Categories, Functors and Natural Transformations

The word class in this development is an abbreviation for a formula in the language of set theory, as described by The first-order language of set theory: $\in$, $=$, formulas with parameters, and class abbreviations. It is not an additional set. A category may therefore have a definable class of objects and a definable class of morphisms. Quantification over such a category is a schema: each use expands to an ordinary formula of ZFC.

A category is small when its objects and morphisms form sets, and locally small when each hom-collection is a set. The category $\mathbf{Cat}$ used below has small categories as objects; its hom-categories are sets or locally small categories as asserted in the relevant result. We do not form a category $\mathbf{CAT}$ of all large categories. Doing so as though all classes were members of a larger set would conflict with the same size obstruction exhibited for the ordinals by Burali-Forti: there is no set of all ordinals. Class recursion, when used, is only the definable schema licensed by Transfinite recursion.

Under the definable-class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed, every class below is a formula and every function whose domain may be proper-class-sized is a definable class-function schema. When its domain is a set, it is an ordinary set-valued function.

A category $\mathcal C$ consists of a class $\operatorname{Ob}(\mathcal C)$ of objects, a class $\operatorname{Mor}(\mathcal C)$ of morphisms, functions $\operatorname{dom},\operatorname{cod}:\operatorname{Mor}(\mathcal C)\to \operatorname{Ob}(\mathcal C)$, an identity morphism $1_A:A\to A$ for every object $A$, and a composite $g\circ f:A\to C$ whenever $f:A\to B$ and $g:B\to C$.

These data satisfy, whenever the composites are defined,

$h\circ(g\circ f)=(h\circ g)\circ f,\qquad 1_B\circ f=f=f\circ1_A.$

We write $\mathcal C(A,B)$, or $\operatorname{Hom}_{\mathcal C}(A,B)$, for the hom-collection of morphisms with domain $A$ and codomain $B$. The object class is allowed to be empty; then the morphism class is empty and all axioms are vacuous.

Morphisms carry their domain and codomain. A category is often presented the other way round, by saying what the morphisms from $A$ to $B$ are for each pair of objects. When it is, $\operatorname{Mor}(\mathcal C)$ is the disjoint union of those hom-collections: a morphism is a triple $(A,B,f)$ with $f$ in the collection assigned to $(A,B)$, and $\operatorname{dom}$ and $\operatorname{cod}$ are the first two projections, which are then functions in the required sense. This is not a technicality that can be dropped. In this library a function is a set of ordered pairs and does not determine a codomain (A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain), so the empty function is a function $\varnothing\to B$ for every $B$ at once; reading the morphisms of $\mathbf{Set}$ as bare functions would give that one set two different codomains and leave $\operatorname{cod}$ undefined. Every concrete category below whose morphisms are described as structure-preserving maps is to be read with this tagging, and each hom-collection is then in canonical bijection with the corresponding collection of untagged maps, so no size or smallness claim is affected.

Let $\mathcal C$ be a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) under the class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed.

• $\mathcal C$ is small when both $\operatorname{Ob}(\mathcal C)$ and $\operatorname{Mor}(\mathcal C)$ are sets.
• $\mathcal C$ is locally small when every $\mathcal C(A,B)$ is a set.
• $\mathcal C$ is large when it is not small.

A small category is locally small because each hom-collection is a subclass of the set of all morphisms. A large category may or may not be locally small.

A subcategory $\mathcal A$ of a category $\mathcal C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) has a subclass of the objects of $\mathcal C$ and, for each $A,B\in\mathcal A$, a subclass $\mathcal A(A,B)\subseteq\mathcal C(A,B)$. It contains the identities of its objects and is closed under the composition inherited from $\mathcal C$.

The subcategory is full when $\mathcal A(A,B)=\mathcal C(A,B)$ for every pair of its objects. Thus a full subcategory is determined entirely by its objects.

Sets as objects and functions as morphisms form a large locally small category $\mathbf{Set}$.

