In the previous section we talked about conditions on the core of the amalgam, $H$, which might imply embeddability. This leads straight into the notion of an amalgamation base.

   Fix a class of groups $\mathfrak{C}$ for context. A group $H\in\mathfrak{C}$ is a weak amalgamation base for $\mathfrak{C}$ if every amalgam of $\mathfrak{C}$-groups with core $H$ is weakly embeddable into a group $M\in\mathfrak{C}$ (we say it is weakly embeddable in $\mathfrak{C}$). Similarly, it is a strong amalgamation base for $\mathfrak{C}$ if every amalgam of $\mathfrak{C}$-groups with core $H$ is strongly embeddable in $\mathfrak{C}$.

   For example, if $\mathfrak{C}$ is closed under finite products, then the trivial group is always a strong amalgamation base, since the amalgam of $G$ and $K$ over $\{e\}$ can be strongly embedded in $G\times K$.

   Again, a strong amalgamation base is necessarily a weak amalgamation base, but the converse need not hold. What is missing in the converse is again the notion of special amalgam. We say that $H$ is a special amalgamation base in $\mathfrak{C}$ if every special amalgam of $\mathfrak{C}$-groups with core $H$ is strongly embeddable in $\mathfrak{C}$. With that, we again have that, in a technical sense, special amalgamation bases measure the "gap" between strong bases and weak bases, and between strong and weak amalgamation:

   Theorem. A group $H$ is a strong amalgamation base if and only if it is both a weak and a special amalgamation base.

   In terms of bases, we want to give a characterization or classification of the bases for a given class. For example, there is D. Saracino's characterization of weak and strong bases for the variety of nilpotent groups of class two (see Amalgamation bases for nil-2 groups, D. Saracino, Algebra Universalis 16 (1983) 47-62), and my own characterization of the special bases for that variety (Absolutely closed nil-2 groups, Algebra Universalis 42 (1999) no. 1-2, 61-77).

   If the class C satisfies some mild closure conditions, then one can prove that in general there will be plenty of amalgamation bases. For example, Isbell proved that in a right-closed category every group can be embedded into a special amalgamation base, so these are particularly plentiful.