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I keep seeing, mostly in homological algebra, the use of "induced

homomorphs" or "induced isomorphisms". I get the idea of what is

going on, but I have not been able to find a formal result that

rigorously explains this, i.e, under what conditions does a map

induce an isomorphism or a homomorphism?. The only patterns

there seems to be in all these induced maps is that they are all

defined in some quotient space of the domain, i.e, if

we have f:X->Y , then the induced maps f* are , or seem to be,

defined in X/~ , for some equiv. relation ~ (e.g, in homotopy and

homology). Also, maybe obviously, f is a continuous map.

Basically: I would like to know a result that would allow me to

give a yes/no answer to the question : does f:X->Y induce an

isomorphism/homomorphism of some sort?

Thanks.