3.5.2. Model Functor

Interpretations of contexts are the **models** in an Institution of Statements.

**Definition 13** (Context interpretations)**.** *An interpretation* (*ι*, U) *of a context K* ∈ Cxt*Obj is given by a* Ξ*-structure* U = (*U*, Φ<sup>U</sup> ) *in* Sem(Ξ) *and a morphism ι* : *K* → *U in* Base*.*

*A morphism ς* : (*ι*, U) → (, V) *between two interpretations of K is given by a morphism ς* : U→V *in* Sem(Ξ) *such that ι*; *ς* = *for the underlying morphism ς* : *U* → *V in* Carr*.*

*For any context K in* Cxt *we denote by* Int(*K*) *the category of all interpretations of K and all morphisms between them and by* Π*<sup>K</sup>* : Int(*K*) → Sem(Ξ) *the obvious projection functor.*

Note that, for an initial object *K* = **0**, the projection functor Π**<sup>0</sup>** : Int(**0**) → Sem(Ξ) is an isomorphism.

For any Ξ-structure U in Sem(Ξ), the corresponding **fiber over** U, i.e., the subcategory of Int(*K*) given by all interpretations of *K* in U, is a discrete category representing the hom-set Base(*K*, *U*).

**Remark 17** (Functorial semantics)**.** *We present in this paper an abstract and general definition of Institutions of Statements covering a brought range of applications. Therefore, we are not assuming any structure on the hom-sets* Base(*K*, *U*)*.*

*In examples, following the path of Functorial Semantics,* Sem(Ξ) *will be constituted by* Ξ*structures* U = (*U*, Φ<sup>U</sup> ) *where U is provided by a category like* Set *or* Par*, for example. In those cases,* Base(*K*, *U*) *will be a category with morphisms reflecting the idea of natural transformations.*

*For those special cases, we can vary Definition 13 in such a way that a morphism between the two interpretations of K is given by a morphism ς* : *U* → *V in* Carr *and a morphism in* Base(*K*, *V*) *from ι*; *ς to . We are convinced that all the following constructions and results can be transferred, more or less straightforwardly, to this extended version of morphisms between interpretations. We let this as a topic of future research.*

Any context morphism *ϕ*: *K* → *G* induces a functor Int(*ϕ*): Int(*G*) → Int(*K*) with:

$$\mathsf{Intr}(\varphi); \Pi\_K = \Pi\_G : \mathsf{Int}(G) \to \mathsf{Sem}(\Xi) \tag{3}$$

defined by simple pre-composition: Int(*ϕ*)(, V) := (*ϕ*; , V) for all interpretations (, V) of *G*, and for any morphism *ς* : (*ι*, U) → (, V) between two interpretations of *G* the same underlying morphism *ς* : *U* → *V* in Carr establishes a morphism Int(*ϕ*)(*ς*) := *ς* : (*ϕ*; *ι*, U) → (*ϕ*; , V) between the corresponding two interpretations of *K*.

It is straightforward to validate that the assignments *K* → Int(*K*) and *ϕ* → Int(*ϕ*) define a functor Int: Cxt*op* <sup>→</sup> Cat. This is the **model functor** of an Institution of Statements.

3.5.3. Satisfaction Relation and Satisfaction Condition

The last two steps, in establishing an institution, are the definition of *satisfaction relations* and the proof of the so-called *satisfaction condition*. The satisfaction relations are simply given by the semantics of features expressions, as described in Definition 10.

**Definition 14** (Satisfaction relation)**.** *For any context K* ∈ Cxt*, any* XE(Ξ)*-statement* (*X*, *Ex*, *γ*) *in K and any interpretation* (*ι*, U) *of context K we define:*

(*ι*, U) |=*<sup>K</sup>* (*X*, *Ex*, *γ*) *iff γ*; *ι* |=<sup>U</sup> *X Ex* (*i.e. γ*; *ι* ∈ [[*Ex*]]<sup>U</sup> *<sup>X</sup>* ) (4) *K ι U X Ex γ* ( *γ*;*ι*

