*2.1. Categories*

In general, we stick to the established category theoretic terminology and notations, such as in [15]. However, unlike there, we prefer to use the diagrammatic notation for compositions of arrows in categories, i.e., if *f* : *A* → *B* and *g* : *B* → *C* are arrows, then *f* ; *g* denotes their composition. The domain of an arrow/morphism *f* is denoted by ✷*f* while its codomain is denoted by *f*✷. **Set** denotes the category of sets and functions and **CAT** the "quasi-category" of categories and functors (this means it is bigger than a category since the hom-sets are classes rather than sets). The class of objects of a category **C** is denoted by |**C**|, and its class of arrows simply by **C** (so by *f* ∈ **C** we mean that *f* is an arrow in **C**).

The *dual* of a category **C** (obtained by formally reversing its arrows) is denoted by **C**-. The following functor from [13] extends the well-known power-set construction from sets to categories. Given a category **C**, the *power-set category* P**C** is defined as follows:


A *partial function f* : *A* → *B* is a binary relation *f* ⊆ *A* × *B* such that (*a*, *b*),(*a*, *b* ) ∈ *f* implies *b* = *b* . The *definition domain* of *f* , denoted dom(*f*) is the set {*a* ∈ *A* | ∃*b* (*a*, *b*) ∈ *f* }. A partial function *<sup>f</sup>* : *<sup>A</sup>* → *<sup>B</sup>* is called *total* when dom(*f*) = *<sup>A</sup>*. We denote by *<sup>f</sup>* <sup>0</sup> the restriction of *f* to dom(*f*) × *B*; this is a total function. Partial functions yield a subcategory of the category of binary relations, denoted **Pfn**. Note that dom(*f* ; *g*) = {*a* ∈ dom(*f*) |

*<sup>f</sup>* <sup>0</sup>(*a*) <sup>∈</sup> dom(*g*)}. If *<sup>A</sup>* <sup>⊆</sup> *<sup>A</sup>* by *<sup>f</sup>*(*<sup>A</sup>* ), we denote the set {*b* | ∃*a* ∈ *A* ,(*a*, *b*) ∈ *f* }. Then, *f*(*A*) is denoted by Im(*f*). It is easy to check the following (though not as immediate as in the case of the total functions): given partial functions *f* : *A* → *B* and *g* : *B* → *C* and *A* ⊆ *A*, we have that (*f* ; *g*)(*A* ) = *g*(*f*(*A* )).

A *3/2-category* is just a category such that its hom-sets are partial orders, and the composition preserves these partial orders. In the literature, 3/2-categories are also called *ordered categories* or *locally ordered categories*. In terms of enriched category theory [16], 3/2-category are just categories enriched by the monoidal category of partially ordered sets.

Given a 3/2-category **C** by **C**, we denote its "vertical" dual which reverses the partial orders, and by **C** its double dual **C**-. Given 3/2-categories **C** and **C** , a *strict 3/2-functor F* : **C** → **C** is a functor **C** → **C** that preserves the partial orders on the hom-sets. *Lax functors* relax the functoriality conditions *F*(*h*); *F*(*h* ) = *F*(*h*; *h* ) to *F*(*h*); *F*(*h* ) ≤ *F*(*h*; *h* ) (when *h*✷ = ✷*h* ) and *F*(1*A*) = 1*F*(*A*) to 1*F*(*A*) ≤ *F*(1*A*). If these inequalities are reversed, then *F* is an *oplax functor*. This terminology complies to [17] and to more recent literature, but in earlier literature [18,19] this is reversed. Note that oplax + lax = strict. In what follows, whenever we say "3/2-functor" without the qualification "lax" or "oplax" we mean a functor which is either lax or oplax.

Lax functors can be composed like ordinary functors; we denote by 3/2**CAT** the category of 3/2-categories and lax functors.

