Download elements of category theory

STELLENBOSCH UNIVERSITY
ELEMENTS OF
CATEGORY THEORY
ZURAB JANELIDZE
INCOMPLETE NOTES OF 4 AUGUST 2013
Zurab Janelidze. Mathematics Division, Department of Mathematical Sciences, Stellenbosch
University.
Contents
Preface
1
Graphs
2
Categories
3
Functors
4
Commutative diagrams
5
Isomorphisms
6
Equivalence of categories
7
Monoids and preorders
8
Duality
1
1
3
5
6
6
7
8
9
Index
11
References
12
i
Preface
These are notes in progress written for the first few weeks of lectures of the Categorical Algebra
honors module at Stellenbosch University, in which we review elementary concepts of category
theory. The exposition avoids proofs — these are left to the reader, in some cases in the form
of exercises. We assume that the reader is well familiar with sets and functions, with universes
in set theory and understands the distinction between sets and classes, and also has encountered
basic mathematical structures such as monoids, groups, ordered sets and structure-preserving maps
between them. In some cases, our terminology differs slightly from the standard one (e.g. the one
used in [3]); in particular, what we call a “category” is more commonly called a “category with
small hom-sets”.
1. Graphs
A graph G consists of a class Ob(G) of objects, and for any two objects X, Y ∈ Ob(G), a set
hom(X, Y ) of morphisms, with each morphism f ∈ hom(X, Y ) displayed as an arrow as in
X
f
/
Y
(1)
or as in f : X → Y . It is customary to further assume that if either X 6= X 0 or Y 6= Y 0 then
hom(X, Y ) ∩ hom(X 0 , Y 0 ) = ∅.
In other words, all hom-sets are disjoint. Then any morphism f determines uniquely the objects X
and Y in the display (1). The object X is called the domain of f , while Y is called the codomain
of f . However, in practice, in many examples of graphs that we construct hom-sets are not disjoint
(or sometimes, whether they are disjoint or not depends on the formal definition of the objects we
put forward as morphisms, which is not necessarily made explicit). We do not fix this problem by
abandoning the assumption that hom-sets must be disjoint — instead, we agree that elements of
hom(X, Y ) are actually triples (f, X, Y ) where f is the original intended element. This will indeed
force all hom-sets to be disjoint. Here is an example of how this convention is used: when we
define the graph of sets as objects and functions between sets as morphisms between objects, we
define hom(X, Y ) to be the set of all functions from X to Y . But, if by a function f : X → Y is
understood a subset of X × Y satisfying the expected condition, then our convention applies and
elements of hom(X, Y ) are actually triples (f, X, Y ) where f is a function from X to Y . In effect,
by a function we mean such a triple.
Notice that while we require that each hom-set hom(X, Y ) is a set, the class of all objects as
well as the class of all morphisms, given by
[
Mor(C) =
hom(X, Y )
X,Y ∈Ob(C)
need not be sets. When they are, we call the graph a small graph. Similarly, a finite graph is a
graph G in which Ob(G) and Mor(G) are finite sets. Any finite graph is evidently a small graph.
A large graph is a graph which is not small.
1
2
ELEMENTS OF CATEGORY THEORY
Here is a simple example of a finite graph:
@2=
=== (2,3)
==
=
/3
(1,2)
1
(2)
(1,3)
It consists of three objects 1, 2, and 3, and three morphisms (1, 2), (2, 3) and (1, 3), where domain
of each morphism (i, j) is its first component i and codomain is its second component j. In this
example, each hom-set hom(X, Y ) is either empty or a singleton.
A path of length n, written as pn , where n is a natural number, is the graph whose objects are
natural numbers 0, . . . , n and morphisms are pairs (i, i + 1) where i is a natural number 0 6 i < n.
