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Algebroids (linear categories)

Idea

A linear category, or algebroid, is a category whose hom-sets are all vector spaces (or modules) and whose composition operation is bilinear. This concept is a horizontal categorification of the concept of (unital associative) algebra.

Definitions

Fix a commutative ring $K$. (Often we want $K$ to be a field, such as the field $\mathbb{C}$ of complex numbers , but we could also chose choose more generaly generally acommutative rig for $K$.)

A $K$-linear category, or $K$-algebroid, is a category enriched over $K\,$Mod, the monoidal category of $K$-modules with the usual tensor product of modules. (Note that one usually speaks of $K\,$Vect instead of $K\,Mod$ when $K$ is a field.)

Just as a $\mathbb{Z}$-algebra is the same thing as a ring, so a $\mathbb{Z}$-algebroid is the same thing as a ringoid.

Remarks

• An algebra is a pointed algebroid with a single object, hence a one-object $K\,Mod$-enriched (or $K\,Vect$-enriched) category. Compare with similar ‘oidfied’ concepts such as groupoid and ringoid.

• Many linear categories are also assumed to be additive. A linear functor (that is, a $K\,Mod$-enriched or $K\,Vect$-enriched functor) between additive linear categories is automatically an additive functor.

• A symmetric monoidal $K$-linear category is a category which is at the same time a $K$-linear category and a symmetric monoidal category and such that the composition and the tensor product on morphisms are bilinear.

• Beware that a Lie algebroid is not a special case of an algebroid in the above sense, just as a Lie algebra is not a unital associative algebra. 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 $Vect$, hence a one-object $Vect$-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 $\infty$-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 (mimicing the extra structure of a $C^*$-algebra), which following the above reasoning could justly be call a $C^*$-algebroid is usually called a $C^*$-category. See spaceoids.

• vertex operator algebroid