Category Theory/Printable version


Category Theory

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Definition, examples

Definition (category):

A category   is a class of objects, together with a class of so-called morphisms, each of which have a domain and a target, and a composition of morphisms, such that the following set of axioms hold, if for any two objects   and   of   the subclass of morphisms with domain   and target   is denoted  :

  1. Whenever either   or  ,   and   are disjoint
  2. For any objects   of   and any morphisms   and  , there exists a morphism  , called the composition of   and  
  3. Composition is associative, ie.  
  4. Whenever   is an object of  , then there exists a unique morphism   that acts as an identity both on the left and on the right for the composition of morphisms.

Exercises

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  1. If both   and   are set mappings such that   is injective, prove that   is injective.


Natural transformations

Definition (natural transformation):

Let   be categories, and let   be two functors. A natural transformation between these two functors is a collection of morphisms of  , one for each object   of  , namely  , such that for all morphisms   of  , the following diagram commutes:

diagram placeholder

Old content

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One further basic notion in the theory of categories, or, as it may be said, a basic item of categorical language, will now be introduced. This is the notion of a natural transformation of functors from one category to another. Indeed, the whole language and apparatus of categories and functors were developed initially by the U.S. mathematicians Samuel Eilenberg and Saunders MacLane in order to render precise the intuitive concept of naturality. First an example will be given, the example that may be said to have motivated the definition.

Motivating example

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Let   be a vector space over some field  , and let   be the dual space of  ; that is, the space of linear functionals on  . There is then a linear transformation   that is given. There is an intuitive feeling that the linear transformation   is natural because its description only involves the terms u and f. Now if   is finite-dimensional, then it is known that   is an isomorphism in the category of vector spaces over   and linear transformations. One way of proving that   is then an isomorphism is to show that   and   are isomorphic and then to observe that   is one–one. The usual proof that   and   are isomorphic would be to proceed by establishing an isomorphism between   and  , in the case when   is finite-dimensional. Now if a base   for   is given, then a basis for   may be set up, called the dual basis, by defining   to be that linear functional on   given by certain rules (see 350). Then the correspondence   sets up an isomorphism between   and  .

On the other hand, this isomorphism does not look natural, because it depends on the choice of bases. Of course, the argument above could be generalized to set up a linear transformation from   to  * even if   is not finite-dimensional over  , but, again, this transformation would not appear to be natural. What is required is a formal and precise expression of the feeling that, for finite-dimensional vector spaces   over the field  ,   and   are naturally isomorphic, while   and   are isomorphic in some unnatural way. Eilenberg and MacLane solved this problem in their seminal article, “General Theory of Natural Equivalences,” published in 1945, which laid the foundation of the theory of categories.

Duality

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For any category   a new category   can be formed by interchanging the domains and codomains of the morphisms of  . More precisely, in the category   the objects are simply those of   and the effect of interchange of domains is expressed in an equation (see 359). Moreover, the composition in   is simply that of  , suitably interpreted.   is called the category opposite to  ; notice that   =  . This apparently trivial operation leads to highly significant results when specific categories are used. In the general setting it enables any concept in the language of categories to be dualized. For example, the coproduct in   is simply the product in  . Any theorem that holds in an arbitrary category has a dual form. For example, the theorem asserting that the product in an arbitrary category is associative may be effectively restated as asserting that the coproduct in an arbitrary category is associative. In the special cases, however, the second statement looks very different from the first. For example, in the category of sets, the coproduct becomes the disjoint union; in the category of groups it is the free product; and in a pre-ordered set regarded as a category, the coproduct is the least upper bound. In particular, for the set of natural numbers, ordered by divisibility, the coproduct is the LCM. Thus, the same universal argument that led to the deduction that the GCD is associative also indicates that the LCM is associative. The duality principle has very wide ramifications indeed. Here it is merely noted that the important concept of a contravariant functor   may be most simply defined as a functor  . Thus the association of the dual vector space   with   yields a contravariant functor from   to   itself.

Definition of natural transformation

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Suppose  ,   are two functors, both from the category   to the category  .

