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My Algebra textbook "Chapter 0" by Aluffi states that the category of a group consists of groups as objects and homomorphisms between them as morphisms.

Then it also gives a commutative diagram to encapsulate the properties of a homomorphism: namely that if $f:G\to H$ is a homomorphism, then $f(a*_G b)=f(a)*_H f(b)$.

Is the commutative diagram part of the category theory definition? Or is it an additional property which along with the category theory definition (taking groups as objects and homomorphisms as morphisms) gives the full definition of a group?

Motivation: I thought categories could only contain objects and morphisms. Hence now it seems rather surprising that you could also include special relationships between objects of a category to encpasulate the properties of an Algebraic structure.

Thank you.

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  1. This is not a category theory definition of a group, but rather the definition of the category of groups.

  2. Thus, one has to specify objects and morphisms. Objects here are groups, morphisms are homomorphisms of groups defined in the usual sense.

  3. That homomorphisms of groups may be described via maps which make a diagram commute, is completely irrelevant to the definition of the category of groups. However, it offers a generalization of groups, see the notion of a group object.

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  • $\begingroup$ So you're saying that the definition of the category of groups does not include the commutative diagram part? $\endgroup$
    – user67803
    May 24, 2014 at 17:19
  • $\begingroup$ This question doesn't make any sense. Apples and Oranges. $\endgroup$
    – Martin Brandenburg
    May 24, 2014 at 17:21
  • $\begingroup$ I'll try to make my doubt clearer, although it may still may not make sense. When we define the category of a structure, can we only declare "what are the objects and what are the morphisms", or can we also encapsulate some additional information in this category? $\endgroup$
    – user67803
    May 24, 2014 at 17:26
  • $\begingroup$ The commutative diagram might be that additional information here. I am new to category theory, and still trying to find my way around. $\endgroup$
    – user67803
    May 24, 2014 at 17:28
  • $\begingroup$ That commutative diagram lives in $\Bbb{Set}$, and it is just another way to describe 'group homomorphisms' which are the arrows of the category $\Bbb{Grp}$. And, indeed, a category only consists of objects and arrows, nothing more. $\endgroup$
    – Berci
    May 24, 2014 at 18:18

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