Joyal's CatLab Categories


Basic category theory

Category theory

Category theory




A category C\mathbf{C} consists of

to indicate that fC(A,B)f\in \mathbf{C}(A,B);



The set C 0\mathbf{C}_0 of objects of a category C\mathbf{C} is often denoted Ob(C)Ob(\mathbf{C}). When the context is clear, we shall often write ACA\in \mathbf{C} instead of AOb(C)A\in Ob(\mathbf{C}); we shall often denote the set C(A,B)\mathbf{C}(A,B) by hom C(A,B)hom_{\mathbf{C}}(A,B), and more simply by hom(A,B)hom(A,B) (hom(A,B)hom(A,B) is the set of homomorphisms ABA\to B). We often use diagrams in category theory. We say that a triangle of arrows commutes if h=gfh=g f, and that a square of arrows commutes if gu=vfg u=v f.

Size issues

The sets involved in the definition of a category can be small? or large?. A category C\mathbf{C} is said to be locally small if the set C(A,B)\mathbf{C}(A,B) is small for every pair of objects A,BCA,B\in \mathbf{C}. A locally small category C\mathbf{C} is said to be [small] if the Ob(C)Ob(\mathbf{C}) is small. A category which is not small is said to be large.


These classical examples are the original motivation for the term “category”: all of the above categories encapsulate one “kind of mathematical structure”. These are often called “concrete” categories (that term also has a technical definition? that these examples all satisfy).

Opposite category

By convention, we shall put (A o) o=A(A^o)^o=A and (f o) o=f(f^o)^o=f for every object ACA\in \mathbf{C} and arrow fCf\in \mathbf{C}. This means that we have

(C o) o=C.(\mathbf{C}^o)^o=\mathbf{C}.

The fact that every category has an opposite is the basis of a fundamental duality principle of category theory. To every statement P(C)P(\mathbf{C}) about the objects and arrows a general category C\mathbf{C} corresponds a dual statement P o(C)P(C o)P^o(\mathbf{C})\equiv P(\mathbf{C}^o) about the objects and arrows of the opposite category. This is analogous to the duality in projective geometrie: to every statement PP about the points and lines of a projective plane Π\Pi corresponds a dual statement P *P^* about the points and lines of the dual projective plane Π *\Pi^*.

Special morphisms




fu=fvu=vf u = f v \Rightarrow u=v

is true for every pair of morphism u,v:ABu,v:A\to B.


uf=vfu=vu f =v f \Rightarrow u=v

is true for every pair of morphism u,v:BCu,v:B\to C. A morphism f:ABf:A\to B is epic iff the opposite morphism f o:B oA of^o:B^o\to A^o is monic. Hence the notion of epimorphism is dual to the notion of monomorphism.

Sections, retractions




Isomorphism classes


Revised on November 20, 2020 at 21:33:37 by Dmitri Pavlov