Exercise 3, § 3

Let $\Sigma$ be a species of structures, let $(\mathrm{A}_\iota)_{\iota \in \mathrm{I}}$ be a family of sets, and for each $\iota \in \mathrm{I}$ let $\mathscr{S}_\iota$ be a structure of species $\Sigma$ on $\mathrm{A}_\iota$. Let E be the set which is the sum of the family $(\mathrm{A}_\iota)_{\iota \in \mathrm{I}}$, each $\mathrm{A}_\iota$ being considered as a subset of E. Suppose that we are given a notion of $\sigma$-morphisms for the species $\Sigma$, and define an $\alpha$-mapping to be a mapping $\varphi$ of E into a $\Sigma$-set F such that, for each $\iota \in \mathrm{I}$, the restriction of $\varphi$ to $\mathrm{A}_\iota$ is a morphism of $\mathrm{A}_\iota$ into F. Show that, if there exists a solution $(\mathrm{F_E}, \varphi_\mathrm{E})$ to the universal mapping problem for E, then the structure of species $\Sigma$ on $\mathrm{F_E}$ is the final structure with respect to the family $(\mathrm{A}_\iota, \mathscr{S}_\iota, \varphi_\iota)_{\iota \in \mathrm{I}}$, where $\varphi_\iota$ denotes the restriction of $\varphi_\mathrm{E}$ to $\mathrm{A}_\iota$.

Furthermore, let F be a set, and for each $\iota \in \mathrm{I}$ let $f_\iota$ be a mapping of $\mathrm{A}_\iota$ into F. If there exists a final structure of species $\Sigma$ on F with respect to the family $(\mathrm{A}_\iota, \mathrm{B}_\iota, f_\iota)_{\iota \in \mathrm{I}}$, then we may write $f_\iota = f \circ \varphi_\iota$, where $f$ is a morphism of $\mathrm{F_E}$ into F, and the structure on F is the direct image under $f$ of the structure on $\mathrm{F_E}$.

Consider the applications to the following cases :

\* (i) $\Sigma$ is a species of algebraic structures, the morphisms being homomorphisms, and the conditions $(\mathrm{CU_I})$ through $(\mathrm{CU_{III}})$ are satisfied. This is so for semi-group structures, group structures, module structures, algebra structures, etc. The $\Sigma$-set $\mathrm{F_E}$ is called the free product of the $\mathrm{A}_\iota$ in the case of groups, direct sum in the case of modules, direct product in the case of algebras.

(ii) $\Sigma$ is the species of topological group structures or the species of topological vector space structures. The conditions $(\mathrm{CU_I})$ through $(\mathrm{CU_{III}})$ are then satisfied. In the case of locally convex topological vector spaces, $\mathrm{F_E}$ is called the topological direct sum of the $\mathrm{A}_\iota$. \*