Exercise 1, § 10

Let $X$ be the direct limit of a direct system $(X_i, f_{ji})$ of sets relative to a right directed set $I$. Let $F_i$ (resp. $F$) be the free group constructed on $X_i$ (resp. on $X$); each $f_{ji}$ extends to a homomorphism $\phi_{ji}: F_i \to F_j$.

(a) Show that $(F_i, \phi_{ji})$ is a direct system of groups.

(b) Let $\varepsilon_i$ be the composite mapping $X_i \to F_i \to \lim \overrightarrow{F_i}$. Show that $\varepsilon_j \circ \phi_{ji} = \varepsilon_i$ if $j \geq i$. Let $\varepsilon$ be the mapping of $X$ into $\lim \overrightarrow{F_i}$ defined by the $\varepsilon_i$. Show that $\varepsilon$ extends to an isomorphism of $F$ onto $\lim \overrightarrow{F_i}$ (so that $F(\lim \overrightarrow{X_i})$ can be identified with $\lim \overrightarrow{F(X_i)}$).

(c) State and prove analogous results for free magmas, free monoids, free commutative groups, free commutative monoids.