Joyal's CatLab Categories

Contents

Categories

Definition

A category $\mathbf{C}$ consists of

• a set $\mathbf{C}_0$ of [objects];

• for each pair of objects $A,B\in \mathbf{C}_0$, a set $\mathbf{C}(A,B)$ of [morphisms] (or [arrows], or maps) $A\to B$ with [domain] $A$ and [codomain] $B$ (or with [source] $A$ and [target] $B$); when the context is clear, we write $f:A\to B$ or write,

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

• for each triples of objects $A,B,C\in \mathbf{C}_0$, a [composition law], $\mathbf{C}(B,C)\times \mathbf{C}(A,B)\to \mathbf{C}(A,C);$
  when the context is clear, we write $g f:A\to C$ for the composite of $f:A\to B$ with $g:B\to C$;

• for each object $A\in \mathbf{C}_0$, a morphism $1_A:A\to A$ called the [unit morphism] or the [identity morphism] of $A$;

• such that the following conditions are satisfied:

  • ([associativity law]) $h (g f)=(h g) f$ for every triple of morphisms $f:A\to B$, $g:B\to C$ and $h:C\to D$;

  • ([the unit law]) $1_B f=f 1_A =f$ for every morphism $f:A\to B$.

=–

Notation

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

Size issues

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

Examples

• We shall denote by $\mathbf{Set}$, the category of small sets: an object of this category is a small set $S$ and a morphism is a function $f:S\to T$; the composition law is the usual composition of functions, and the unit morphisms are the identity functions. The category $\mathbf{Set}$ is large but locally small.

• Let $R$ be a ring. We shall denote by $Mat(R)$ the category of matrices with coefficients in $R$; an object of this category is a natural number $n\geq 0$ and a morphism $m\to n$ is a $n\times m$ matrix with coefficients in $R$; the composite of a matrix $A:m\to n$ with a matrix $B:n\to p$ is their product $B A:m\to p$; the unit of $n$ is the $n\times n$ unit matrix. The category $Mat(R)$ is small.

• $\mathbf{Grp}$ - groups? as objects, group homomorphisms as morphisms.

• $\mathbf{Ab}$- abelian groups? as objects, group homomorphisms as morphisms.

• $\mathbf{Ring}$ - rings? as objects, ring homomorphisms as morphisms.

• $K\mathbf{Vect}$ - K-vector spaces? as objects, $K$-linear maps as morphisms.

• $\mathbf{Top}$ - topological spaces? as objects, continuous maps as morphisms.

• $\mathbf{SMan}$ - smooth manifolds? as objects, smooth maps as morphisms.

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).

• Poset A poset? can be thought of as a category with its elements as objects and one morphism in each $hom(x,y)$ if $x$ is less than or equal to $y$, but none otherwise.

• Monoid A monoid $M$ can be thought of as a category $C$ with a single object $\star$ and with $C(\star,\star)=M$. It is sometime useful to distinguish between a monoid $M$ and the associated category with a single object by denoting the latter $\mathbf{B}M$. The category $\mathbf{B}M$ is often called the classifying category of $M$.

• The simplicial category $\Delta$ has objects the ordered sets $[n]=\{0,\ldots,n\}$ for $n\geq 0$, and for morphisms the order preserving maps $[n]\to [m]$.

Opposite category

• The opposite of a category $\mathbf{C}$ is the category $\mathbf{C}^o$ obtained by formally reversing the direction of the arrows of $\mathbf{C}$. It is sometimes useful to distinguish between the objects of $\mathbf{C}$ and of $\mathbf{C}^o$ by denoting the object of $\mathbf{C}^o$ which corresponds to an object $A\in \mathbf{C}$ by $A^o\in \mathbf{C}^o$ and by denoting the arrow which corresponds to an arrow $f\in \mathbf{C}(A,B)$ by $f^o\in \mathbf{C}^o(B^o,A^o)$. If $f\in \mathbf{C}(A,B)$ and $g\in \mathbf{C}(B,C)$, then the composite of $g^o\in \mathbf{C}^o(C^o,B^o)$ with $f^o\in \mathbf{C}^o(B^o,A^o)$ is defined by putting $f^o g^o= (g f)^o$,

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

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

Special morphisms

Isomorphisms

• Let $f:A\to B$ and $g:B\to A$ be two morphisms in a category. If $g f=1_A$ and $f g=1_B$ we say that $g$ is the inverse of $f$. The inverse is unique when it exists and we write $g=f^{-1}$.
  A morphism which admits an inverse is said to be [invertible] , or to be an [isomorphism].

• A category $\mathbf{C}$ is said to be a groupoid if every morphism of $\mathbf{C}$ is invertible.

Endomorphisms

• An endomorphism of an object $A$ is a morphism $f:A\to A$. The composition law gives the set of endomorphisms $End(A)=hom(A,A)$ the structure of a monoid, the monoid of endomorphisms of $A$.

• An automorphism of an object $A$ is an endomorphism of $A$ which is invertible. The composition law gives the set of automorphisms of $A$ the structure of a group, the group of automorphisms of $A$.

• An endomorphism $e:A\to A$ is said to be idempotent if $e e=e$.

