**3. The Canonical Stratified Institution Associated to a 3/2-Institution**

The representation of 3/2-institutions as stratified institutions is in general partial in the sense that the signature morphisms that are subject to the representation have to satisfy certain technical conditions. Two of these are defined below. The second one appears as a 3/2-institution theoretic replica of a property from ordinary institution theory [3] with the same name but in a somewhat reverse form. While the former concept is a lifting concept, the 3/2-institution theoretic one may have an opposite appearance because it goes along the direction of the model reduct. However, this is misleading because in 3/2-institutions, due to the implicit partiality of the signature morphisms, reducts also have a nature of expansion. Towards the end of this section, we discuss what these two properties mean in concrete situations. The constructions and the results in this section are developed at the abstract level. It would be helpful if the reader would interpret them in the context of the examples of 3/2-institutions listed above. This should be a rather straightforward

exercise, especially if one considers the discussion on the technical conditions at the end of this section.

**Definition 1.** *In any 3/2-institution, a signature morphism χ*


We now fix a 3/2-institution I = (*Sign*, *Sen*, *Mod*, |=) and gradually build the entities that define its associated stratified institution <sup>I</sup><sup>s</sup> = (*Sign*<sup>s</sup> , *Sen*<sup>s</sup> , *Mod*<sup>s</sup> , [[\_]], <sup>|</sup>=s). The main idea of this representation is that the reducts of a model *M* are considered to be its states. In order to make precise sense of this idea, we have to change the concept of signature: in the stratified institution, a signature is a certain signature morphism *χ* in I, such that *M* is a *χ*✷-model. It is the abstract nature of the concept of institution that allows for such a conceptual twist.

**Definition 2** (The category of the signatures)**.** *The category Sign*<sup>s</sup> *has the objects the fiber-small quasi-representable signature morphisms <sup>χ</sup> of Sign. The arrows <sup>χ</sup>* <sup>→</sup> *<sup>χ</sup> in Sign*<sup>s</sup> *are pairs of signature morphisms* (*ϕ*, *θ*)*, such that*

• *Both ϕ and θ are total and Mod-strict;*

*.*

• *χ*; *θ* ≤ *ϕ*; *χ* 

*The composition in Sign*<sup>s</sup> *is defined as pairwise composition in Sign, i.e.,* (*ϕ*, *θ*);(*ϕ* , *θ* ) = (*ϕ*; *ϕ* , *θ*; *θ* )*, as shown in the following diagram:*

*An arrow* (*ϕ*, *θ*) : *χ* → *χ is* strict *when χ*; *θ* = *ϕ*; *χ .*

We have the correctness of definition 1:

**Proposition 1.** *Sign*<sup>s</sup> *is a category.*

**Proof.** We have to prove that the composition preserves the preorder property. This follows from the monotonicity of the composition in *Sign* (we use the notations from (7)):

$$
\chi; \theta; \theta' \le \varphi; \chi'; \theta' \le \varphi; \varphi'; \chi'.
$$

It remains to note that totality and *Mod*-strictness when considered together are preserved by the composition of the signature morphisms. (*Mod*-strictness supports the preservation of *Mod*-maximality by the composition of signature morphisms.)

**Definition 3** (The sentence translation functor)**.** *For any Sign*<sup>s</sup> *signature χ, we define Sen*<sup>s</sup> (*χ*) = *Sen*(✷*χ*) *and for any Sign*<sup>s</sup> *-morphism* (*ϕ*, *θ*)*, we define Sen*<sup>s</sup> (*ϕ*, *θ*) = *Sen*(*ϕ*)*.*

**Proposition 2.** *Sen*<sup>s</sup> *is a functor Sign*<sup>s</sup> <sup>→</sup> **Set***.*

**Proof.** This is an immediate consequence of the functoriality of *Sen* and of the totality hypothesis, which guarantees that *Sen*(*ϕ*) is indeed a total function.

**Definition 4** (The model reduct functor)**.** *For any Sign*<sup>s</sup> *signature χ, we define Mod*<sup>s</sup> (*χ*) = *Mod*(*χ*✷) *and for any Sign*<sup>s</sup> *-morphism* (*ϕ*, *θ*) *we define Mod*<sup>s</sup> (*ϕ*, *θ*) = *Mod*(*θ*)*.*

**Proposition 3.** *Mod*<sup>s</sup> *is a functor* (*Sign*<sup>s</sup> )-→ **CAT***.*

**Proof.** This is an immediate consequence of the functoriality of *Mod* and of the *Mod*maximality and *Mod*-strictness hypothesis (on *θ*).