In particular, when n = 0 this is the graph with no morphisms and 0 as the unique object. Thus,
these graphs look as follows:
p0 :
0
p1 :
0
p2 :
0
pn :
0
(0,1)
/
1
(0,1)
/
1
(0,1)
/
1
(1,2)
/
2
(1,2)
/
2
(2,3)
/
···
(n−1,n)
/
n
In the above examples, hom-sets are either always empty or are singletons. Small graphs of
this kind are essentially the same as sets equipped with a binary relation. An example of a graph
which is not of this kind is given by the graph, already referred to above, whose objects are all sets
and morphisms between them are functions between sets. In this graph, the display (1) obtains
a familiar meaning: f is a function from a set X to a set Y . This graph is large since the class
of all sets is not a set. We will work with both finite graphs of the previous simple type, as
well as more complex large graphs of mathematical structures (of a given type) as objects, and
structure-preserving maps between them as morphisms.
A graph morphism D : G → C from a graph G to a graph C consists of two separate maps
Ob(G) → Ob(C),
Mor(G) → Mor(C)
called the object function and the morphism function of the graph morphism, although both are
written by the same letter D, such that for any morphism (1) from the graph G, we have
D(X)
D(f )
/
D(Y )
where D(f ) is the morphism corresponding to f by the morphism function, and D(X) and D(Y )
are objects corresponding by the object function to the domain X and the codomain Y of f ,
respectively. Less formally, a graph morphism G → C maps objects of G to objects of C, and
maps morphisms of G to morphisms of C, so that the domain and the codomain of morphisms are
preserved. Graph morphisms between graphs should not be confused with morphisms in a given
graph.
For example, if T denotes the graph (2), then it is not difficult to see that there is a unique
graph morphism T → T, and namely, the one which maps every object and morphism to itself.
There are of course infinitely many graph morphisms from the same graph T to the large graph of
ELEMENTS OF CATEGORY THEORY
3
sets and functions. Such a graph morphism amounts to a display
Y
~> @@@ g
@@
~~
~
@@
~~
@
~~
/Z
f
X
(3)
h
of sets and functions. More generally, any display of objects and morphisms in a graph C, such
as the one above, can be seen as a graph morphism from a certain graph whose objects are
represented by “places” for objects in the display, and morphisms are represented by the arrows in
the display. The graph morphism is then given by the labeling of the places and arrows by objects
and morphisms of C.
Exercise 1.1. Display all possible graph morphisms T → T0 where T0 is the graph
(2,2)
@2=
(1,2)
(1,1)
&
1
(1,3)
== (2,3)
==
==
/3
(4)
f
(3,3)
obtained from the graph T as in (2), by adding to T three morphisms (1, 1) : 1 → 1, (2, 2) : 2 → 2,
and (3, 3) : 3 → 3.
In mathematics, structures of a given type usually come with a corresponding notion of a
structure-preserving map between them. For graphs, this is the notion of a graph morphism. Two
graph morphisms
F
A
/
G
B
/
(5)
C
can be composed to give a graph morphism G ◦ F : A → C. As expected, the composite G ◦ F
is defined as the graph morphism whose object and morphism functions are the composites of
object and morphisms functions of G and F . Composition of graph morphisms is associative in
an appropriate sense: for any three graph morphisms
A
F
/
B
G
/
C
H
/
D
we have
H ◦ (G ◦ F ) = (H ◦ G) ◦ F.
The identity morphism of a graph B, written as 1B , is the graph morphism B → B whose object
and morphism functions are identity functions. Identity graph morphisms play the role of identities
for composition: for any two graph morphisms (5) we have 1B ◦ F = F and G ◦ 1B = G.
2. Categories
With every graph of mathematical structures and structure preserving maps there is usually
an assigned additional structure of associative composition of the structure preserving maps and
identity maps for the composition. For example, composition of graph morphisms between small
graphs give such structure for the graph of small graphs, which consists of small graphs as objects
and graph morphisms between small graphs as morphisms between objects. Notice however that
here we have to use the convention adopted in Section 1: each hom-set hom(A, B), where A and
B are small graphs, consists of not just graph morphisms D from A to B, but triples (D, A, B)
4
ELEMENTS OF CATEGORY THEORY
where D is a graph morphism from A to B, since a graph morphism D : A → B defined as a pair
of object and morphism functions does not determine the graphs A and B uniquely.
