Category of vector spaces

In algebra, given a ring R {\displaystyle R} , the category of left modules over R {\displaystyle R} is the category whose objects are all left modules over R {\displaystyle R} and whose morphisms are all module homomorphisms between left R {\displaystyle R} -modules. For example, when R {\displaystyle R} is the ring of integers Z {\displaystyle \mathbb {Z} } , it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way.

One can also define the category of bimodules over a ring R {\displaystyle R} but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R {\displaystyle R} (or over the opposite of that).

Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action.^[1]

Properties

The categories of left and right modules are abelian categories. These categories have enough projectives^[2] and enough injectives.^[3] Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules over some ring.

Projective limits and inductive limits exist in the categories of left and right modules.^[4]

Over a commutative ring, together with the tensor product of modules ⊗ {\displaystyle \otimes } , the category of modules is a symmetric monoidal category.

Objects

A monoid object of the category of modules over a commutative ring R {\displaystyle R} is exactly an associative algebra over R {\displaystyle R} .

A compact object in R {\displaystyle R} - M o d {\displaystyle \mathbf {Mod} } is exactly a finitely presented module.

Category of vector spaces

The category K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } (some authors use V e c t K {\displaystyle \mathbf {Vect} _{K}} ) has all vector spaces over a field K {\displaystyle K} as objects, and K {\displaystyle K} -linear maps as morphisms. Since vector spaces over K {\displaystyle K} (as a field) are the same thing as modules over the ring K {\displaystyle K} , K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } is a special case of R {\displaystyle R} - M o d {\displaystyle \mathbf {Mod} } (some authors use M o d R {\displaystyle \mathbf {Mod} _{R}} ), the category of left R {\displaystyle R} -modules.

Much of linear algebra concerns the description of K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } . For example, the dimension theorem for vector spaces says that the isomorphism classes in K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } correspond exactly to the cardinal numbers, and that K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } is equivalent to the subcategory of K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } which has as its objects the vector spaces K n {\displaystyle K_{n}} , where n {\displaystyle n} is any cardinal number.

Generalizations

The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).

See also

• Algebraic K-theory (the important invariant of the category of modules.)
• Category of rings
• Derived category
• Module spectrum
• Category of graded vector spaces
• Category of representations
• Change of rings
• Morita equivalence
• Stable module category
• Eilenberg–Watts theorem

References

External links

• Mod at the nLab