**Definition 5** (The stratification)**.** *For any Sign*<sup>s</sup> *signature χ, we define*


*For each signature morphism* (*ϕ*, *<sup>θ</sup>*) : *<sup>χ</sup>* <sup>→</sup> *<sup>χ</sup> in Sign*<sup>s</sup> *, we define:*

• [[*M* ]](*<sup>ϕ</sup>*,*θ*)*<sup>N</sup>* <sup>=</sup> *Mod*(*ϕ*)*<sup>N</sup> for any M* ∈ |*Mod*<sup>s</sup> (*χ* )| *and any N* ∈ [[*M* ]]*<sup>χ</sup> .*

**Proposition 4.** [[*\_*]] *is a lax natural transformation Mod*<sup>s</sup> <sup>⇒</sup> *SET.*

**Proof.** The correctness of definition 5 is justified as follows:


$$\begin{array}{llll} \left[M'\right]\_{(\varphi,\theta)}(\left[M'\right]\_{X'}) &=& \operatorname{Mod}(\varphi)(\operatorname{Mod}(\chi')M') & \text{ definition of } \left[\Box\right] \\ &\subseteq& \operatorname{Mod}(\varphi;\chi')M' & \text{Mod} \,\operatorname{lux} \\ &\subseteq& \operatorname{Mod}(\chi;\theta)M' & \chi;\theta \le \varphi;\chi',\operatorname{Mod} \,\operatorname{monotone} \\ &=& \operatorname{Mod}(\chi)(\operatorname{Mod}(\theta)M') & \theta \operatorname{Mod}\text{-strict} \\ &=& \left[\operatorname{Mod}(\theta)M'\right]\_{X} & \text{definition of } \left[\Box\right]\_{X'}\theta \operatorname{Mod}\text{-maximal} \\ &=& \left[\operatorname{Mod}^{\mathfrak{s}}(\varphi,\theta)M'\right]\_{X} & \text{definition of } \operatorname{Mod}^{\mathfrak{s}}. \end{array}$$

The *functoriality of* [[*\_*]]*<sup>χ</sup>* : *Mod*<sup>s</sup> (*χ*) → **Set** means two things:


$$\begin{array}{ccccc} h^N; h\_0^{N\_0} \in & \operatorname{Mod}(\chi)h \; ; \; \operatorname{Mod}(\chi)h\_0 & & \text{definitions of } \; h\_0^N, h\_0^{N\_0} \\ \subseteq & \operatorname{Mod}(\chi)(h; h\_0) & & \text{Mod}(\chi) \; \operatorname{lax} .\end{array}$$

From the uniqueness aspect of the quasi-representability property of *χ*, it follows that

$$h^N; h\_0^{N\_0} = (h; h\_0)^N. \tag{8}$$

Hence,

$$\begin{aligned} \left[\![h;h\_0]\!\right]\_X &N = \begin{array}{c} \left(h;h\_0\right)^N \Box \end{array} &\text{definition of } \left[\![h;h\_0]\!\right]\_X N = \left[\![h^N;h\_0^N]\!\right]\_Y\\ &= \left(\!(h^N;h\_0^N)\!\right) \Box &\text{(8)}\\ &= \left[\![h\_0]\!\right]\_X \left(\![h]\!\right]\_X N\text{)} &\text{definition of } \left.N\_0, \left\[\![h\_0]\!\right]\_X N\_0, \left\[\![h]\!\right]\_X N\right] \end{aligned}$$

For proving the *lax natural transformation property of* [[*\_*]] (relation ((3))), we consider a composition of signature morphisms in *Sign*<sup>s</sup> such as in diagrams (7), an Ω -model *M* , and *N* ∈ [[*M* ]]*<sup>χ</sup>* = *Mod*(*χ* )*M* . Note that since *N* ∈ [[*M* ]]*<sup>χ</sup>* , we have that

$$\mathrm{Mod}(\varrho')N'' = \left\lbrack \mathrm{M}' \right\rbrack \mathbb{I}\_{\left(\varrho',\theta'\right)} \mathrm{N}'' \in \left\llbracket \mathrm{Mod}"\left(\varrho',\theta'\right)\mathrm{M}'' \right\rbrack\_{\chi'} = \left\llbracket \mathrm{Mod}(\theta')\mathrm{M}'' \right\rbrack\_{\chi'} \tag{9}$$

