# nLab CRing

### Context

#### Algebra

higher algebra

universal algebra

# The category $CRing$

## Definition

$CRing$ is the category of commutative rings and ring homomorphisms.

A commutative ring is a commutative monoid object in Ab, so $CRing = CMon(Ab)$. As for commutative monoid objects in any symmetric monoidal category, the tensor product of commutative rings is again a commutative ring, and is the coproduct in $CRing$; thus $CRing$ is cocartesian monoidal.

The opposite category $CRing^{op}$ is the category of affine schemes.

## Generalizations

• The slice category of $CRing$ under a ring $R$ is the category $R$CAlg of commutative associative algebras over $R$.

• There is a “smooth” version of $CRing^{op}$: the category of smooth loci.

• There is a higher category theory version of $CRing$: the $(\infty,1)$-category of $E_\infty$-rings.

category: category

Revised on September 12, 2014 20:37:17 by Hew Wolff? (4.35.166.120)