Groups and group homomorphisms form a large locally small category $\mathbf{Grp}$.

Unital rings and unit-preserving ring homomorphisms form a large locally small category $\mathbf{Ring}$.

For a fixed field $F$, vector spaces over $F$ and linear maps form a large locally small category $\mathbf{Vect}_F$.

For a fixed ring $R$, left $R$-modules and module homomorphisms form a large locally small category $R\text{-}\mathbf{Mod}$.

Topological spaces and continuous maps form a large locally small category $\mathbf{Top}$.

Partially ordered sets and monotone maps form a large locally small category $\mathbf{Poset}$.

Every monoid is a one-object category. Under this identification, the monoid is a group exactly when every morphism is invertible.

A preorder on a set $P$ is a relation $\le$ that is reflexive and transitive. Unlike a partial order (Partial order and partially ordered set), it need not be antisymmetric.

A map $f:P\to Q$ between preorders is monotone when $x\le_P y$ implies $f(x)\le_Q f(y)$. Every partial order is a preorder, and the definition of monotone map agrees with the usual one for partially ordered sets.

A preorder determines a category with at most one morphism between any two objects, and functors between such categories are exactly monotone maps.

For a category $\mathcal C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), the opposite category $\mathcal C^{\mathrm{op}}$ has the same objects and reverses every morphism:

$\mathcal C^{\mathrm{op}}(A,B)=\mathcal C(B,A).$

The identity at $A$ remains $1_A$. If $f^{\mathrm{op}}:B\to A$ and $g^{\mathrm{op}}:C\to B$ correspond to $f:A\to B$ and $g:B\to C$ in $\mathcal C$, define $f^{\mathrm{op}}\circ_{\mathrm{op}}g^{\mathrm{op}}=(g\circ f)^{\mathrm{op}}$. Associativity and the identity laws follow directly from those of $\mathcal C$, so this prescription is a category. Moreover $(\mathcal C^{\mathrm{op}})^{\mathrm{op}}=\mathcal C$ strictly.

A formal theorem derivable solely from the category axioms has a formal dual: reverse every morphism, reverse the order of every composite, and exchange each defined notion with its opposite-category version.

In a category $\mathcal C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), a morphism $f:A\to B$ is an isomorphism if there is a morphism $g:B\to A$ with $g\circ f=1_A$ and $f\circ g=1_B$. Such a $g$ is unique and is denoted $f^{-1}$.

A groupoid is a category in which every morphism is an isomorphism. A category is connected when it is nonempty and any two objects can be joined by a finite zigzag of morphisms, with successive arrows allowed to point in either direction. In a groupoid this is equivalent to being nonempty and having an isomorphism between every ordered pair of objects. Nonemptiness cannot be dropped: the empty groupoid has an isomorphism between every ordered pair of its objects, vacuously, and is not connected.

For every category $\mathcal C$, the subcategory with all objects and exactly the isomorphisms as morphisms is a groupoid, and it contains every subgroupoid of $\mathcal C$.

For a morphism $f:A\to B$, the following are equivalent: $f$ is an isomorphism; for every object $X$, postcomposition $f\circ-:\mathcal C(X,A)\to\mathcal C(X,B)$ is bijective; and for every $X$, precomposition $-\circ f:\mathcal C(B,X)\to\mathcal C(A,X)$ is bijective.

Let $f:A\to B$ be a morphism in a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).

• $f$ is a monomorphism, or monic, when $f\circ g=f\circ h$ implies $g=h$ for every parallel pair $g,h:X\to A$.
• $f$ is an epimorphism, or epic, when $g\circ f=h\circ f$ implies $g=h$ for every parallel pair $g,h:B\to Y$.

Thus monomorphisms are left-cancellable and epimorphisms are right-cancellable. The definitions are formally dual.

Let $f:A\to B$ be a morphism.

