In this paper we consider the classes of all continuous $\mathcal {L}$-(pre-)structures for a continuous first-order signature $\mathcal {L}$. We characterize the moduli of continuity for which the classes of finite, countable, or all continuous $\mathcal {L}$-(pre-)structures have the amalgamation property. We also characterize when Urysohn continuous $\mathcal {L}$-(pre)-structures exist, establish that certain classes of finite continuous $\mathcal {L}$-structures are countable Fraïssé classes, prove the coherent EPPA for these classes of finite continuous $\mathcal {L}$-structures, and show that actions by automorphisms on finite $\mathcal {L}$-structures also form a Fraïssé class. As consequences, we have that the automorphism group of the Urysohn continuous $\mathcal {L}$-structure is a universal Polish group and that Hall’s universal locally finite group is contained in the automorphism group of the Urysohn continuous $\mathcal {L}$-structure as a dense subgroup.

A category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category I, the following are equivalent: (i) every I-shaped diagram in a category with the AP and the JEP has a cocone; (ii) every I-shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category ${\cal L}\left( {\bf{I}} \right)$ of “paths” in I has only idempotent endomorphisms. When I is a finite poset, these are further equivalent to: (iv) every upward-closed subset of I is simply-connected; (v) I can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite I.

In this paper we shall prove that any 2-transitive finitely homogeneous structure with a supersimple theory satisfying a generalized amalgamation property is a random structure. In particular, this adapts a result of Koponen for binary homogeneous structures to arbitrary ones without binary relations. Furthermore, we point out a relation between generalized amalgamation, triviality and quantifier elimination in simple theories.

Certain classes of smoothly approximable structures — the class of affine covers of Lie geometries — are shown to have the amalgamation property. In particular, this shows that any affine cover of a Lie geometry has the small index property.

2000 Mathematical Subject Classification: 03C45.