My textbook keeps saying that maps "descend to" other maps. I have never encountered this terminology before (at least not in English). What does it mean?

Here is an example:

The function $H$ is a smooth invariant function on $M = \mathbb{C}^2$, and therefore descends to a smooth function on $M_0$.

In general, how do I interpret "descends to"?

  • $\begingroup$ I typically see it used with regards to restricting a function or with regards to equivalence relations, though there might be other instances in which the terminology is used that I can't immediately recall right now. For example, if you have a linear map $\varphi: V\rightarrow V$, then this `descends' to a map $\varphi: \mathbb{P}(V)\rightarrow \mathbb{P}(V)$ (where $\mathbb{P}(V)$ is the projectivization of $V$) because if $v=\lambda w$, then $\varphi(v)=\lambda \varphi(w)$. More generally, the equivalence relations used in quotients, as in Internet Sheriff's answer. $\endgroup$ – Hayden Feb 28 '15 at 11:12
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    $\begingroup$ Is it the opposite, or more properly the categorical dual, of "lifts to"? That is, if a map $f: A \to B$ is given, and if $f = p \circ q$, then we say that $f$ "descends to" $p$, if $q$ is given, or that $f$ "lifts to" $q$, if $p$ is given. (I'm just wondering, not answering.) $\endgroup$ – Calum Gilhooley Feb 28 '15 at 11:50

It usually has the same meaning as "induces a function on". It is often used in connection with the universal property of a quotient (group/space/etc.), which explains the "descend".


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