A category 1 C is a graph with an assigned structure of associative composition of morphisms
and identity morphisms. In detail, a category is a graph in which for any two morphisms as in the
display
f
X
/
g
Y
/
Z
a third morphism is defined, called the composite of g and f and written as g ◦ f or also as gf .
When g and f can be arranged as in the above display (that is, when the codomain of f is the
same as the domain of g), we say that g and f are composable. The composite g ◦ f is then a
morphism
g◦f
X
/
Z
having the same domain as f and the same codomain as g. The operation which produces composite
of two composable morphisms is called composition. Composition is associative in the following
sense: for any three morphisms
W
e
/
f
X
/
g
Y
/
Z
we have g ◦ (f ◦ e) = (g ◦ f ) ◦ e. In addition, in a category for each object X there is a morphism
1X
X
/
X
called the identity morphism of X, such that for any morphism f as above we have f ◦ 1X = f =
1Y ◦ f . Note that although the choice of identity morphisms is part of the structure of a category,
it is uniquely determined by composition since if 10X is another identity morphism for the object
X we must have 10X = 10X ◦ 1X = 1X . Because of this, a category can be specified by its objects,
morphisms between objects, and composition only.
If C denotes a category, we write the same C for its underlying graph. We say that C is a
finite/small/large category when its underlying graph is finite/small/large.
Some graphs have unique category structure. For instance, the graph (4) is such. In the
corresponding category (i, i) is an identity morphism of object i, and composition is given by
(i, j) ◦ (k, l) = (min(i, k), max(j, l)).
The graph of sets and functions is a category, denoted by Set and called the category of sets,
under the ordinary composition of functions. Small graphs and graph morphisms between them
form a category under composition of graph morphisms. This category will be denoted by Graph.
Other categories of familiar mathematical structures can be obtained in a similar way.
We now describe a canonical way of producing a small category starting with any small graph,
called the category of paths of the graph. Let X and Y be objects in a small graph G. In G, a path
of length n, from X to Y , is a graph morphism P : pn → G such that pn (0) = X and pn (n) = Y .
It can be given be described by a display
X = P (0)
/
P (1)
/
P (2)
/
···
/
P (n) = Y
of objects and morphisms in G, where the label of each arrow P (i) → P (i+1) is given by P ((i, i+1))
— which we can write as P (i, i+1), avoiding unnecessary double brackets. For a given small graph
G, the category of paths of G, written as Paths(G), is obtained as follows:
1The
notion of a category was introduced by Eilenberg and Mac Lane in [1]. This same paper lay foundations
to what later became known as category theory.
ELEMENTS OF CATEGORY THEORY
5
Objects of Paths(G) are the same as objects of G. For any two objects X and Y in G, a
morphism from X to Y is a path P from X to Y , of arbitrary length n. For any two paths
P (0)
/
P
Q
P (n) = Q(0)
/
Q(m)
where P is path of length n, and Q is path of length m, as suggested in the above display, the
composite Q ◦ P is a path P (0) → Q(m) of length m + n described by the display
P (0)
P (0,1)
/
···
P (n−1,n)
/
Q(0,1)
P (n) = Q(0)
/
···
Q(m−1,m)
/
Q(m)
For any object X, the identity morphism 1X is the unique path of length 0 from X to X.
Exercise 2.1. Verify that Paths(G) is indeed a (small) category.
Exercise 2.2. Describe the category of paths of the graph (2).