Monomorphisms

• A morphism $f:A\to B$ in a category is said to be [monic] , or to be a [monomorphism] if the implication

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

Epimorphisms

• A morphism $f:A\to B$ in a category is said to be epic , or to be an epimorphism if the implication

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

Sections, retractions

• Let $f:A\to B$ and $g:B\to C$ be two morphisms in a category. If $g f =1_A$, we say that $g$ is a left inverse of $f$ and that $f$ is a right inverse of $g$. A morphism which admits a left inverse is monic, it is called a split monomorphism. A morphism which admits a right inverse is epic, it is called a split epimorphism. The notion of split epimorphism is dual to the notion of split monomorphism. A left inverse of a morphism $f$ is sometimes called a retraction of $f$, and a right inverse a section.

• If $f:A\to B$, $g:B\to A$ and $g f=1_A$, then the composite $f g:B\to B$ is idempotent.

• An idempotent $e:B\to B$ is said to be split if there exists an object $A$ together with a pair of morphism $f:A\to B$ and $g:B\to A$ such that $g f=1_A$ and $f g=e$.

Examples

• A map $f:S\to T$ in the category of sets $\mathbf{Set}$ is invertible iff it is bijective.

• An isomorphism in the category of topological spaces $\mathbf{Top}$ is said to be a homeomorphism.

• An isomorphism in the category of smooth manifolds $\mathbf{SMan}$ is said to be a diffeomorphism.

• The isomorphisms in a category $\mathbf{C}$ form a groupoid $Iso(\mathbf{C})$ with the same objects as $\mathbf{C}$. It is the groupoid of isomorphisms of $\mathbf{C}$.

• An automorphism of a set is often called a permutation. The group of permutations of the set ${\underline n}=\{1,\ldots, n\}$ is the symmetric group? $\Sigma_n$ of degree $n$.

• Let $R$ be a ring. In the category of $Mat(R)$, the group of automorphisms of the object $n$ is the general linear group? $GL(n,R)$.

• In the category of sets $\mathbf{Set}$, a map is monic iff it is injective, and it is epic iff it is surjective.

• In a poset, every morphism is monic and epic.

Groupoids

• A [groupoid] is a category in which every morphism is invertible.

Examples

• Every type of mathematical structure carries with it a notion of isomorphism. Hence the collection of all structures of a given type form a groupoid.

• The isomorphisms in a category $\mathbf{C}$ form a groupoid $Iso(\mathbf{C})$ having the same objects as $\mathbf{C}$, the groupoid of isomorphisms of $\mathbf{C}$.

• A group is essentially a groupoid with a single object. The classifying category of a group $G$ is a groupoid with a single object $\mathbf{B}G$. In fact, groupoids are the many object version of groups.

Isomorphism classes

• Two objects $A$ and $B$ of a category $\mathbf{C}$ are said to be isomorphic, $A\simeq B$, if there exists an isomorphism $A\to B$. The relation $A\simeq B$ is symmetric, transitive and reflexive. It is thus an equivalence relation between the objects of $\mathbf{C}$. An equivalence class for this equivalence relation is said to be an isomorphism class, or an isomorphism type.

• In the category of sets $\mathbf{Set}$, two sets $X$ and $Y$ are isomorphic iff they have the same cardinality. In fact, the notion of cardinal number as defined by Cantor is an isomorphism class of sets.

• Let us denote by $\mathbf{Ord}$ the category of ordered sets and order preserving maps. Two ordered sets $X$ and $Y$ are said to have the same order type if they are isomorphic in the category $\mathbf{Ord}$. An ordinal number is defined to be the order type of a well-ordered set.

• Let us denote by $K\mathbf{Vect}$ the category of (left or right) vector spaces over a field $K$. Every $K$-vector space $V$ has a basis, and any two basis have the same cardinality. The dimension of $V$ is defined to be the cardinality of a basis. Two objects of $K\mathbf{Vect}$ are isomorphic iff they the same dimension.

• Let us denote by $K\mathcal {F}$ the category of algebraically closed field extensions of a commutative field $K$. Every algebraically closed field extension $F$ of $K$ has a transcendance basis over $K$, and any two basis have the same cardinality. The transcendance dimension of $F$ over $K$ is defined to be the cardinality of a basis. Two objects of $K\mathcal {F}$ are isomorphic iff they have the same transcendance dimension over $K$.

Exercises

• Prove that a morphism which is left and right invertible is invertible.

• Prove that the inverse of a morphism is unique when it exists.

• Show that the composite of two isomorphisms $f: A \to B$ and $g:B \to C$ is an isomorphism and that we have $(g f)^{-1}=f^{-1}g^{-1}$.

• Show that a category is a groupoid iff every morphism has a left inverse.

• Show that the composite of two monomorphisms (resp. epimorphisms) is a monomorphism (resp. an epimorphism).

• Show that if the composite $g f$ of two morphisms $f:A\to B$ and $g:B\to C$ is monic, then then so is $f$. Dually, show that if $g f$ is epic, then so is $g$.

• Show that a morphism which is right invertible is epic, and that a morphism which is left invertible is monic.

• Give an example of a morphism in a category which is both monic and epic without been invertible.

• Show that if a monomorphism has a right inverse, then it is invertible. Dually, show that if an epimorphism has a left inverse, then it is invertible.

• Show that if an idempotent is monic or epic, then it is a unit.