Category of sets

The full description is available here in Russian.

Sample objects and morphisms

In the category of sets objects are sets, morphisms are total functions, and a morphism composition is a function composition.

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Epimorphism

In the category of sets, the epimorphism is the surjective function.

Commutative diagram for epimorphism e and two morphisms f and g.

The diagram commutes (f h = g h) but f != g, because h is not epimorphism.

Monomorphism

In the category of sets, the monomorphism is the injective function.

Commutative diagram for monomorphism m and two morphisms f and g.

Isomorphism

In the category of sets, the isomorphism is the bijective function.

Commutative diagram for isomorphism f.

Terminal, initial and null object

In the category of sets, the terminal object is the singleton set.

There are exists non-unique morphisms with a terminal object domain.

In the category of sets, the initial object is the empty set.

Product

Commutative diagram.

In the category of sets, the product is the cartesian product.

The universal property.

Coproduct

Commutative diagram.

In the category of sets, the coproduct is the disjoint union.

The universal property.

Complement.

Equalizer

Coequalizer

Pullback

Pushout