Then a natural transformation   is a rule that assigns to each object A of category   a morphism  .

The morphisms  involved must be subject to the condition that the diagram

 

should be commutative for every   (note   is a morphism in the category  ); that is,   (note the commutative diagram is drawn in category  ).

Natural isomorphisms

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Further, if   is invertible for each A, then   is said to be natural isomorphism (or a natural equivalence). It is clear that if   is a natural equivalence from functors   to  , then  , given by an equation (see 352), is a natural equivalence from functors   to  . Thus the term equivalence used here is fully justified. Indeed, the functors from   to   may be collected into equivalence classes according to the existence of a natural equivalence between them.

Examples

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This definition can be tested against the example. Consider two functors from K-Vect to K-Vect, in which K-Vect is the category of vector spaces over the field K and linear transformations. One functor is the identity functor. The other functor is the double dual functor ** that associates with every vector space V its double dual V** and with every linear transformation f : UV in the linear transformation f**: U** → V** (see 353). A linear transformation T: VV** was described above. It is easy to check that T is a natural transformation from the identity functor to the functor **. If the subcategory f of K-Vect that consists of finite-dimensional vector spaces over K and their linear transformations is considered, then it turns out that the functor ** transforms f into itself; and the natural transformation T, restricted to f, is then a natural equivalence. Further examples of natural transformations of functors can be given:

  • The category of Abelian groups and homomorphisms is considered. With every Abelian group may be associated its torsion subgroup. The torsion subgroup AT of the Abelian group A consists of those elements of A that are of finite order. A homomorphism from A to B must necessarily send AT to BT. Thus a functor f is obtained from to (or to T, the category of torsion Abelian groups and their homomorphisms), by associating with every Abelian group A the Abelian group FA = AT. Now AT is a subgroup of A. Thus there is always an embedding iA of AT in A. It is easy to see that i is a natural transformation from the torsion functor f to the identity functor. Further, the quotient group Afr = A/AT may be considered. It is a torsion-free Abelian group. This gives a functor g from to (or from to fr, the category of torsion-free Abelian groups) by associating with the Abelian group A the Abelian group GA = Afr. Then the projection of A onto Afr yields a natural transformation from the identity functor to the torsion-free functor g.
  • With every group may be associated its commutator subgroup. It is then not difficult to see that the embedding of the commutator subgroup in the group is a natural transformation from the commutator subgroup functor to the identity functor. On the other hand, the centre of every group may be associated with the group. Here, however, there is not a functor because a homomorphism from one group to another does not necessarily map the centre of the first group to the centre of the second. On the other hand, if the category of groups and surjective homomorphisms (a surjective homomorphism is one in which the image coincides with the codomain) is considered, then in this category the centre is a functor. It is a functor, however, from the category of groups and surjective homomorphisms to the category of groups and all homomorphisms, because a surjective homomorphism does not necessarily map the centre surjectively. Then the embedding of the centre of a group in the group may be regarded as a natural transformation from the centre functor to the inclusion functor, both of which are functors from the category of groups and surjective homomorphisms to the category of groups and homomorphisms.
  • In algebraic topology, the singular homology groups and the homotopy groups of a pointed topological space (X, x) are considered. A Hurewicz homomorphism (see 354) exists, from the homotopy groups to the homology groups. Then pn and hn, n ≥ 2, are functors from the category of pointed spaces and pointed continuous functions to the category of Abelian groups, and the Hurewicz homomorphism is a natural transformation of functors.

The Yoneda lemma

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Let   be a locally small category, let   be an object in  , let  , let   denote the covariant Hom functor, and let   denote the natural transformations from   to  . Then  . In addition, if   is another Hom functor  , then  .


(Co-)cones and (co-)limits

Definition (cone):

Let   be a category and let   be a diagram in  . A cone over   is an object   of  , together with morphisms   for each  , so that for each   (so that  ) we have

 .

Definition (limit):

Let   be a category, let   be a diagram in   and let

Definition (diagonal):

Let   be a category, and let   be an object of  . Further, suppose that the object   and the morphisms   and   constitute a product of   with itself in  . Then the diagonal (also called "diagonal morphism") is the unique morphism

 

such that  .

