# What is the equivalent of a group in category theory?

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I understand that in group theory, a group consists of a set and a binary operation for the elements in the set, and of course all the group axioms. But if we move away from set theory into category theory, is a group defined on a category?

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fresh_42
Mentor
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.

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

fresh_42
Mentor
Should I learn category theory? Sounds like it could be useful.
... to achieve what? Should I and useful depend on a system of values that I don't know. It is partly useful to better understand the nature of constructions like tensor products, direct sums or what is meant, if someone says exact differential form, and why they are so widely used. I wouldn't go as far as to say it is useful in a general case, except for the study of say cosmology or other realms where homological constructions and topology play a major role. In my opinion it is useful to read a bit about the terms I've mentioned in a book about homological algebra, esp. about the universal property. I would not go as far as to say study it, because it is a rather abstract point of view.

lavinia
Gold Member
I understand that in group theory, a group consists of a set and a binary operation for the elements in the set, and of course all the group axioms. But if we move away from set theory into category theory, is a group defined on a category?
A group is a category with one object in which all of the morphisms are isomorphisms.

mathwonk
WWGD
Gold Member
2019 Award
I understand that in group theory, a group consists of a set and a binary operation for the elements in the set, and of course all the group axioms. But if we move away from set theory into category theory, is a group defined on a category?
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.

mathwonk
Homework Helper
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.

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lavinia
lavinia
Gold Member
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.
Similar to the idea of a morphism, is the idea of a functor. It applies to many different areas of mathematics.

mathwonk
mathwonk