Isn't the concept of homomorphism and isomorphism in abstract algebra analogous to functions and invertible functions in set theory respectively? That's one way to quickly grasp the concept into the mind?
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Related, maybe helpful: https://math.stackexchange.com/questions/2039702/what-is-an-homomorphism-isomorphism-saying/2039715#2039715 – Ethan Bolker Aug 26 '17 at 13:26
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Homomorphisms are not just functions. A group homomorphism $\phi:G\rightarrow H$, for example, has to satisfy an additional law: $\phi(xy)=\phi(x)\phi(y)$ for all $x,y\in G$. But the condition to be bijective for an isomorphism is indeed like functions and invertible functions.
Dietrich Burde
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Yeah, apart from the identity that preserves the operation, they're pretty much like functions, aren't they? – Nishant Garg Aug 26 '17 at 11:12
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The identity may not be a homomorphism, because the composition in $G$ need not be the same as the composition in $H$. Take $G=(\mathbb{R},+)$ and $H=(\mathbb{R}^+,\cdot)$. – Dietrich Burde Aug 26 '17 at 11:15
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Isn't the concept of homomorphism and isomorphism in abstract algebra analogous to functions and invertible functions in set theory respectively?
In category theory, we describe the category of sets as having objects which are sets and arrows which are functions between the sets. In the category of $F$-vector spaces, the objects are $F$-vector spaces, and the arrows are $F$-linear homomorphisms between the vector spaces.
A similar description exists for other categories, where the arrows are the desired type of homomorphism for the category. So yes, they are analogous when you look at them from the viewpoint of basic category theory. In each category, an arrow may or may not be invertible. In the category of sets, the invertible arrows are called "bijections" and in the category of $F$-vector spaces they are "isomorphisms of $F$-vector spaces."
Another way to think of it is that sets and functions between them are like the basic thing you can start with. When you enrich the sets you're talking about (by, say, giving them operations or topologies) then you also would like to enrich the requirements on the functions between them (so that they respect the operations, or respect the topology).
rschwieb
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Thank you! This is precisely what I was thinking, I thought of it like an extension of sets and functions. It's much easier to grasp the concept that way I think. – Nishant Garg Aug 26 '17 at 14:00