next up previous
Next: fiber product Up: Elementary category theory Previous: universe

examples of categories.

In this section we fix a sufficiently large universe $ U$ . For some of readers it may be happier to neglect the term ``U-small''.

EXAMPLE 5.5   The category $ (\operatorname{SETS})$ of $ U$ -small sets.

$\displaystyle \operatorname{Ob}(\operatorname{SETS})=\{$$U$-small sets$\displaystyle \}.
$

For any $ X,Y\in \operatorname{Ob}(\operatorname{SETS})$ , we put

$\displaystyle \operatorname{Hom}_{(\operatorname{SETS})}(X,Y)=($the set of all maps from $X$ to $Y$.$\displaystyle )
$

EXAMPLE 5.6   The category $ (\operatorname{GROUPS})$ of $ U$ -small groups (that means, groups that are $ U$ -small as sets).

$\displaystyle \operatorname{Ob}(\operatorname{GROUPS})=\{$$U$-small groups$\displaystyle \}.
$

For any $ X,Y\in \operatorname{Ob}(\operatorname{GROUPS})$ , we put

$\displaystyle \operatorname{Hom}_{(\operatorname{GROUPS})}(X,Y)
=($the set of all group homomorphisms from $X$ to $Y$.$\displaystyle )
$

Likewise, we may easily define categories such as the category $ (\operatorname{RINGS})$ of $ U$ -small-rings, the category $ (R-\operatorname{ALG})$ of algebras over a ring $ R$ , the category $ (k-\operatorname{VS})$ of $ U$ -small vector spaces over a field $ k$ , and so on.

EXAMPLE 5.7   The category $ (\operatorname{TOP})$ of $ U$ -small topological space

$\displaystyle \operatorname{Ob}(\operatorname{GROUPS})=\{$$U$-small topological space$\displaystyle \}.
$

For any $ X,Y\in \operatorname{Ob}(\operatorname{TOP})$ , we put

$\displaystyle \operatorname{Hom}_{(\operatorname{TOP})}(X,Y)
=($the set of all continuous maps from $X$ to $Y$.$\displaystyle )
$

One may also consider the category of $ C^\infty$ -manifolds, the category of $ C^1$ -manifolds, and so on.

Of course, the category of schemes (with morphisms the ones we defined in the previous part) is very important category for us.


next up previous
Next: fiber product Up: Elementary category theory Previous: universe
2007-12-11