If there is $r:B\to A$ with $r\circ f=1_A$, then $f$ is a split monomorphism and a section, while $r$ is a retraction of $f$. If there is $s:B\to A$ with $f\circ s=1_B$, then $f$ is a split epimorphism and a retraction, while $s$ is a section of $f$.

When both identities hold, $f$ and its splitting are inverse isomorphisms in the sense of Isomorphism, groupoid, and connected category.

Identity morphisms are monic and epic; composites of monomorphisms are monic, composites of epimorphisms are epic; every split monomorphism is monic, and every split epimorphism is epic.

In $\mathbf{Set}$, a function is monic exactly when it is injective, and it is epic exactly when it is surjective.

Assume the Axiom of Choice. Every epimorphism in $\mathbf{Set}$ is a split epimorphism.

In $\mathbf{Ring}$, the canonical inclusion $i:\mathbb Z\hookrightarrow\mathbb Q$ is monic and epic, but its underlying function is not surjective and it is not an isomorphism.

In a category $\mathcal C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), an object $I$ is initial when for every object $A$ there is exactly one morphism $I\to A$. An object $T$ is terminal when for every $A$ there is exactly one morphism $A\to T$. These notions are dual under Every theorem about categories has a formal dual obtained by reversing morphisms and composition.

A zero object is an object that is both initial and terminal. The definition does not choose a particular zero object when several distinct but isomorphic ones occur.

A category with zero morphisms is a category $\mathcal C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) equipped with a specified morphism $0_{A,B}:A\to B$ for every ordered pair of objects such that, for all $f:A'\to A$ and $g:B\to B'$,

$0_{A,B}\circ f=0_{A',B},\qquad g\circ0_{A,B}=0_{A,B'}.$

The family is part of the structure. Merely naming an isolated morphism as zero does not supply zero morphisms.

A chosen zero object $0$ in a category supplies a unique compatible system of zero morphisms.

For categories $\mathcal C,\mathcal D$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), a covariant functor $F:\mathcal C\to\mathcal D$ assigns an object $FA$ to every object $A$ and a morphism $Ff:FA\to FB$ to every $f:A\to B$, satisfying

$F(1_A)=1_{FA},\qquad F(g\circ f)=Fg\circ Ff.$

The identity functor acts identically on objects and morphisms. For $F:\mathcal C\to\mathcal D$ and $G:\mathcal D\to\mathcal E$, the composite functor $GF$ has $(GF)A=G(FA)$ and $(GF)f=G(Ff)$.

A contravariant functor from $\mathcal C$ to $\mathcal D$ means a covariant functor $\mathcal C^{\mathrm{op}}\to\mathcal D$, using Opposite category $\mathcal C^{\mathrm{op}}$.

Every functor sends isomorphisms to isomorphisms.

Let $F:\mathcal C\to\mathcal D$ be a functor (Covariant functor, identity functor, composite functor, and contravariant functor). For each $A,B$, it induces

$F_{A,B}:\mathcal C(A,B)\to\mathcal D(FA,FB),\qquad f\mapsto Ff.$

The functor is faithful when every $F_{A,B}$ is injective, full when every $F_{A,B}$ is surjective, and fully faithful when every $F_{A,B}$ is bijective (Injection, surjection, bijection).

It is essentially surjective when every object $D$ of $\mathcal D$ is isomorphic to some $FC$ (Isomorphism, groupoid, and connected category). It is split essentially surjective when the data include, for every $D$, a specified object $C_D$ and specified isomorphism $\varepsilon_D:FC_D\to D$. The word split records these witnesses and is strictly stronger as data than their mere existence.

An embedding of categories is a functor that is faithful and injective on objects. A full embedding is an embedding that is also full. These terms use the notions of Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors.

Thus a fully faithful functor need not be an embedding: it may send distinct but isomorphic objects to the same object. Conversely, an embedding need not be full.

If $F:\mathcal C\to\mathcal D$ is fully faithful and $F(f)$ is an isomorphism, then $f$ is an isomorphism.