Most typical examples of a 3/2-category are **Pfn**—the category of partial functions in which the ordering between partial functions *A* → *B* is given by the inclusion relation on the binary relations *A* → *B* and **PoSET**—the category of partially ordered sets (with monotonic mappings as arrows) with orderings between monotonic functions being defined point-wise (*f* ≤ *g* if and only if *f*(*p*) ≤ *g*(*p*) for all *p*).

The following 3/2-category of [13] is instrumental for the concept of 3/2-institution. The category **CAT**<sup>P</sup> has categories as objects and has arrows/morphisms **C** → **C** as mappings **C** → P**C** . The composition in **CAT**<sup>P</sup> is defined as follows: given *F* : **C** → **C** and *F* : **C** → **C** in **CAT**P, then their composition is the mapping **C** → P**C** that maps each arrow *<sup>f</sup>* ∈ **<sup>C</sup>** to the set *<sup>f</sup>* <sup>∈</sup>*F f F f* .

By considering the point-wise partial order on the class of the mappings **C** → P**C** , we obtain a 3/2-category denoted 3/2(**CAT**P). Note that in the above definition, we do not require that the mappings **C** → P**C** are functors of any kind, not even morphisms of graphs, they are just mappings between classes of arrows. In fact, the above composition in general does not preserve functoriality properties.

#### *2.2. Institutions*

The original standard reference for institution theory is [1]. An *institution*

$$\mathcal{T} = (\operatorname{Sign}^{\mathcal{T}}, \operatorname{Sen}^{\mathcal{T}}, \operatorname{Mod}^{\mathcal{T}}, \vdash^{\mathcal{T}})$$

consists of:


$$M' \left| = \_{\Sigma'}^{\mathcal{T}} \operatorname{Sen}^{\mathcal{T}}(\boldsymbol{\varrho}) \boldsymbol{\varrho} \text{ if and only if } \operatorname{Mod}^{\mathcal{T}}(\boldsymbol{\varrho}) \\ M' \left| = \_{\Sigma}^{\mathcal{T}} \boldsymbol{\varrho} \right. \tag{1}$$

holds for each *M* ∈ |*Mod*<sup>I</sup> (*ϕ*✷)| and *ρ* ∈ *Sen*<sup>I</sup> (✷*ϕ*). This can be expressed as the satisfaction relation |= being a natural transformation:

$$\begin{array}{ccc} \Box \mathcal{q} & \operatorname{Sen}^{\mathcal{T}}(\Box \mathcal{q}) \xrightarrow{\mid \vdash^{\mathcal{L}\_{\mathcal{Q}}}} [|\operatorname{Mod}^{\mathcal{T}}(\Box \mathcal{q})| \to 2] \\ \varphi & \operatorname{Sen}^{\mathcal{T}}(\mathcal{q}) \end{array} \Big|\begin{array}{c} [\operatorname{Mod}^{\mathcal{T}}(\Box \mathcal{q})| \to 2] \\\\ \operatorname{Sen}^{\mathcal{T}}(\mathcal{q} \sqcap) \xrightarrow[|\rightsquigarrow]{} [|\operatorname{Mod}^{\mathcal{T}}(\mathcal{q} \sqcap)| \to 2] \end{array} \Big| $$

([|*Mod*(Σ)| → 2] represents the "set" of the "subsets" of |*Mod*(Σ)|).

We may omit the superscripts or subscripts from the notations of the components of institutions when there is no risk of ambiguity. For example, if the considered institution and signature are clear, we may denote |=<sup>I</sup> <sup>Σ</sup> just by |=. For *M* = *Mod*(*ϕ*)*M* , we say that *M* is the *ϕ-reduct* of *M* . The institution is called *discrete* when the model categories *Mod*(Σ) are discrete (i.e., do not posses nonidentity arrows).

The literature (e.g., [2,3]) shows myriads of logical systems from computing or from mathematical logic captured as institutions. In fact, an informal thesis underlying institution theory is that any "logic" may be captured by the above definition. While this should be taken with a grain of salt, it certainly applies to any logical system based on satisfaction between sentences and models of any kind.