3. Functors
A functor F : C → D from a category C to a category D is a graph morphism F : C → D from
the underlying graph of C to the underlying graph of D, which preserves composition and identity
morphisms: for any two composable morphisms g and f in C, we have F (g ◦ f ) = F (g) ◦ F (f ),
and for any object X in C, we have F (1X ) = 1F (X) . For instance, the display
1Y
~~
~~
~
~~
f
1X
(
X
Y
~> @@
g◦f
@@ g
@@
@@
/Z
g
(6)
1Z
where f and g are two morphisms in any category D describes a graph morphism from the underlying graph of the category (4) to the underlying graph of some category D, which constitutes a
functor between these two categories.
It is not difficult to see that given two functors
C
F
/
D
G
/
E
(7)
their composite G ◦ F as graph morphisms is again a functor C → E. We will refer to the functor
G ◦ F : C → E as the composite of the two functors G and F . Since composition of graph
morphisms is associative, so is composition of functors. Small categories with functors between
categories and composition of functors form a category which will be denoted by Cat and will be
called the category of small categories.
In many important examples, objects of a category are mathematical structures of a given
type and morphisms are suitable structure-preserving maps. We have already encountered such
examples of categories: Set, Graph, and Cat. Functors between such categories arise naturally
from ways of producing structures of a given type from structures of another or same type. For
example, to each small graph we can associate the set of its objects. This gives rise to a functor
Graph → Set, under which a graph morphism is mapped to its object function. Similarly, there
is a functor Graph → Set which maps each small graph to its set of morphisms, and each graph
morphism to its morphism function.
Mathematical structures of a given type usually come equipped with the corresponding notion
of a “substructure”. In the case of categories, this is the notion of a subcategory defined here.
6
ELEMENTS OF CATEGORY THEORY
A subcategory of a category C is a category B such that Ob(B) ⊆ Ob(C) and for any two
objects X, Y ∈ Ob(B), any morphism X → Y in B is also a morphism X → Y in C (and thus,
in particular, also Mor(B) ⊆ Mor(C)); moreover, composition of morphisms in B is defined in the
same way as in C, and all identity morphisms in B are identity morphisms in C.
When a category B is a subcategory of a category C, there is an obvious functor B → C which
maps every objects and every morphism of B to the same object and the same morphism in C.
This functor is called the subcategory inclusion functor .
Exercise 3.1. Describe all subcategories of the category (4).
4. Commutative diagrams
A graph morphism G → C from a graph G to the underlying graph of a category C is called
a diagram in C. Diagrams are often given by the display describing the diagram as a graph
morphism.
For any graph G there is an obvious diagram D : G → Paths(G), which maps every object to
itself, and every morphism f : X → Y to the path P of length 1 from X to Y , given by P (0, 1) = f .
This diagram has the following “universal property”: given any other diagram D0 : G → C to a
category C, there is a unique functor F : Paths(G) → C, such that mapping objects and morphisms
from G to C consecutively along the two upper sides of the triangle
Paths(G)
G
u:
D uuu
u
uu
uu
D0
HH
HH F
HH
HH
HH
$
/C
gives the same result as mapping them directly from G to C along the base of the triangle. In
other words, F is unique with the property F ◦ D = D0 .
Exercise 4.1. Verify the universal property described above.
Exercise 4.2. Show that the assignment G 7→ Paths(G) gives an object function of a functor
Graph → Cat. Hint: use the universal property established in Exercise 4.1.
A commutative diagram in a category C is a diagram D : G → C such that for any two
morphisms f, g : X → Y in Paths(G) the unique functor Paths(G) → C induced by the diagram
D carries f and g to the same morphism in C. Thus, commutativity of a diagram states that in
its display any two composites of morphisms in two paths with a common start and a common
end, must be equal to each other, and moreover, when the path ends at the same place where it
started, its composite must be equal to the identity morphism of the object in that place. For
example, any diagram of the form (6) in any category is a commutative diagram. Commutative
diagrams provide an effective tool for working in a category.