Let $F,G:\mathcal C\to\mathcal D$ be functors (Covariant functor, identity functor, composite functor, and contravariant functor). A natural transformation $\alpha:F\Rightarrow G$ is a family of morphisms $\alpha_A:FA\to GA$, one for each object $A$ of $\mathcal C$, such that every $f:A\to B$ satisfies the naturality equation

$Gf\circ\alpha_A=\alpha_B\circ Ff.$

The morphism $\alpha_A$ is the component of $\alpha$ at $A$.

For a functor $F:\mathcal C\to\mathcal D$, the identity natural transformation $1_F:F\Rightarrow F$ has component $(1_F)_A=1_{FA}$.

For natural transformations $\alpha:F\Rightarrow G$ and $\beta:G\Rightarrow H$ (Natural transformation and its components), their vertical composite $\beta\circ\alpha:F\Rightarrow H$ is defined componentwise by

$(\beta\circ\alpha)_A=\beta_A\circ\alpha_A.$

The fact that this componentwise family is natural is discharged by Vertical composites of natural transformations satisfy naturality ↗.

The vertical composite of two natural transformations satisfies the naturality equation.

Let $\alpha:F\Rightarrow G:\mathcal C\to\mathcal D$ be a natural transformation (Natural transformation and its components), and let $H:\mathcal D\to\mathcal E$ and $K:\mathcal B\to\mathcal C$ be functors (Covariant functor, identity functor, composite functor, and contravariant functor). The left whiskering $H\alpha:HF\Rightarrow HG$ has components $H(\alpha_A)$, and the right whiskering $\alpha K:FK\Rightarrow GK$ has components $\alpha_{KB}$.

For $\beta:H\Rightarrow L:\mathcal D\to\mathcal E$, the horizontal composite $\beta*\alpha:HF\Rightarrow LG$ has either equal component formula

$(\beta*\alpha)_A=\beta_{GA}\circ H(\alpha_A)=L(\alpha_A)\circ\beta_{FA}.$

Their equality is the naturality equation for $\beta$ at $\alpha_A$. Naturality of the resulting family is proved in Horizontal composites of natural transformations satisfy naturality ↗.

The horizontal composite of natural transformations satisfies the naturality equation.

Whenever the expressions are defined,

$(\beta'\circ\beta)*(\alpha'\circ\alpha)=(\beta'*\alpha')\circ(\beta*\alpha).$

Thus horizontal and vertical composition of natural transformations satisfy the interchange law.

For categories $\mathcal C$ and $\mathcal D$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), their product category $\mathcal C\times\mathcal D$ has objects $(C,D)$, morphisms $(f,g):(C,D)\to(C',D')$, componentwise identities, and componentwise composition

$(f',g')\circ(f,g)=(f'\circ f,g'\circ g).$

The category axioms hold componentwise. The projection functors $\pi_{\mathcal C}$ and $\pi_{\mathcal D}$ send an object or morphism to its corresponding component, and satisfy the functor laws of Covariant functor, identity functor, composite functor, and contravariant functor componentwise.

Within the definable-class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed, the construction below is formed when the source category $\mathcal C$ is small. Then a functor out of $\mathcal C$ and a natural transformation between two such functors are set-coded data, so they can be objects and morphisms of a category in ZFC.

For categories $\mathcal C,\mathcal D$, the functor category $[\mathcal C,\mathcal D]$ has functors $\mathcal C\to\mathcal D$ as objects and natural transformations as morphisms (Natural transformation and its components). Its identities and composition are the identity transformations and vertical composition. These are the operations of Identity natural transformation and vertical composition.

Closure under composition is Vertical composites of natural transformations satisfy naturality. Associativity and the identity laws hold at each component because they hold in $\mathcal D$. Further smallness and local-smallness properties of this category are stated separately.

For an arbitrary large source $\mathcal C$, the same notation may be used only as metatheoretic shorthand for functors and natural transformations; this definition does not form those proper-class-sized data into a category.