Definition (anti-diagonal):

Let   be a category, and let   be an object of  . Further, suppose that the object   and the morphisms   and   constitute a coproduct of   with itself in  . Then the anti-diagonal (also called "anti-diagonal morphism") is the unique morphism

 

such that  .

Exercises

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    1. Let   be a category such that every two objects of   have a product. Suppose further that   is another category, and that   is a functor. Let   be an object of  . Use the universal property of the product in order to show that there exists a functor   that sends an object   of   to the object   of  .
    2. Prove that any morphism   in   gives rise to a natural transformation  .
    3. Can we weaken the assumption that every two objects of   have a product? (Hint: Consider the image of the class function on objects associated to the functor  .)


Adjoint functors

Definition (adjoint functors):

Let   be categories. A pair of adjoint functors consists of two functors   and   (where   is the left adjoint and   is the right adjoint) such that the two bifunctors

  and  

from   to   are naturally isomorphic to each other.

Proposition (left adjoint functors preserve epimorphisms):

Let   be categories, and let   and   be an adjoint pair of functors. Suppose that   and   is an epimorphism. Then   is also an epimorphism.

Proof: Let   be arrows in   so that  .  



Proposition (right adjoint functors preserve monomorphisms):


Subcategories

Definition (subcategory):

Let   be a category. Then a subcategory   of   is a category such that   and  .

Definition (full):

A subcategory   of a category   is called full iff for all  , we have

 .

Proposition (limits are preserved when restricting to a full subcategory):

Let   be a category, let   be a diagram in  , and let   be a full subcategory of  . Suppose that   is a limit over   in   such that   and all targets of the   are in  . Then   is a limit over   in  .

Proof: Certainly, the underlying cone of   is contained within  , because the subcategory is full. Now let another cone   in   over the diagram   (which, analogously, is a diagram in  ) be given. By the universal property of   in  , there exists a unique morphism   which satisfies   for all  . Since   is full,   is in  .  

Analogously, we have:

Proposition (colimits are preserved when restricting to a full subcategory):

Let   be a category, let   be a diagram in  , and let   be a full subcategory of  . Suppose that   is a colimit over   in   such that   and all domains of the   are in  . Then   is a colimit over   in  .

Proof: This follows from its "dual" proposition, reversing all arrows in its statement and proof except the direction of the diagram functor.  


Categories of ordered sets

Definition (category of preordered sets):

The category   of preordered sets is the category whose objects   are given by the preordered sets and whose morphisms are given by the order homomorphisms.

Definition (category of posets):

The category   of posets is the category whose objects are the posets and whose morphisms are the order homomorphisms.

Directly from the definitions, we have:

Proposition (posets form full subcategory of preordered sets):

  is a full subcategory of  .

Proposition (products in the category of preordered sets):

Let   be a family of preordered sets. Then a product of this family in the category   is given by the set   together with the product order, where the projections are given by the functions

 .

Proof: The functions   are order homomorphisms, because if   and  , then   by definition. Thus, we have a cone. Moreover, if the order homomorphisms   define another cone, then as for the set product, the function given by

 

is the unique function from   to   such that   for all  , and it is an order homomorphism, because if  , then for all   we have  .  

Proposition (coproducts in the category of preordered sets):

Let   be a family of preordered sets. Then a coproduct of this family in the category   is given by the parallel composition of the  , where the inclusions are given by the functions

 .

Proof: The   are order homomorphisms by definition of the parallel order, so that we do have a cocone. Suppose now that the maps   define another cocone in the category  . Then the unique function   such that   for all   is given by

 

as in set theory, and it is an order homomorphism because if  , then   by def. of the parallel order and consequently  .  

Proposition (products and coproducts in the category of posets):

In the category  , products and coproducts are given by the respective products and coproducts in the category  .

Proof: This follows since limits and colimits are preserved when restricting to a full subcategory.  


