*4.3. Model Amalgamation*

Model amalgamation is one of the most important concepts/properties in institution theory. The institution theory literature contains numerous works where model amalgamation is used decisively. Refs. [2,26], etc., are representative for computing science-oriented works, especially in the area of software modularisation, while in [3] and many other articles, one may find an abundance of uses of model amalgamation in institution-independent model theory. Regarding its role in 3/2-institution theory and applications, in [13], it is argued that model amalgamation squares in 3/2-institutions constitute a superior approach to the categorical modeling of conceptual blending than 3/2 or lax colimits.

The most notorious form of model amalgamation comes from ordinary institution theory. Given a diagram of signature morphisms, a model of that is a family (*Mi*)*i*∈*<sup>I</sup>* of models, indexed by the nodes of the diagram, such that *Mi* is a Σ*i*-model, where Σ*<sup>i</sup>* is the signature at node *i*, and such that for each signature morphism *ϕ* : Σ*<sup>i</sup>* → Σ*<sup>j</sup>* in the diagram, *Mi* = *Mod*(*ϕ*)*Mj*. A cocone *μ* of the diagram has the *model amalgamation* property when for each model (*Mi*)*i*∈*<sup>I</sup>* of the diagram there exists an unique model *<sup>M</sup>* of the vertex of the colimit, such that *Mod*(*μi*)*<sup>M</sup>* = *Mi*, *<sup>i</sup>* ∈ *<sup>I</sup>*. Then *<sup>M</sup>* is called the *amalgamation* of (*Mi*)*i*∈*I*.

The most frequent use of model amalgamation is for cocones of spans of signature morphisms (which are in fact commutative squares of signature morphisms). There are also variations of the concept of model amalgamation: when we do not require the uniqueness of the amalgamation *M* (called weak model amalgamation), or when we refer only to colimits (called exactness) or even to particular colimits such as pushout squares (called semiexactness).

In stratified institution theory, there is a specific concept of model amalgamation called *stratified model amalgamation*, which corresponds to model amalgamation in the flattening <sup>S</sup> of the respective stratified institution S. This has been introduced in [14]. When the stratified institution is strict, stratified model amalgamation collapses to ordinary model amalgamation. Though in our context, this does not happen because the stratifications of the representations of 3/2-institutions as stratified institutions are proper lax natural transformations.

The 3/2-institution theoretic concept of model amalgamation [13] represents another refinement of the ordinary concept of model amalgamation. Its definition just replaces the ordinary definition of model amalgamation equalities relations with membership relations (for instance *Mi* = *Mod*(*ϕ*)*Mj* becomes *Mi* ∈ *Mod*(*ϕ*)*Mj*) and strict commutativity with lax commutativity. 3/2-institutional model amalgamation goes at the heart of the applications of 3/2-institutions because the 3/2-institution theoretic approach to conceptual blending comes with the proposal [13] to replace the original approach based on 3/2-categorical colimits [10,11] with model amalgamation cocones.

The following result establishes an equivalence relationship between stratified model amalgamation in <sup>I</sup>*<sup>s</sup>* and 3/2-model amalgamation in <sup>I</sup>.
