What is the exact definition of "induced homomorphism"?

Question

I am working through Hatcher on my own, but having trouble finding a general definition of "induced homomorphism," at least without resorting to category theory (of which I have no knowledge). I understand Hatcher's specific uses, for example with fundamental groups (p. 34) and homology groups (p.111). But in his discussions, he also seems to imply that there are certain properties of induced homomorphisms, generally. Is this correct?

Any (non-category-theory) definition and properties would be much appreciated.

Answer 1

The phrase "induced homomorphism" is really informal; it refers to any context where we can "put together" a homomorphism from given information, in a way that doesn't involve making any choices (so we're not "missing" any information needed). Below I describe one way this crops up; Zev Chonoles' answer describes another (which is probably more common in algebraic topology):

A homomorphism from $A$ to $B$ can be induced by a partial homomorphism. Given two structures $A$ and $B$ and a map (not necessarily a homomorphism) $m$ from $X$ to $Y$ (where $X\subseteq A$ and $Y\subseteq B$, but are not necessarily substructures), we say $m$ induces a homomorphism if there is exactly one homomorphism extending $m$ - that is, exactly one homomorphism $f: A\rightarrow B$ such that $f(x)=m(x)$ for all $x\in X$.

One standard example is if $A, B$ are vector spaces (over a field $k$), $X$ is a basis for $A$, and $m$ is any map from $X$ to $B$, there is a unique homomorphism from $A$ to $B$ extending $m$. More generally, if a subset $X$ generates all of $A$ in the appropriate sense, then any map out of $X$ induces a homomorphism from $A$.

Another standard example - from topology, this time, so "homomorphism" should be replaced with "continuous map" - is the following. Let $A, B$ be $\mathbb{R}$ with the usual topology, and take $X=\mathbb{Q}$. Then

If $m: \mathbb{Q}\rightarrow \mathbb{R}$, there is at most one continuous map $f: \mathbb{R}\rightarrow\mathbb{R}$ extending $m$.

For example, if $m:q\mapsto q$ is the identity on $\mathbb{Q}$, then $m$ induces the identity map on $\mathbb{R}$ $f:r\mapsto r$, and if $m:q\mapsto q^2$ then $m$ induces $f: r\mapsto r^2$; by contrast, the map $m(q)=0$ if $q^2<2$ and $m(q)=1$ if $q^2>2$, while well-defined (and in fact continuous on $\mathbb{Q}$!) does not extend to any continuous map from $\mathbb{R}$ to $\mathbb{R}$.

Answer 2

A different sense of the word "induced homomorphism" than the one Noah refers to is when you have a map $f:A\to B$ between two structures, and a substructure $C$ of $A$ such that $f$ can be written as a composition $$A\xrightarrow{\;\text{quotient}\;} A/C\xrightarrow{\;\;\;\widetilde{f}\;\;\;}B$$ for some other map $\widetilde{f}:A/C\to B$. In this situation, we also usually say that $\widetilde{f}$ is induced by $f$.

Answer 3

A comment that would have helped me years ago when I first encountered induced homomorphisms: you can think of the word "induced" as simply meaning "associated" or "derived". An induced homomorphism is one that we associate to, or derive from, a given homomorphism. Although the English language definition of "to induce" is simple (it just means "to bring about"), I think "to associate" and "to derive" are more familiar to most people. This was certainly the case for me; "induced" sounded like an unfriendly Chemistry term. But now, I'm super comfortable with the term and do think it more accurately captures the essence of the phenomenon than the other two terms do.

Here's an example to add to those given in other answers: Suppose that $f:V \longrightarrow W$ is a linear transformation between vector spaces $V$ and $W$. We can define a new linear transformation $f^*:W^* \longrightarrow V^*$ between the dual spaces $W^*$ and $V^*$ in terms of $f$ as follows: set $f^*(\lambda)(v) = \lambda(f(v))$. Because of its close association to $f$ (its very definition depends on $f$) we call the map $f^*$, which is to say, we decorate the name of the original map with a superscripted asterisk. You'll notice that we also denote the dual spaces with superscripted asterisks as well, and none of this is accidental; there's a contravariant functor underlying all of this, but you don't have to know any of that just yet.

Finally, there certainly are abstract properties that induced maps share in general (they all come from functors), but their context matters significantly. Knowing the definition of the induced dual map in the preceding paragraph doesn't tell you how to define, for example, the push-forward in differential geometry (induced on tangent spaces by a smooth map between manifolds). BUT! :) by understanding the categorical framework for induced maps in general, one can make natural and often correct guesses at definitions. And, of course, you also develop a sense for when to expect induced maps to be lurking around the corner.