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My question is rather simple.

I'm a bit confused about the fact that all texts I am finding on the topic talk about morphisms between varieties in particular, and do not touch upon morphisms between algebraic sets.

Is it fair to say that what we do is that we define morphisms between varieties, and then simply use that to define morphisms between algebraic sets?

For example, a morphism $\varphi: M \rightarrow N$ between two algebraic sets $M = A \cup B$ and $N = C \cup D$, where $A,B,C,D$ are all varieties, is simply a set of morphisms $\varphi_{AC}:A \rightarrow C$, $\varphi_{AD}:A \rightarrow D$, $\varphi_{BC}:B \rightarrow C$, and $\varphi_{BD}:B \rightarrow D$ that are such that on $A \cap B$, $\varphi_{AC} = \varphi_{BC}$ and $\varphi_{AD} = \varphi_{BD}$?

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  • $\begingroup$ What is your defintion of a morphism between varieties? What goes wrong if you try to apply it verbatim to algebraic sets? The point is that somehow nothing should be different, and all you should need to do is some definition-pushing. $\endgroup$ – KReiser Aug 31 '17 at 4:24

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