5. Isomorphisms
A morphism f : X → Y in a category is said to be an isomorphism if there is a morphism
g : Y → X such that g ◦ f = 1X and f ◦ g = 1Y . When it exists, such g is necessarily unique.
Indeed, if g 0 is also such then we have
g 0 = g 0 ◦ 1Y = g 0 ◦ (f ◦ g) = (g 0 ◦ f ) ◦ g = 1X ◦ g = g.
It is easy to see that when the morphism g exists, it is also an isomorphism. For an isomorphism
f the corresponding g is called the inverse inverse of f and is denoted by g = f −1 . If f is an
ELEMENTS OF CATEGORY THEORY
7
isomorphism then f = (f −1 )−1 . Any identity morphism is obviously an isomorphism whose inverse
is itself. When there is an isomorphism X → Y , we say that X and Y are isomorphic. In Set
isomorphisms are the bijections. In Graph isomorphisms are those graph morphisms whose object
function and morphism functions are both bijections. Similarly, in Cat isomorphisms are those
functors whose object and morphism functions are bijections. In general, when we work with
mathematical structures of a given type, we seldom distinguish between those structures which are
isomorphic in the corresponding category of structures. The isomorphism and its inverse allow to
transfer any property of the mathematical structure to its isomorphic structure, and vice versa.
Exercise 5.1. Show that for any isomorphism f : X → Y in a category, the diagram
f
1X
6X k
*
Y
v
1Y
f −1
commutes.
Exercise 5.2. Show that in a commutative diagram
Y
~> @@@ g
~
@@
~~
@@
~~
@
~~
/Z
f
X
h
in any category, if two of the three morphisms f , g and h are isomorphisms, then the third
morphism is also an isomorphism.
Exercise 5.3. Show that the relation “X is isomorphic to Y ” is an equivalence relation.
Exercise 5.4. Show that a functor always preserves isomorphisms, i.e. if F : C → C0 is a
functor, and f is an isomorphism in C, then F (f ) is an isomorphism in C0 .
6. Equivalence of categories
A functor F : C → D is said to be faithful/full if for any two objects X and Y in C, the map
hom(X, Y ) → hom(F (X), F (Y )),
f 7→ F (f )
is injective/surjective. F is said to be a category equivalence, when F is both full and faithful, and
any object in D is isomorphic to F (C) for some object C of C. Two categories C and D are said
to be equivalent when there is a category equivalence C → D.
Exercise 6.1. Show that equivalence of categories is an equivalence relation (i.e. it is reflexive,
symmetric and transitive).
The functor Cat → Graph which maps a category to its underlying graph, and maps a functor
to the same functor regarded as a graph morphism, is evidently faithful. However, it is not full
since not every graph morphism between categories preserves composition and identity morphisms.
Exercise 6.2. Show that the functor Cat → Set which maps every small category to its set
of objects (and which maps a functor to its object function), is neither faithful nor full. What
about the functor Cat → Set which maps every small category to its set of morphisms (and which
maps a functor to its morphism function)?
Any subcategory inclusion functor B → D is faithful. When it is full, B is said to be a full
subcategory of D. Note that specifying a full subcategory of a category D simply amounts to
specifying a class of objects of D. For instance, finite sets form a full subcategory of the category
8
ELEMENTS OF CATEGORY THEORY
Set of sets: objects in this subcategory are all finite sets, and morphisms between finite sets are
all functions between them, with composition of functions defined in the ordinary way (that is, in
the same way as in Set).
Any full and faithful functor F : C → D gives rise to a category equivalence F 0 : C → D0 , where
0
D is the full subcategory of D consisting of those objects D which are isomorphic to an object
F (C) for some object C of C. For example, let C be the full subcategory of Set consisting of sets
of the form {1, . . . , n}, where n is any natural number (when n = 0 we set {1, . . . , n} = ∅). The
subcategory inclusion C → Set = D is then full and faithful. It restricts to a category equivalence
C → D0 where D0 is the full subcategory of Set consisting of all finite sets. Here is another example.
