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Section
Foundations of category theory: categories, functors, natural transformations, limits, adjunctions, and abelian categories.
This section covers the foundations of category theory, providing a unified language for describing mathematical structures and the relationships between them.
A category consists of:
These data satisfy the category axioms (associativity and identity/unit laws); see category axioms.
This abstracts the behavior of functions and their composition.
Let be a category and . A morphism is an element of .
Special cases:
Let be a category. If
are morphisms in , their composition is a morphism
Composition is required to satisfy:
This abstracts composition of functions in set theory.
Let be a category and let be an object of . An identity morphism on is a morphism such that:
(Here is composition.)
Uniqueness: If also satisfies these unit laws, then .
This generalizes the identity function on a set.
Let be a category. A morphism is an isomorphism if there exists a morphism such that
where and are the identity morphisms and is composition.
The morphism is unique if it exists, and is denoted . In this case, and are said to be isomorphic (in ).
Related notions:
Let be a category. A morphism is a monomorphism (or mono) if for every object and all morphisms ,
where denotes composition.
Let be a category and let be a morphism in .
The morphism is an epimorphism (or epi) if it is right-cancellative with respect to composition: for every object of and all morphisms ,
Let be a category.
A subcategory of consists of:
such that:
In this situation, is itself a category, with composition and identities inherited from .
A particularly important case is a full subcategory, where contains all morphisms in between its objects.
Let be a category and a subcategory of .
The subcategory is full if for every pair of objects ,
In words: once you decide which objects to keep, you keep all morphisms between them.
Let be categories.
A (covariant) functor consists of:
such that the following axioms hold:
Let be categories.
A contravariant functor consists of:
such that:
Let be categories, and let be functors.
A natural transformation assigns to every object a morphism
in , such that for every morphism in , the naturality condition
holds (composition in ; see composition).
Let be functors.
A natural isomorphism is a natural transformation such that for every object , the component
is an isomorphism in .
Equivalently, is a natural isomorphism iff there exists a natural transformation such that for all (and hence , ).
Let be categories.
A functor is an equivalence of categories if there exists a functor and natural isomorphisms
In this case is called a quasi-inverse (or weak inverse) of , and we say and are equivalent.
Let be a category and let be an indexing category. A diagram of shape in is a functor
A cone to consists of an object of together with morphisms
such that for every morphism in ,
(using composition).
A limit of is a cone such that for every other cone to , there exists a unique morphism with
One writes (or ). If a limit exists, it is unique up to unique isomorphism.
Let be a category and let be an indexing category. A diagram of shape in is a functor
A cocone from consists of an object of together with morphisms
such that for every morphism in ,
(using composition).
A colimit of is a cocone such that for every other cocone from , there exists a unique morphism with
One writes (or ). If a colimit exists, it is unique up to unique isomorphism.
Let be a category and let be objects of .
A (binary) categorical product of and is a triple consisting of an object and morphisms
such that for every object and every pair of morphisms , , there exists a unique morphism
making the equations
hold (see composition).
In diagram form:
A product is unique up to unique isomorphism: if and are both products of , there is a unique isomorphism compatible with projections.
This is a special case of a limit (the limit of the discrete diagram ).
Let be a category and let be objects of .
A (binary) coproduct of and is a triple consisting of an object and morphisms
such that for every object and every pair of morphisms , , there exists a unique morphism
making
hold (see composition).
Coproducts are unique up to unique isomorphism.
Coproduct is the dual notion to product: a coproduct in is a product in the opposite category . It is a special case of a colimit.
Let be a category and let
be parallel morphisms (same domain and codomain).
A coequalizer of is a morphism such that:
Let be a category and let
be a cospan of morphisms.
A pullback (or fiber product) of is an object together with morphisms
such that
(using composition), and satisfying the following universal property:
For any object and morphisms , with , there exists a unique morphism such that
When it exists, the pullback is unique up to unique isomorphism and is often denoted .
Let be a category and let
be a span of morphisms.
A pushout of is an object together with morphisms
such that
(using composition), and satisfying the following universal property:
For any object and morphisms , with , there exists a unique morphism such that
When it exists, the pushout is unique up to unique isomorphism and is often denoted .
Let be a category such that each hom-class is a set (i.e. is locally small).
A covariant functor is representable if there exists an object and a natural isomorphism
A contravariant functor is representable if there exists an object and a natural isomorphism
where is the opposite category.
The object is called a representing object for . If is representable, then any two representing objects are uniquely isomorphic.
Let be a locally small category. The Yoneda embedding is the functor
defined as follows:
Here is the opposite category and is the functor category of presheaves on .
Let and be functors with an adjunction .
Let and be functors with an adjunction .
An additive category is a category in which morphisms can be added and finite direct sums exist.
A category is preadditive if:
A preadditive category is additive if, in addition:
Kernels are a categorical version of “solutions of ”, defined using universal properties.
Throughout, assume is a category with a zero object (e.g. any additive category), so that for any objects there is a distinguished zero morphism .
Cokernels are dual to kernels, capturing “quotienting out by the image of ” in contexts where that makes sense.
Throughout, assume is a category with a zero object (e.g. any additive category), hence zero morphisms .
An abelian category is a category in which one can do “linear algebra + exact sequences” abstractly.
A category is abelian if:
Exactness is a condition on a composable string of morphisms measuring “no loss, no redundancy” at each stage.
Because “image” behaves best in an abelian category, the standard categorical definition of exact sequences is given in that setting.
Let be abelian categories and let be an additive functor.
The functor is left exact if it preserves finite limits; equivalently (in abelian categories), if it preserves kernels.
A standard “exact sequence” formulation is:
> For every short exact sequence in , >
> the sequence >
> is exact in .
Let be abelian categories and let be an additive functor.
The functor is right exact if it preserves finite colimits; equivalently (in abelian categories), if it preserves cokernels.
A standard “exact sequence” formulation is:
> For every short exact sequence in , >
> the sequence >
> is exact in .
Let be abelian categories and let be an additive functor.
The functor is exact if it preserves short exact sequences: whenever
is exact in , then
is exact in .
A category consists of:
The category axioms are the following two laws.
An abelian category is most often defined as an additive category in which kernels/cokernels exist and monomorphisms/epimorphisms are “normal.” One standard axiom list is:
Let be a category such that each hom-class is a set (i.e. is locally small). Fix an object and a functor .
Write the representable functor
which is a representable functor.
These notes are a scratch space for category-theory diagrams. The mathematical content is intentionally compact; the main point is to exercise the renderer on commutative diagrams, universal properties, adjunction triangles, and string-like composition pictures.