**Remark 18** (Validity of Closed Formulas)**.** *In case X* = *K* = **0***, we do have for any* Ξ*-structure* U = (*U*, Φ<sup>U</sup> ) *in* Sem(Ξ) *exactly one interpretation* (!*U*, U) *thus for any closed formula* (**0**, *Ex*, *id***0**) *(see Remark 15)* (!*U*, U) |=**<sup>0</sup>** (**0**, *Ex*, *id***0**) *means nothing but that the closed formula* (**0**, *Ex*, *id***0**) *is* valid in U *in the traditional sense. Therefore, we will also write* U |= (**0**, *Ex*, *id***0**) *instead of* (!*U*, U) |=**<sup>0</sup>** (**0**, *Ex*, *id***0**)*.*

*Moreover, the validity of closed formulas is* context independent *in the following sense: For any context K and any closed expressions* **0** *Ex, we have:*

$$(\iota, \mathcal{U} \urcorner) \vdash\_K (\mathbf{0}, \operatorname{Ex} \iota \mathbf{1}\_K) \quad \text{iff} \quad \mathsf{l}\_K \urcorner \iota = \mathsf{l}\_{\mathcal{U}} \urcorner \vdash^{\mathcal{U}} \mathbf{0} \rhd \mathbf{Ex} \quad \text{iff} \quad \|\operatorname{Ex}\|\_{\mathbf{0}}^{\mathsf{l}} = \{\mathsf{l}\_{\mathcal{U}}\} \quad \text{iff} \quad \mathsf{l} \urcorner = (\mathsf{0}, \operatorname{Ex}, \operatorname{id}\_{\mathbf{0}}) $$

After we developed everything in a systematic modular way, we obtain the satisfaction condition nearly "for free".

**Corollary 1** (Satisfaction condition)**.** *For any morphism ϕ*: *K* → *G in* Cxt*, any* XE(Ξ)*-statement* (*X*, *Ex*, *γ*) *in K and any interpretation* (, U) *of context G we have:*

$$\mathsf{Int}(\varphi)(\varrho, \mathcal{U}) \left| =\_{\mathcal{K}} (X, \mathcal{E}x, \gamma) \right. \quad \left. \left. \begin{array}{c} \left(\varrho, \mathcal{U}\right) \left| =\_{\mathcal{G}} \mathsf{Stm}(\varphi)(X, \mathcal{E}x, \gamma) . \right. \end{array} \right. \right|$$

**Proof.** Due to the definition of the functors Int : Cxt*op* <sup>→</sup> Cat and Stm : Cxt <sup>→</sup> Set, we obtain the commutative diagram, above on the right, thus the satisfaction condition follows immediately from Definition 14.

**Remark 19** (Satisfaction Condition)**.** *As mentioned in the introductory Section 1.1.6, the finding of* corresponding assignments *and* corresponding evaluations *enabled us to prove in [29] the satisfaction condition for four formalisms in a systematic, uniform and straightforward way. The proof of Corollary 1 mirrors the essence of this uniform and straightforward way at a very high abstraction level.*

Summarizing all definitions and results, we obtain the main result of this section:

**Theorem 1** (Institution of Statements)**.** *Any choice of a category* Base*, of subcategories* Var*,* Cxt*,* Carr *of* Base*, of a footprint* Ξ *over* Var*, of a category* Sem(Ξ) *of* Ξ*-structures and of an* Var*Obj-indexed family* XE(Ξ) *of first-order* Ξ*-expressions establishes a corresponding Institution of Statements* IS = (Cxt, Stm, Int, |=)*.*

**Remark 20** (Indexed institutions)**.** *We come back to the discussion in Remark 5. If we consider a category of footprints over* Var *we will obtain, due to Theorem 1, for each footprint a corresponding institution of statements. To lift morphisms between footprints to corresponding morphisms between institutions of statements, we have, however, to coordinate somehow the construction of the different institutions (consult Figure 1).*

*All institutions should share, besides* Base *and* Var *also the same categories* Carr *and* Cxt*. We have to show that this assumption ensures that the assignments* Ξ → Str(Ξ) *can be lifted to a functor* Str*. Analogously, the assignments* Ξ → FE(Ξ) *should also provide a functor* FE*. Finally, the choices of* Sem(Ξ) *and* XE(Ξ) *have to be aligned in such a way that we obtain corresponding restrictions of the functors* Str *and* FE*, respectively.*

*Under these assumptions, we will hopefully be able to establish a category of institutions of statements indexed by the category of footprints; thus, we can benefit from all the nice results and constructions in [2]. In particular, the construction of the corresponding Grothendieck institution will surely become relevant.*
