- #1

- 35

- 13

- I
- Thread starter Krunchyman
- Start date

- #1

- 35

- 13

- #2

fresh_42

Mentor

- 13,457

- 10,515

A category is defined by objects and morphisms, which are mappings between the objects that respect the structure of the objects.

Groups and (group) homomorphisms are an example of a category.

Sets and functions are another example of a category.

Others are (vector spaces, linear transformations), (topological spaces, continuous functions), (rings, ring homomorphisms), (##\mathbb{C}^n##, differentiable functions) and so on.

Some objects can even belong to more than one category, e.g. ##\mathbb{R}^n## is a group, a ring, an algebra, a vector space and a topological space and all are usually always sets. The functions between categories are called functors. They map objects from one category to another, e.g. a group to its underlying set, and also the morphisms: homomorphisms to functions. This particular functor is called forgetful-functor, because it forgets the structure of the group.

Groups and (group) homomorphisms are an example of a category.

Sets and functions are another example of a category.

Others are (vector spaces, linear transformations), (topological spaces, continuous functions), (rings, ring homomorphisms), (##\mathbb{C}^n##, differentiable functions) and so on.

Some objects can even belong to more than one category, e.g. ##\mathbb{R}^n## is a group, a ring, an algebra, a vector space and a topological space and all are usually always sets. The functions between categories are called functors. They map objects from one category to another, e.g. a group to its underlying set, and also the morphisms: homomorphisms to functions. This particular functor is called forgetful-functor, because it forgets the structure of the group.

Last edited:

- #3

- 35

- 13

Should I learn category theory? Sounds like it could be useful.A category is defined by objects and morphisms, which are mappings between the objects that respect the structure of the objects.

Groups and (group) homomorphisms are an example of a category.

Sets and functions are another example of a category.

Others are (vector spaces, linear transformations), (topological spaces, continuous functions), (rings, ring homomorphisms), (##\mathbb{C}^n##, differentiable functions) and so on.

Some objects can even belong to more than one category, e.g. ##\mathbb{R}^n## is a group, a ring, an algebra, a vector space and a topological space and all are usually always sets. The functions between categories are called functors. They map objects from one category to another, e.g. a group to its underlying set, and also the morphisms: homomorphisms to functions. This particular functor is called forgetful-functor, because it forgets the structure of the group.

- #4

fresh_42

Mentor

- 13,457

- 10,515

... to achieve what?Should I learn category theory? Sounds like it could be useful.

- #5

lavinia

Science Advisor

Gold Member

- 3,236

- 623

A group is a category with one object in which all of the morphisms are isomorphisms.

- #6

WWGD

Science Advisor

Gold Member

2019 Award

- 5,341

- 3,295

A group requires structure beyond that of a set, where you have the barebones structure of whether an element belongs to a set or not; it requires algebraic structure: an operation, I believe binary from pairs of the group into the group. Category theory, as I understand it, is the perspective that you can gain understanding of structures by understanding the elements as well as mappings between them that preserve structure in a precise sense/definition.

- #7

mathwonk

Science Advisor

Homework Helper

- 10,990

- 1,176

to me the lesson from category theory is that morphisms are more important than objects. so in any subject you study, learn what a morphism is, and especially learn what an isomorphism is. learn a few things from a category perspective, such as a "product" of two objects X,Y is not necessarily the set iof all pairs (x,y) with x in X and y in Y, but rather it is an object Z together with a pair of morphisms Z-->X and Z-->Y, such that, for any W, a morphism W-->Z is equivalent to two morphisms W-->X and W-->Y.

Last edited:

- #8

lavinia

Science Advisor

Gold Member

- 3,236

- 623

Similar to the idea of a morphism, is the idea of a functor. It applies to many different areas of mathematics.to me the lesson from category theory is that morphisms are more importamnt than objects. so in any subject you study, learn what a morphism is, and especially learn what an isomorphism is. learn a few things from a category perspective, such as a "product" of two objects X,Y is not necessarily the set iof all pairs (x,y) with x in X and y in Y, but rather it is an object Z together with a pair of morphisms Z-->X and Z-->Y, such that, for any W, a morphism W-->Z is equivalent to two morphisms W-->X and W-->Y.

- #9

mathwonk

Science Advisor

Homework Helper

- 10,990

- 1,176

- Replies
- 11

- Views
- 3K

- Replies
- 17

- Views
- 10K

- Last Post

- Replies
- 3

- Views
- 2K

- Last Post

- Replies
- 1

- Views
- 1K

- Last Post

- Replies
- 10

- Views
- 2K

- Replies
- 4

- Views
- 4K

- Replies
- 22

- Views
- 1K

- Last Post

- Replies
- 2

- Views
- 2K

- Replies
- 7

- Views
- 11K

- Last Post

- Replies
- 2

- Views
- 1K