Hi, everyone: 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.