Then, we have:

$$\begin{array}{ll} \left[M^{\theta}\right]\_{(\varphi,\mathfrak{g}',\mathfrak{g}')}N^{\prime\prime} = & \\ = & \left[M^{\prime\prime}\right]\_{(\varphi,\mathfrak{g}',\mathfrak{g}')}N^{\prime\prime} & \text{ definition of }\operatorname{Sign}^{\mathfrak{g}} \\ = & \operatorname{Mod}(\varphi,\mathfrak{g}')N^{\prime\prime} & \text{definition of }\left[\ldots\right] \\ = & \operatorname{Mod}(\mathfrak{g})(\operatorname{Mod}(\mathfrak{g}')N^{\prime\prime}) & \text{Mod functor},\ \mathfrak{g}\operatorname{Mod}\operatorname{cst}\operatorname{cst}\operatorname{of} \\ = & \left[\operatorname{Mod}(\mathfrak{g}')M^{\prime\prime}\right]\_{(\varphi,\mathfrak{g})}(\operatorname{Mod}(\mathfrak{g}')N^{\prime\prime}) & \text{(9), definition of }\left[\ldots\right] \\ = & \left[\operatorname{Mod}^{\mathfrak{s}}(\mathfrak{g}',\mathfrak{g}')M^{\prime\prime}\right]\_{(\varphi,\mathfrak{g})}(\operatorname{Mod}(\mathfrak{g}')N^{\prime\prime}) & \text{defination of }\operatorname{Mod}^{\mathfrak{s}} \\ = & \left[\operatorname{Mod}^{\mathfrak{s}}(\mathfrak{g}',\mathfrak{g}')M^{\prime\prime}\right]\_{(\varphi,\mathfrak{g})}(\left[\operatorname{Mod}^{\mathfrak{s}}\right]\_{(\varphi,\mathfrak{g}')}N^{\prime\prime}) & \text{defination of }\left[\ldots\right]. \end{array}$$

For proving the *natural transformation property of* [[*\_*]](*<sup>ϕ</sup>*,*θ*), under the notations from diagrams (7) by considering an Ω -model homomorphism *h* : *M* → *M* <sup>0</sup>, we have to show that the diagram below commutes:

$$\begin{array}{cc} M' & \operatorname{Mod}(\chi')M' & \xrightarrow{\operatorname{[Lh']}\_{(q,\theta)}} \operatorname{Mod}(\chi)(\operatorname{Mod}(\theta)M')\\\ h' & & [h']\_{\chi'} \end{array} \tag{10}$$
 
$$\begin{array}{cc} M'\_0 & \xrightarrow{\operatorname{[Lh']}\_{\chi'}} & \operatorname{Mod}(\chi')M'\_0 \xrightarrow{\operatorname{[Lhd}(\theta)h']\_{\chi}} \operatorname{Mod}(\chi)(\operatorname{Mod}(\theta)M'\_0) \end{array} \tag{11}$$

Let *N* ∈ *Mod*(*χ* )*M* , and let us denote *N* = *Mod*(*ϕ*)*N* and *h* = *Mod*(*θ*)*h* . Let us first prove that

$$\operatorname{Mod}(\varphi)h^{\prime N^{\prime}} = h^N. \tag{11}$$

On the one hand, we have:

$$\begin{array}{ccc} \mathit{Mod}(\mathfrak{q})h^{\mathcal{N}^{\mathcal{N}}} \in & \mathit{Mod}(\mathfrak{q})(\mathit{Mod}(\chi')h') & \mathrm{definit}(\mathfrak{q})\ h'^{\mathcal{N}^{\mathcal{N}}}\\ & \subseteq & \mathit{Mod}(\mathfrak{q};\chi')h' & & \mathit{Mod} \text{ lax} \\ & \subseteq & \mathit{Mod}(\chi;\theta)h' & & \mathit{Mod} \text{ monotone} \text{one}, \ \chi;\theta \leq \mathfrak{q};\chi' \\ & = & \mathit{Mod}(\chi)(\mathit{Mod}(\theta)h') & & \theta \text{ Mod}\text{-strict} \\ & = & \mathit{Mod}(\chi)h. & & \end{array}$$

On the other hand, we have:

$$\Box \mathcal{M} \text{od}(\varrho) h^{\prime N^{\prime}} = \mathcal{M} \text{od}(\varrho) (\Box h^{\prime N^{\prime}}) = \mathcal{M} \text{od}(\varrho) N^{\prime} = N \Box$$

Then, (11) follows from *Mod*(*ϕ*)*h <sup>N</sup>* <sup>∈</sup> *Mod*(*χ*)*<sup>h</sup>* and ✷*Mod*(*ϕ*)*<sup>h</sup> <sup>N</sup>* = *N* and from the uniqueness aspect of the quasi-representability property of *χ*. Now, the following argument completes the proof of the natural transformation property of [[\_]](*<sup>ϕ</sup>*,*θ*):

