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Is it possible to define a ring as a category? For example, a group can be defined as a category with just one objet and all morphisms being iso.

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A monoid is a category with exactly one object. A homomorphism of monoids is just a functor of the corresponding categories.

A ring is a linear category with exactly one object. A homomorphism of rings is just a linear functor between the corresponding linear categories.

More generally, if $R$ is a commutative ring, then an $R$-algebra is an $R$-linear category with exactly one object, and $R$-algebra homomorphisms correspond to $R$-linear functors.

This offers a compact definition of modules: A left module over $R$ is just a linear functor $R \to \mathsf{Ab}$. Actually this has lead to the following more general notion ( which has found applications in category theory and algebraic topology): If $C$ is any linear category, then a $C$-module is a linear functor $C \to \mathsf{Ab}$. Homomorphisms of $C$-modules are natural transformations. Thus, we optain a category $C$-modules.

Besides, it shows that the category of monoids or rings is actually a bicategory: If $f,g : M \to N$ are homomorphisms of monoids (or rings), then a $2$-morphism $f \to g$ is an element $n \in N$ such that $n f(m) = g(m) n$ for all $m \in M$.

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  • $\begingroup$ I suppose that would make an algebra an $R$-linear category. $\endgroup$ – Baby Dragon Mar 28 '13 at 18:47
  • $\begingroup$ Yes, I've added this above. Thanks. $\endgroup$ – Martin Brandenburg Mar 28 '13 at 18:50

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