Consider the category Matr whose objects are natural numbers, and a morphism n → m is an
m × n matrix, where composition of morphisms is given by matrix multiplication. Each natural
number n gives rise to the vector space Rn , and each m × n matrix gives rise to a linear map
Rn → Rm . This defines a full and faithful functor from Matr to the category Vec of all vector
spaces and linear maps between them, where composition of linear maps is defined in the ordinary
way. This gives rise to the equivalence of Matr and the full subcategory of Vec consisting of finite
dimensional vector spaces.
7. Monoids and preorders
The category of monoids, denoted by Mon, has monoids as objects and monoid homomorphisms as morphisms between monoids, where composition of monoid homomorphisms is the
ordinary one. Notice that any object X in a category gives rise to a monoid: the set of all morphisms X → X, with composition of morphisms as multiplication and identity morphism as the
identity of the monoid. This monoid is called the endomorphism monoid of the object X and is
denoted by End(X). In fact, any monoid M is an endomorphism monoid of some object in some
category. Indeed, it is the endomorphism monoid of the unique object in a single-object category
whose set of morphisms is given by the set M , and where composition of morphisms is defined
by monoid multiplication. In particular, we can set this unique object to be the identity of M .
Moreover, for any homomorphism f from a monoid M to a monoid N there is a unique functor
between the corresponding single-object categories, whose morphism function is f . This defines a
full and faithful functor Mon → Cat, which gives rise to an equivalence between Mon and the
full subcategory of Cat consisting of single-object categories.
An element of a monoid is invertible if and only if in the corresponding category it is an
isomorphism. The full subcategory of Mon consisting of groups (i.e. those monoids where all
elements are invertible), which we denote by Grp and call the category of groups, is equivalent to
the full subcategory of Cat consisting of those single-object categories where every morphism is
an isomorphism.
An ordered set (X, 6) gives rise to a small category whose objects are the elements of the
ordered sets, while a morphism x → y is a pair (x, y) such that x 6 y. This gives a graph which
has unique category structure. The possibility of defining composition is given by transitivity
of the order relation, while reflexivity gives identity morphisms. Once composition and identity
morphisms are defined, associativity and identity laws will hold trivially since in this graph each
hom-set is either a singleton or empty. In fact, any category in which each hom-set is either a
singleton or empty is equivalent to a category arising in this way from an ordered set. Categories
in which each hom-set is either a singleton or empty are called preorders. A preorder which is a
small/finite category is called a small/finite preorder . Thus, any ordered set gives rise to a small
preorder, and any preorder is equivalent to such.
Ordered sets and order-preserving maps between them constitute a category, denoted by Ord
and called the category of ordered sets, where composition of order-preserving maps is defined
ELEMENTS OF CATEGORY THEORY
9
in the usual way. Any order-preserving map between ordered sets is the object function of a
unique functor between the small preorders that these ordered sets give rise to. This defines a
full and faithful functor Ord → Cat, which gives rise to an equivalence between Ord and the
full subcategory of Cat consisting of those small preorders where identity morphisms are the only
isomorphisms.
This, in a category, restricting the number of objects leads to the notion of a monoid, whereas
restricting the number of morphisms leads to the notion of a preorder. Then, groups are monoids
with most number of isomorphisms and ordered sets are small preorders with least number of
isomorphisms.
8. Duality
Let G be a graph. Its dual graph, written as Gop , has the same class of objects and the same
class of morphisms as G. However, a morphism f : X → Y in G is a morphism f : Y → X in Gop .
For example, dual of the graph (2) is the graph
(1,2)
1o
2 ^=
== (2,3)
==
==
3
(1,3)
A graph morphism between graphs A → B gives rise to a graph morphism Aop → Bop between dual
graphs, called the dual of the graph morphism — it is a graph morphism with the same object and
morphism functions as the original one. To display the dual of a graph morphism, simply invert
the direction of all morphisms in the display of the original graph morphism.
