{C}$ over $\Sigma$. To see this, suppose $G:\mathcal{C}\to\mathcal{E}$ is another functor with $G(f)$ an isomorphism. Define $H:\mathcal{D}\to\mathcal{E}$ given by $H(A):=G(A),H(B):=G(A)$ and $H(1_{A})=G(1_{A}),H(1_{B})=G(1_{B}),H(g)=G(f)$, and $H(g^{{-1}})=G(f)^{{-1}}$. It is clear that $H$ is a functor with $H\circ F=G$, and that $H$ is uniquely determined.

In fact, one can prove the following existence property:

###### Proposition 1.

$\mathcal{C}\Sigma^{{-1}}$ exists if $\Sigma$ is a set. Furthermore, $\mathcal{C}\Sigma^{{-1}}$ is small if $\mathcal{C}$ is.

## Mathematics Subject Classification

18A32*no label found*

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