Examples

Almost every known example of a mathematical structure with the appropriate structure-preserving map yields a category.

1. The category Set with objects sets and morphisms the usual functions. There are variants here: one can consider partial functions instead, or injective functions or again surjective functions. In each case, the category thus constructed is different
2. The category Top with objects topological spaces and morphisms continuous functions. Again, one could restrict morphisms to open continuous functions and obtain a different category.
3. The category hoTop with objects topological spaces and morphisms equivalence classes of homotopic functions. This category is not only important in mathematical practice, it is at the core of algebraic topology, but it is also a fundamental example of a category in which morphisms are not structure preserving functions.
4. The category Vec with objects vector spaces and morphisms linear maps.
5. The category Diff with objects differential manifolds and morphisms smooth maps.
6. The categories Pord and PoSet with objects preorders and posets, respectively, and morphisms monotone functions.
7. The categories Lat and Bool with objects lattices and Boolean algebras, respectively, and morphisms structure preserving homomorphisms, i.e., (⊤, ⊥, ∧, ∨) homomorphisms.
8. The category Heyt with objects Heyting algebras and (⊤, ⊥, ∧, ∨, →) homomorphisms.
9. The category Mon with objects monoids and morphisms monoid homomorphisms.
10. The category AbGrp with objects abelian groups and morphisms group homomorphisms, i.e. (1, ×, ?) homomorphisms
11. The category Grp with objects groups and morphisms group homomorphisms, i.e. (1, ×, ?) homomorphisms
12. The category Rings with objects rings (with unit) and morphisms ring homomorphisms, i.e. (0, 1, +, ×) homomorphisms.
13. The category Fields with objects fields and morphisms fields homomorphisms, i.e. (0, 1, +, ×) homomorphisms.
14. Any deductive system T with objects formulae and morphisms proofs.

These examples nicely illustrates how category theory treats the notion of structure in a uniform manner. Note that a category is characterized by its morphisms, and not by its objects. Thus the category of topological spaces with open maps differs from the category of topological spaces with continuous maps — or, more to the point, the categorical properties of the latter differ from those of the former.

We should underline again the fact that not all categories are made of structured sets with structure-preserving maps. Thus any preordered set is a category. For given two elements p, q of a preordered set, there is a morphism f : p → q if and only if p ≤ q. Hence a preordered set is a category in which there is at most one morphism between any two objects. Any monoid (and thus any group) can be seen as a category: in this case the category has only one object, and its morphisms are the elements of the monoid. Composition of morphisms corresponds to multiplication of monoid elements. That the monoid axioms correspond to the category axioms is easily verified.

Hence the notion of category generalizes those of preorder and monoid. We should also point out that a groupoid has a very simple definition in a categorical context: it is a category in which every morphism is an isomorphism, that is for any morphism f : X → Y, there is a morphism g : Y → X such that f ○ g = id_X and g ○ f = id_Y.

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