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There's a categorification of what it means to be an element of a set; indeed, from Goldblatt's Topoi: The Categorial Analysis of Logic, we have this

Definition: In any category $\mathbb{C}$ with terminal object $1_{\mathbb{C}}$, an element of a $\mathbb{C}$-object $a$ is a $\mathbb{C}$-arrow $1_{\mathbb{C}}\stackrel{x}{\to}a$.

The idea of categorification is, that of replacing sets with categories, elements with objects, relations between elements with morphisms between objects, etc., so perhaps the above isn't quite a categorification of an element.

There's the category of rings and ring homomorphisms.

So are there categorifications of the notions of prime or irreducible elements (of a ring, say)?

Mike Pierce
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Shaun
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    This is not so much categorification as it is rephrasing in category-theoretic language. – Zhen Lin Jun 12 '14 at 00:26
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    Agreed, that's not a categorification. A categorification should tell you what a "categorified element" of a "categorified set" is. (There's a silly answer here, which is that a "categorified set" is a category and a "categorified element" is an object. Categorification is not always so useful.) – Qiaochu Yuan Jun 12 '14 at 01:06
  • @ZhenLin Thank you. Does this warrant me rewording the question? I'm not sure how to fix this. – Shaun Jun 13 '14 at 10:17

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An object $X$ of a category is called indecomposable if $X$ not initial and for every coproduct decomposition $X = A \oplus B$ we have that either $A$ or $B$ is initial. Similarly, $X$ is called simple (or irreducible) if $X$ has exactly two quotients. Both notions are very common in representation theory. In Krull-Schmidt categories we have categorified prime factorizations.

Martin Brandenburg
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