$$\begin{array}{rclcrcl} \left[ \operatorname{Mod}(\theta)h' \right]\_{\mathbb{X}} (\left[ \big[ M' \big]\_{(\ \rho, \theta)} \big] \operatorname{N}') &=& \left[ \operatorname{Mod}(\theta)h' \big]\_{\mathbb{X}} (\operatorname{Mod}(\rho) \operatorname{N}') & \text{ definition of } \left[ \big[ M' \big]\_{(\ \rho, \theta)} \right] \\ &=& h^{N} \big[ \big[ \begin{array}{c} \operatorname{definit}(\ \rho \end{array} \text{ definition of } \left[ \operatorname{Mod}(\theta)h' \right] \big]\_{\mathbb{X}} = \left[ \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{definition of } \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \right] \\ &=& \left( \operatorname{Mod}(\rho)h'^{N} \right) \big[ \big[ \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{definition of } \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{definition of } \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{)} \\ &=& \left( \operatorname{Mod}(\rho)h'^{N'} \right) \big[ \big[ \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{definition of } \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{)} \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{\mathbb{X}} \end{array} \text{\mathbb{X}} \end{array} \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{\mathbb{X}} \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{array} \text{\mathbb{X}} \end{array} \begin{array}{c} \operatorname{Id}\_{\mathbb{X}} \end{$$

**Definition 6** (The satisfaction relation)**.** *For each signature χ in Sign*<sup>s</sup> *, each χ*✷*-model M, each* ✷*χ-model N* ∈ [[*M*]]*χ, and each* ✷*χ-sentence ρ,*

$$M(\vdash^s)^N\_{\mathcal{X}}\rho \quad \text{if and only if} \quad \mathcal{N} \vdash\_{\bigcirc \mathcal{X}} \rho.$$

**Proposition 5.** *For any signature morphism* (*ϕ*, *<sup>θ</sup>*) : *<sup>χ</sup>* <sup>→</sup> *<sup>χ</sup> in Sign*<sup>s</sup> *, any χ -model M , any N* ∈ [[*M* ]]*<sup>χ</sup> , and any χ-sentence ρ:*

$$M' \equiv\_{\chi'}^{N'} \operatorname{Sen}^{\mathfrak{s}}(\mathfrak{q}, \theta)\mathfrak{p} \text{ if and only if } \operatorname{Mod}^{\mathfrak{s}}(\mathfrak{q}, \theta)M' \mid\_{X} \mathfrak{p}.$$

**Proof.** By similarity to (9), we have that:

$$\operatorname{Mod}(\mathfrak{q})\mathcal{N}' \in \left\| \operatorname{Mod}(\mathfrak{\theta})\mathcal{M}' \right\|\_{\mathcal{X}} \tag{12}$$

Then, we have:

$$\begin{array}{ccccc} M' \vert \vert =^{s} \rangle\_{\chi}^{\mathcal{N}'} Sm^{s} (\varphi, \theta) \rho & \Leftrightarrow & N' \vert =\_{\Box \chi'} Sm (\varphi) \rho & \text{definition of } \mid =^{s} \simeq \\ & & \Leftrightarrow & Mod (\varphi) N' \vert =\_{\Box \chi} \rho & \text{Satisfation Condition of } \mathcal{Z} \\ & & \Leftrightarrow & Mod (\theta) M' \langle \vert =^{s} \rangle\_{\chi}^{Mod (\varphi) N'} \rho & \text{(12), definition of } \mid =^{s} \text{s} \\ & & \Leftrightarrow & Mod (\theta) M' \langle \vert =^{s} \rangle\_{\chi}^{\left[ \!\!\!M \"\!\rceil\_{(\varphi \theta)} \right]^{N'}} & & \text{definition of } \mid\_{\Box} \text{ (1)} \\ & & & \Leftrightarrow & Mod ^{\mathfrak{s}} (\varphi, \theta) M' \langle \vert =^{s} \rangle\_{\chi}^{\left[ \!\!\!M \"\!\rceil\_{(\varphi \theta)} \right]^{N'}} & & \text{definition of } \mathit{Mod} \, ^{\mathfrak{s}}. \\ \end{array}$$

By putting together propositions 1–5, we obtain:

**Corollary 1.** <sup>I</sup><sup>s</sup> = (*Sign*<sup>s</sup> , *Sen*<sup>s</sup> , *Mod*<sup>s</sup> , [[*\_*]], <sup>|</sup>=s) *is a stratified institution.*

The technical conditions underlying the construction of <sup>I</sup><sup>s</sup> imposes some restriction both on the <sup>I</sup> signature morphisms *<sup>χ</sup>* that play the role of *Sign*<sup>s</sup> -signatures and on the I signature morphisms that make up the *Sign*<sup>s</sup> morphisms. Let us see their significance and what they might mean in concrete situations.


context doe it amount to a certain restriction, namely that there is no partiality of the translation of the sorts.

	- **–** As it is about (proper) model homomorphisms, it holds trivially in their absence. This degenerated situation is in fact the norm in the applications of the 3/2 institutions, as until now there are not known applications that involve proper model homomorphisms.
	- **–** When *χ* admits partiality only on the constants, then the quasi-representable holds.
	- **–** When *χ* admits partiality only on the relation symbols and the model homomorphisms are "strong" (in the sense of [3]), then *χ* is quasi-representable, too.