Additive categories

Definition (biproduct):

Let   be a category, and let   be a family of objects in  . A biproduct of   is an object of   that is usually denoted as

 

and for which there exist arrows

  and  

for all   that have the following properties:

  1.   and  
  2.  , together with the morphisms  , constitutes a coproduct in the category  
  3.  , together with the morphisms  , constitutes a product in the category  

Definition (additive category):

An additive category is a category   that satisfies each of the following requirements:

  1. Every morphism in   has a kernel and a cokernel
  2. For every two objects   of  , there exists a biproduct  
  3. For every two objects   of  , the assignment  , where   is the morphism that arises from postcomposing the morphism   (where   shall denote the diagonal) with the anti-diagonal  , turns   into an abelian group


Abelian categories

Proposition (object in abelian category is decomposed into sum by subobject):

Let   be an abelian category, and let   be an object. Let   be a subobject, and let   be the corresponding quotient object. Moreover, denote  . Then there exists a unique isomorphism   such that

  and  .

Proof:   is a biproduct. First we apply the universal property of a product in order to obtain a morphism

  such that   and  .

Then we apply the universal property of a coproduct in order to obtain a morphism

  such that   and  .

Moreover, we get a morphism   from the projection to  , and a morphism   from the inclusion of  . The latter morphism is the kernel of  , and the cokernel of that kernel is  


Categories

This is the Categories chapter of Category Theory.

Definition

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A category   consists of four kinds of data subject to three axioms, as listed below:

Data

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Objects
  has objects denoted by  ,  ,  ,…
Morphisms
For each ordered pair of objects  ,   in  , there is a class of morphisms or arrows from   to  . The notation   means that   is a morphism from   to  . The class of all morphisms from   to   is denoted by   or sometimes simply  .
Composition
For each ordered triple of objects  ,  ,   in  , there is a law of composition: If   and  , then the composite of   and   is a morphism  
Identity
For each object   there is a designated identity morphism on  , notated as  , from   to  .

Axioms

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These data satisfy the following three axioms, of which the first is in the nature of a convention, while the remaining two are more substantial:

Unique typing
  and   are disjoint unless  ,  .
Associative Law
  if the composites are defined. Note that if one composite is defined, the other is necessarily defined.
Identity is a “neutral element”
For the identity morphism   associated to each object  , two equations must hold for each pair of objects   and   and each pair of arrows  ,  :
  •  
  •  

Terminology and fine points

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  • If   in a category,   is called the domain or source of  , and   is called the codomain or target of  .
  •   is called a hom class (or a hom set if it is indeed a set). In general a hom set may be empty, but for any object  ,   is not empty because it contains the identity morphism.
  • The hom class   may be denoted by   or   if it is necessary to specify which category is referred to.
  • An object in a category need not be a set; the object need not have anything called elements.
  • Morphisms may also be called maps. This does not mean that every morphism in any category is a set function (see #Baby examples and #Preorders). Arrow is a less misleading name.
  • The composite   may be written  .
  • It might be more natural to write the composite of   and   as   instead of   but the usage given here is by far the most common. This stems from the fact that if the arrows are set functions and  , then  . Thus   is best read as "do   after  ".

Large and small

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The definition says that a category 'has' objects and 'has' morphisms. This means that for any category   and   is any mathematical object, the statement '  is an object of  ' is either true or false, and similarly for the statement '  is a morphism of  '. The objects (or arrows) of a category need not constitute a set. If they do, the category is said to be small. If they don't, the category is large.

The requirement that the collection of morphisms from   to   be a set makes a category locally small. In this book, all categories are locally small.

Discrete

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A category is discrete if every morphism is an identity.

Preorder

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A category   is a preorder if for every pair of objects  , there exists at most one morphism  .

Examples of categories

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Baby examples

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These examples are trivial and maybe uninteresting. But do not underestimate the power of baby examples. For one thing, they are sometimes counterexamples to possible theorems.