Now let C be a category. The dual category of C, written as Cop , is defined as a category whose
underlying graph is the dual graph of C, and in which a composite f ◦ g is defined as the composite
g ◦ f in C. Notice that if g and f are composable in C, say
X
f
/
Y
g
/
Z
then in Cop we have
f
g
Xo
Y o
Z
op
so in C the morphisms f and g are indeed composable. It is not difficult to see that the
composition in Cop is indeed associative and the identity morphisms of C are still the identity
morphisms in Cop . In the same way as any graph morphism between graphs determined a graph
morphism between dual graphs, any functor between categories determined a functor between the
dual categories, called the dual functor .
The process of assigning to a graph morphism/functor its dual preserves composition and
identities. Thus, if we restrict to small graphs/categories, we get functors Graph → Graph and
Cat → Cat.
One of the aims of category theory is to study mathematical structures structures of a give type,
abstractly via the category they form. Here “abstractly” means that one develops such a study in
an abstract category and then applies it to the concrete category of interest. Often this process
leads to the discovery of new conceptual connections between different mathematical structures,
and sometimes they lead to unifications of different results for structures of different types. Such
unification can also occur for two different results on structures of the same type, when we apply
the same result to a category of structures of a given type, and to its dual category2. It is then
useful to identify those notions expressed in the language of a category which are invariant under
2This
point was first emphasized by Mac Lane in [2]
10
ELEMENTS OF CATEGORY THEORY
switching to the dual category. An example of such notion is that of an isomorphism: a morphism
f in a category C is an isomorphism if and only if it is an isomorphism in Cop .
Dual of a preorder is a preorder. If we take an ordered set X = (X, 6), view it as a preorder
and then take the dual preorder, we get a preorder isomorphic to the dual ordered set X op =
(X, >). Dual of a monoid regarded as a single-object category is isomorphic to the single-object
category corresponding to the dual monoid — the same set, with the same identity, and with the
multiplication order reversed. Notice that a monoid is commutative if and only if it coincides with
its dual monoid.
Index
1- , 3
Cop , 9
Gop , 9
Cat, 5
Grp, 8
Mon, 8
Ord, 8
Set, 4
Mor(-), 1
Ob(-), 1
Paths(G), 4
hom(-, -), 1
pn , 2
- ◦ -, 4
dual of a — morphism, 9
finite —, 1
identity morphism of a —, 3
large —, 1
morphism function of a — morphism, 2
object function of a — morphism, 2
small —, 1
group
category of —s, 8
category, 4
category of small categories, 5
dual —, 9
equivalence of —s, 7
finite —, 4
large —, 4
small —, 4
sub—, 6
codomain
— of a morphism, 1
composite
— of morphisms in a category, 4
composition
— of morphisms in a category, 4
monoid
category of —s, 8
endomorphism —, 8
morphism
— in a graph, 1
composable —s, 4
identity —, 4
inverse, 6
isomorphic
— objects in a category, 7
isomorphism, 6
object
— of a graph, 1
ordered set
category of —s, 8
path, 2
a — between objects in a graph, 4
category of —s, 4
preorder, 8
finite —, 8
small —, 8
diagram, 6
commutative —, 6
domain
— of a morphism, 1
set
the category of —s, 4
subcategory
— inclusion functor, 6
full subcategory, 7
functor
composition of —s, 5
dual —, 9
faithful —, 7
full —, 7
graph, 1
— morphism, 2
composition of — morphisms, 3
dual —, 9
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References
[1] Samuel Eilenberg and Sunders Mac Lane. General theory of natural equivalences. Trans. Amer. Math. Soc.,
58:231–294, 1945.
[2] Saunders Mac Lane. Duality for groups. Bull. Amer. Math. Soc., 56:485–516, 1950.
[3] Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in
Mathematics. Springer, 2nd edition, 1998.
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