Understanding the Definition of Categories: A Brief Introduction

[gotjrgkr]

Homework Statement

Hi!
I'd like to ask you a definition of a morphism which is used to define the category.
I refer to a book "introduction to topological manifolds" by author Lee.
In his book, the concept of the category is introduced in p. 170.
I'll write it below.

A category C consists of the following:
1. a class (not necessarily a set)of objects;
2. for each pair of objects X,Y, a set Hom$_{C}$(X,Y) of morphisms; and
3. for each triple X,Y,Z of objects a function called composition :
Hom$_{C}$(X,Y)xHom$_{C}$(Y,Z)$\rightarrow$Hom$_{C}$(X,Z), written (f,g)$\rightarrow$g$\circ$f;
such that the following axioms are satisfied;
(i) Composition is associative: (f$\circ$g)$\circ$h=f$\circ$(g$\circ$h).
(ii) For each object X there exists an identity morphism Id$_{X}$$\in$
Hom$_{C}$(X,X) such that for any morphism f$\in$Hom$_{C}$(X,Y) we have Id$_{Y}$$\circ$f=f=f$\circ$Id$_{X}$.

In the above definition, I don't know what morphisms actually are...
I expect that it should be a function which is defined between classes. As you know, like functions between two sets in ZFC set theory, a function also can be defined between two classes if we admit NBG set theory. Do you think I am right? If not, what are exactly morphisms in the above definition??
Thanks a lot for reading my questions!

Homework Equations

The Attempt at a Solution

 

[Office_Shredder]

In any reasonable category the morphisms are the reasonable functions. Note each morphism is only mapping between two chosen objects, it's not a map on the category itself. For example if you have the category of groups you would define hom(G,H) would be the group homomorphisms between G and H

 

[gotjrgkr]

Then, is it right if I say that a morphism is a kind of a function from one class into another class??

 

[Deveno]

In any reasonable category the morphisms are the reasonable functions. Note each morphism is only mapping between two chosen objects, it's not a map on the category itself. For example if you have the category of groups you would define hom(G,H) would be the group homomorphisms between G and H

this, strictly speaking, isn't true. some categories have morphisms that don't even resemble functions. for example:

let C have as objects natural numbers (not including 0).

let Hom_C(m,n) be the set of all mxn matrices with integer entries.

the axioms are satisfied, but a matrix isn't a function from m to n.

another example:

let C have objects {1,2,3,4,6,12,24}

define f:a→b as the only arrow that exists, only if a|b.

then the axioms are satisfied, but arrows (morphisms) don't even remotely look like functions.

a "more natural" example:

let Obj(C) = G, a group.

define g:G→G for each g in G, and set composition of arrows to be group multiplication.


*******

there is a reason morphisms are left "vague". because many kinds of things might be morphisms: edges on a directed graph, functions, homotopy class mappings (the last example should make you think, especially if one is looking at manifolds).

basically, anything which can be "composed" in an associative way, can be a morphism. ANYTHING. that said, the "canonical examples" are sets-with-structure (as objects), and the structure-preserving maps between them (as objects). furthermore, one usually isn't even interested in the sets-with-structure themselves, but only in characterizing morphisms. for example, in topology one might be interested in knowing which identity functions from a set with one topology to the same set with a different topology, are continuous (which tells us "how fine" in comparison the topologies are), or in knowing what well-understood space another (perhaps less-understood) space is homeomorphic to. as another example, with a path, it's not the interval of definiton, or the ambient space the image of the path is in, that is of interest, it's the path mapping itself.

for manifolds, these (morphisms) are typically (n-times) differentiable maps, which can be thought of as "preserving (partial) information about the manifolds they transform". for example, when talking about regions in the plane under two different coordinate systems (like cartesian and polar coordinates), what we're really interested in is the Jacobian matrix of the transform from one system to the other. if the Jacobian is singular (0 determinant), we "lose information" somewhere along the way.

Then, is it right if I say that a morphism is a kind of a function from one class into another class??

no, the concept you are looking for here is called a functor. functors are, curiously enough, the morphisms in the category Cat, whose objects are all (small, or in some texts, locally small) categories.

 

[gotjrgkr]

this, strictly speaking, isn't true. some categories have morphisms that don't even resemble functions. for example:

let C have as objects natural numbers (not including 0).

let Hom_C(m,n) be the set of all mxn matrices with integer entries.

the axioms are satisfied, but a matrix isn't a function from m to n.

another example:

let C have objects {1,2,3,4,6,12,24}

define f:a→b as the only arrow that exists, only if a|b.

then the axioms are satisfied, but arrows (morphisms) don't even remotely look like functions.

a "more natural" example:

let Obj(C) = G, a group.

define g:G→G for each g in G, and set composition of arrows to be group multiplication.


*******

there is a reason morphisms are left "vague". because many kinds of things might be morphisms: edges on a directed graph, functions, homotopy class mappings (the last example should make you think, especially if one is looking at manifolds).

basically, anything which can be "composed" in an associative way, can be a morphism. ANYTHING. that said, the "canonical examples" are sets-with-structure (as objects), and the structure-preserving maps between them (as objects). furthermore, one usually isn't even interested in the sets-with-structure themselves, but only in characterizing morphisms. for example, in topology one might be interested in knowing which identity functions from a set with one topology to the same set with a different topology, are continuous (which tells us "how fine" in comparison the topologies are), or in knowing what well-understood space another (perhaps less-understood) space is homeomorphic to. as another example, with a path, it's not the interval of definiton, or the ambient space the image of the path is in, that is of interest, it's the path mapping itself.

for manifolds, these (morphisms) are typically (n-times) differentiable maps, which can be thought of as "preserving (partial) information about the manifolds they transform". for example, when talking about regions in the plane under two different coordinate systems (like cartesian and polar coordinates), what we're really interested in is the Jacobian matrix of the transform from one system to the other. if the Jacobian is singular (0 determinant), we "lose information" somewhere along the way.



no, the concept you are looking for here is called a functor. functors are, curiously enough, the morphisms in the category Cat, whose objects are all (small, or in some texts, locally small) categories.

Do you mean that a morphism is just a set related with a pair of objects A,B in the class, is just denoted f:A$\rightarrow$B, and could be anything (not necessarily a function) satisfying certain axioms in the category??

 

[Deveno]

Do you mean that a morphism is just a set related with a pair of objects A,B in the class, is just denoted f:A$\rightarrow$B, and could be anything (not necessarily a function) satisfying certain axioms in the category??

yup.

if the category is Set, then functions fit the bill (they work as morphisms). stranger categories are possible, though.

 

[gotjrgkr]

Thanks! It really helps me a lot!