nLab
algebroid

Idea

The concept algebroid is the many object version (or oidification) of the concept unital associative algebra.

Definition

An algebroid is a category enriched over (Vect,$\otimes$).

Remarks

• An algebra is an algebroid with a single object. Hence a one-object $\mathrm{Vect}$-enriched category. See horizontal categorification.

• Compare with similar concepts such as groupoid and ringoid.

• Beware that a Lie algebroid is not a special case of an algebroid in the above sense. The point is that there is a restrictive and a general sense of “algebra”. In the restrictive sense an algebra is an associative unital algebra, hence a monoid in $\mathrm{Vect}$, hence a one-object $\mathrm{Vect}$-enriched category. But in a more general sense an algebra is an algebra over an operad. It is this more general sense in terms of which Lie algebras are special cases of algebras and Lie algebroids their horizontal categorification.

Generalizations

• Replacing plain vector spaces with chain complexes of vector spaces leads to an $\mathrm{\infty }$-version of algebroids: a category enriched in chain complexes, which following the above reasoning could justly be called a DG algebroid is usually called a DG-category.

• Replacing plain vector spaces with Banach spaces leads to a $C^{*}$-version of algebroids: a category enriched in Banach spaces with some extra structure, which following the above reasoning could justly be call a $C^{*}$-algebroid is usually called a $C^{*}$-category. See spaceoids.