2. Manes collection monads

We recall the two equivalent definitions of the collection monad on the category ${\mathscr{S}}{\kern.8pt}$ of sets and some properties of ${\mathscr{S}}{\kern.8pt}$ and finitary monads on ${\mathscr{S}}{\kern.8pt}$ (for proofs the reader should refer to Manes (Reference Manes1998)).

We would like to replace the category ${\mathscr{S}}{\kern.8pt}$ of sets with other categories, which must be at least lextensive (i.e., they have finite limits and well-behaved finite sums; see Carboni et al. (Reference Carboni, Lack and Walters1993)). Therefore, one cannot rely on the peculiar properties of ${\mathscr{S}}$ . For instance, in a category ${\mathscr{C}}{\kern.8pt}$ with finite products, one can define the category ${\mathscr{A}}$ of algebras in ${\mathscr{C}}{\kern.8pt}$ for an equational theory $(\Omega ,E)$ , but the forgetful functor $U:\mathscr{A}\xrightarrow {\,\,\,\hspace{6pt}}{\mathscr{C}}{\kern.8pt}$ may not have a left adjoint (i.e., free algebras may not exist). However, the second characterization of collection monads generalizes to any lex category ${\mathscr{C}}$ , provided one has a substitute for the finite powerset monad $\mathbb{P}_f$ , which plays the role of the mother of all collection monads on ${\mathscr{C}}$ .

There are full sub-categories of ${\mathscr{S}}{\kern.8pt}$ that better resemble the categories we are looking for.

3. Lextensive categories

Lextensive categories are categories with finite limits and well-behaved finite sums (aka coproducts), as formalized by the notion of extensive category in Carboni et al. (Reference Carboni, Lack and Walters1993). By lextensive functor, we mean a functor between lextensive categories that preserves finite limits and finite sums. Similarly, we call a lex category a category with finite limits and a lex functor a functor between lex categories that preserves finite limits.

Any (locally small) category ${\mathscr{C}}{\kern.8pt}$ with a terminal object $1$ has a global section functor ${\mathscr{C}}{\kern.5pt}(1,-):{\mathscr{C}}\xrightarrow {\,\,\,\hspace{6pt}}\mathscr{S}$ , which preserves all small limits existing in ${\mathscr{C}}$ . We are mainly interested in (essentially small) lextensive categories with the property that the global section functor is faithful and lextensive. The following result gives sufficient conditions (in fact also necessary) on a lextensive category to have such a property.

Proof. Since ${\mathscr{C}}{\kern.8pt}$ has enough points, the global section functor is faithful and lex. To prove that it is extensive (i.e., it preserves finite sums), it suffices to prove that when $X_0\xrightarrow {\,\,\,\hspace{6pt}} 1\xleftarrow {\,\,\,\,\textrm {id}\,\,\,\,} X_1$ is a coproduct diagram, exactly one of the $X_i$ is initial.

• • The coproduct injections $X_i\xrightarrow {\,\,\,\hspace{6pt}} 1$ are monic (because ${\mathscr{C}}{\kern.8pt}$ is extensive); thus, for any $X\in {\mathscr{C}}$ , the hom-set ${\mathscr{C}}{\kern.5pt}(X,X_i)$ has at most one arrow (because $1$ is terminal).

• • If $|X_i|={\mathscr{C}}{\kern.5pt}(1,X_i)$ has one arrow, then $X_i$ is terminal (i.e., $X_i\xrightarrow {\,\,\,\hspace{6pt}} 1$ is an iso).

• • If $|X_i|={\mathscr{C}}{\kern.5pt}(1,X_i)$ is empty, then for any $Y\in {\mathscr{C}}$ the hom-set ${\mathscr{C}}{\kern.5pt}(X_i,Y)$ has at most one arrow (because $|-|$ is faithful); thus, $X_i\xrightarrow {\,\,\,\,\textrm {id}\,\,\,\,} X_i\xleftarrow {\,\,\,\,\textrm {id}\,\,\,\,} X_i$ is a coproduct, which implies $X_i$ initial (because in an extensive category coproduct injections are disjoint).

• • In conclusion, each $X_i$ is either initial or terminal.

• • It cannot be that both $X_0$ and $X_1$ are initial; otherwise, $1$ would be initial, contradicting the assumption $0\not \cong 1$ .

• • It cannot be that both $X_0$ and $X_1$ are terminal; otherwise, $1\xrightarrow {\,\,\,\hspace{6pt}}^{\textrm {id}_{}} 1\xleftarrow {\,\,\,\,\textrm {id}\,\,\,\,}^{} 1$ is a coproduct, contradicting the assumption $0\not \cong 1$ .

If $X\cong \sum _{i\in n}1$ (i.e., $X$ is the sum of $n$ copies of the terminal object), then we have the following sequence of isomorphisms natural in $Y$ (because $|-|$ is lextensive)

\begin{equation*}{\mathscr{C}}{\kern.5pt}(X,Y)\cong \prod _{i\in n}{\mathscr{C}}{\kern.5pt}(1,Y)=|Y|^n\cong \prod _{i\in n}\mathscr{S}{\kern.5pt}(1,|Y|)\cong \mathscr{S}{\kern.5pt}(\sum _{i\in n}1,|Y|)\cong \mathscr{S}{\kern.5pt}(|X|,|Y|)\end{equation*}

Lextensivity is preserved by several constructions, for example:

• • a slice category ${\mathscr{C}}/I$ of a lextensive category is lextensive (Carboni et al., Reference Carboni, Lack and Walters1993, Proposition 4.8),

• • a product $\prod _{i\in I}{\mathscr{C}}_i$ of lextensive categories is lextensive, since limits and colimits in $\prod _{i\in I}{\mathscr{C}}_i$ are computed pointwise,

• • a functor category ${\mathscr{C}}^{{\kern1.5pt}\mathscr{A}}$ is lextensive when ${\mathscr{C}}{\kern.8pt}$ is lextensive, since limits and colimits in ${\mathscr{C}}^{{\kern1.5pt}\mathscr{A}}$ are computed pointwise.