0 (the empty category)
This category has no objects and no morphisms.
1
 
The category 1 has one object and one morphism, which must necessarily be the object's identity arrow.
1+1
 
This category has two objects and two morphisms: the identities on each object.
2
 
This category has two objects and three morphisms. The third morphism goes from one object to the other.
Remarks
  • The objects of these baby categories are nodes in a graph (not sets) and the morphisms are arrows in the graph (not functions).
  • For these baby categories we don't have to say what the composition operation does: it is always forced.
  • It is impolite to say that categorists think that 1 + 1 is not equal to 2.

The category of sets

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The category of sets, denoted by Set, is this category:

  • The objects are all sets
  • A morphism from a set   to a set   is a function with domain   and codomain  .
  • The composition is the usual composition: If   and   then   is defined by   for all  .
  • The identity morphism   on a set   is the identity function defined by   for  .
Terminology and fine points
  • In order to preserve the unique typing in a function definition, it is necessary to include its codomain. For example,   is a different function from the inclusion function   to some set   properly including  .
  • In most approaches to the foundations of math, the collection of all sets is not a set. This makes Set a large category. However, it is still locally small since the class of all functions between two sets,   and   is a subclass of the power set of their Cartesian product   which is by definition a set.

Mathematical structures as categories

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Preorders

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A preorder   on a set   is a reflexive and transitive relation on  , which means that for all  ,   and for all  ,  ,   in  , if   and  , then  .

A preorder "is" a category in the following sense: Given a preorder ( ,  ) the category structure is this:

  • The objects of the category are the elements of  .
  • There is exactly one morphism from   to   if and only if  .

The existence of identities is forced by reflexivity and the composition law is forced by transitivity. It follows that the category structure has the property that there is at most one morphism from any object   to any object  .

Conversely, suppose you have a category with set   of objects, with the property that there is at most one morphism between any two objects. Define a relation   on   by requiring that   if and only if there is a morphism from   to  . Then ( ,  ) is a preorder.

The statements in the two preceding paragraphs describe an equivalence of categories between the category of small categories with at most one morphism between any two objects and all functors between such categories, and the category of preorders and order-preserving maps.

Remark: Given a preorder, the morphisms of the corresponding category exist by definition. There is exactly one morphism from   to   if and only if  . This is an axiomatic definition; in a model a morphism from   to   could be anything, for example the pair ( ,  ). In no sense is the morphism required to be a function.

Groups

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Every group   can be viewed as a category   as follows:   has one single object; call it  . Therefore it has only one homset  , which is defined to be the underlying set of the group   (in other words, the arrows are the group elements.) We take as composition the group multiplication. It follows that the identity element of   is  . Notice that in the category  , every morphism is an isomorphism (invertible under composition). Conversely, any one-object category in which all arrows are isomorphisms can be viewed as a group; the elements of the group are the arrows and the multiplication is the composition of the category. This describes an equivalence between the category of groups and homomorphisms and the category of small categories with a single object in which every morphism is an isomorphism.

This can be generalized in two ways.

A category   is called a groupoid if every morphism is an isomorphism. Thus a groupoid can be called "a group with many objects."

A monoid is a set with an associative binary operation that has an identity element. By the same technique as for groups, any monoid "is" a category with exactly one object and any category with exactly one object "is" a monoid.

Matrices

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This is a good example of a category whose objects are not sets and whose arrows are not functions.   is the category whose objects are the positive integers and whose arrows   are   matrices where composition is matrix multiplication, for any commutative ring  . For any object  ,  , the   identity matrix.

Categories of sets with structure

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finite sets and functions; denoted FinSet.
monoids and morphisms; denoted Mon.
groups and homomorphisms; denoted Grp.
abelian groups and homomorphisms; denoted Ab.
rings and unit-preserving homomorphisms; denoted Rng.
commutative rings and unit-preserving homomorphisms; denoted CRng.
left modules over a ring   and linear maps; denoted  -Mod.
right modules over a ring   and linear maps; denoted Mod- .
modules over a commutative ring   and linear maps; denoted  -Mod.
subsets of Euclidean space of 3 dimensions and Euclidean movements
subsets of Euclidean space of n dimensions and continuous functions
topological spaces and continuous functions; denoted Top.
topological spaces and homotopy classes of functions; denoted Toph.
Remarks

The law of composition is not specified explicitly in describing these categories. This is the custom when the objects have underlying set-structure, the morphisms are functions of the underlying sets (transporting the additional structure), and the law of composition is merely ordinary function-composition. Indeed, sometimes even the specification of the morphisms is suppressed if no confusion would arise—thus one speaks of the category of groups.

