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.

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

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

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

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

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

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

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

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

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

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

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

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.