The examples of sets with structure suggest a conceptual framework. For example, the concept of group may be regarded as constituting a first-order abstraction or generalization from various concrete, familiar realizations such as the additive group of integers, the multiplicative group of nonzero rationals, groups of permutations, symmetry groups, groups of Euclidean motions, and so on. Then, again, the notion of a category constitutes a second-order abstraction, the concrete realizations of which consist of such first-order abstractions as the category of groups, the category of rings, the category of topological spaces, and so on.

Properties of objects and morphisms

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Isomorphisms

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A morphism   in a category is said to be an isomorphism if there is a morphism   in the category with  ,  . It is easy to prove that   is then uniquely determined by  . The morphism   is called the inverse of f, written  . It follows that  . If there is an isomorphism from   to  , we say   is isomorphic to  , and it is easy to prove that "isomorphism" is an equivalence relation on the objects of the category.

Examples

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  • A function from   to   in the category of sets is an isomorphism if and only if it is bijective.
  • A homomorphism of groups is an isomorphism if and only if it is bijective.
  • The isomorphisms of the category of topological spaces and continuous maps are the homeomorphisms. In contrast to the preceding example, a bijective continuous map from one topological space to another need not be a homeomorphism because its inverse (as a set function) may not be continuous. An example is the identity map on the set of real numbers, with the domain having the discrete topology and the codomain having the usual topology.

Monomorphisms and Epimorphisms

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A morphism   in a category   is a monomorphism if, for any morphisms   and  , if   and   are defined and   then  .


A morphism   in a category   is an epimorphism if, for any morphisms   and  , if   and   are defined and   then  .

Initial and terminal objects

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  is said to be a terminal (or final) object when   is a unique morphism for any   in  . The law of composition ensures that if   and   are terminal objects in  , they are isomorphic, i.e.,   is unique up to isomorphism. In the categories of sets, groups, and topological spaces, the terminal objects are singletons, trivial groups, and one-point spaces, respectively. "b" is the terminal object in 2 as depicted above.

Constructions on categories

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Subcategories

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A subcategory   of a category   is a category in which:

  • The class of objects of   is contained in the class of objects of  .
  • The class of arrows of   is contained in the class of arrows of  .
  • For every arrow   in  , the domain and codomain of   are in  .
  • For every object   in  , the identity arrow   is in  .
  • For every pair of arrows   in  , the arrow   is in   where it is defined.

Opposite category

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Given a category  , the opposite (or dual) category   has the same objects as   and for every arrow   in  , the arrow   is in  . In other words, it has the same objects, and arrows are reversed.

The product of two categories

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Given two categories   and  , the product category, denoted  , is given by the following data:

  • The objects of   are   where   is an object of   and   is an object of  .
  • The arrows of   are   where   is an arrow of   and   is an arrow of  .
  • Composition is given by  .

The product   is called the cylinder category, denoted  .

Arrow categories

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  • A functor category   is a category whose objects are functors   and arrows are natural transformations.
  • A functor category   is called an arrow category, denoted  . Its objects are arrows   in   and its arrows are pairs of arrows   such that  .

Comma categories

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  • Given categories and functors   and  , the comma category   has objects   where   is an object in  ,   is an object in  , and   is an arrow in  . Its arrows are   where   is an arrow in   and   is an arrow in   such that  . Composition is given by  . The identity is  .

Special types of comma categories

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Using the definition of comma category above, assume that we have  ,  , and  . Let   denote the object in  . Then   for some object   in  . In this case, we write the comma category as   and call it the slice category of   (or the category of objects over  ).

Now assume, instead, that we have  ,  , and  . Let   denote the object in  . Then   for some object   in  . In this case, we write the comma category as   and call it the coslice category of   (or the category of objects under  ).

Finally, if we have   and  , then the comma category  .