On the contrary, the property that the global section functor is lextensive may not be preserved, for example, $\mathscr{S}^{{\kern1.5pt}2}$ (or equivalently $\mathscr{S}/2$ ) does not have such property, since the sum $1+1$ in $\mathscr{S}^{{\kern1.5pt}2}$ has 4 global elements instead of 2.

4. Display maps and dominions

We recall the definitions of class of display maps from Taylor (Reference Taylor1987, Reference Taylor1999), dominion from Rosolini (Reference Rosolini1986) and Robinson and Rosolini (Reference Robinson and Rosolini1988), and properties of endo-functors and natural transformations w.r.t. a dominion (or a class of display maps) from Moggi (Reference Moggi1995). For simplicity we cast these definitions in the setting of a lex category.

The original definition of CDM in Taylor (Reference Taylor1987) does not require closure w.r.t. identities and composition, while the definition of dominion does. If a classifier $p:E\xrightarrow {\,\,\,\hspace{6pt}} U$ is a dominance, then $E$ is necessarily a terminal object; see (Moggi, Reference Moggi1995, Lemma 3.11).

The composition $\mathscr{D}\circ \mathscr{E}\stackrel {\Delta }{=}\{f|\exists d\in \mathscr{D}.\exists e\in \mathscr{E}.f=d\circ e\}$ of two CDM may not be a CDM, but $(\mathscr{D}\cup \mathscr{E})^+$ is the smallest CDM containing it, and $\mathscr{D}\circ \mathscr{E}=(\mathscr{D}\cup \mathscr{E})^+$ when $\mathscr{E}\circ \mathscr{D}\subseteq \mathscr{D}\circ \mathscr{E}$ .

If Every endo-functor on a lex category ${\mathscr{C}}{\kern.8pt}$ with enough points has at most one tensorial strength. Therefore, one has the following inclusions ${\textbf{Endo}}_{s\mathscr{E}}{\kern1pt}({\mathscr{C}})\subseteq {\textbf{Endo}}_{\mathscr{E}}{\kern1pt}({\mathscr{C}})\subseteq {\textbf{Endo}}_{\mathscr{D}}{\kern.8pt}({\mathscr{C}})$ among strict monoidal categories, where $\mathscr{D}\subseteq \mathscr{E}$ are CDM in ${\mathscr{C}}$ .

A lextensive category ${\mathscr{C}}{\kern.8pt}$ may not have a NNO nor enough dependent types to mimic the definitions given in ${\mathscr{S}}{\kern.8pt}$ for the maps $p:\sum n:\mathbb{N}.n\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{N}$ , $u:1\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{N}$ , $a:\sum n:\mathbb{N}.\mathbb{N}^n\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{N}$ , and the list monad ${\mathbb{L}} X\stackrel {\Delta }{=}\sum n:\mathbb{N}.X^{n}$ . However, when ${\mathscr{C}}{\kern.8pt}$ is small, one can replace ${\mathscr{C}}{\kern.8pt}$ with the Gaeta topos $G({\mathscr{C}})$ , that is, the topos of sheaves for the sub-canonical topology $J_G$ of the finite coproduct diagrams, since the Yoneda embedding from ${\mathscr{C}}{\kern.8pt}$ to the topos $G({\mathscr{C}})$ is lextensive. Therefore a map $p:E\to U$ in ${\mathscr{C}}{\kern.8pt}$ induces a polynomial functor $P$ on $G({\mathscr{C}})$ . The following result, which we state without proof, ensures that when $p$ is a classifier for a CDM ${\mathscr{D}}$ in ${\mathscr{C}}$ , then $Yp$ is a classifier for a CDM $\mathscr{D}^{{\kern1pt\prime}}$ in $G({\mathscr{C}})$ , which coincides with ${\mathscr{D}}{\kern1pt}$ when restricted to (the image of) ${\mathscr{C}}$ .

5. Evaluation logic in lextensive categories

In conformity with categorical semantics for fragments of predicate logic, Evaluation Logic (see Moggi (Reference Moggi1995) can be interpreted in any lex category ${\mathscr{C}}{\kern.8pt}$ after choosing a dominion ${\mathscr{M}}$ and a strong ${\mathscr{M}}$ -monad ${\mathbb{M}}$ , namely

• • A type $\tau \in T_{EL}::=1\ |\ \tau _0\times \tau _1\ |\ \ldots \ |\ {\mathbb{M}}\tau$ (and a context $\Gamma$ ) is interpreted by an object in ${\mathscr{C}}{\kern.8pt}$

• • a well-formed term $\Gamma \vdash _{} \tau :{e}$ is interpreted by an arrow in ${\mathscr{C}}{\kern.8pt}$ from (the interpretation of) $\Gamma$ to (that of) $\tau$ , where ${e}\in E_{EL}::=x\ |\ f({e}_i|i:n)\ |\ \ldots \ |\ [{e}]_{}\ |\ \textrm {let}_{}\,x{\Leftarrow }{e}_0\,\textrm {in}\,{e}_1$ and the formation rules for $[{e}]_{}$ and $\textrm {let}_{}\,x{\Leftarrow }{e}_0\,\textrm {in}\,{e}_1$ are

  

• • a well-formed formula $\Gamma \vdash _{} \phi :\mathsf{prop}$ is interpreted by an ${\mathscr{M}}$ -subobject of (the interpretation of) $\Gamma$ , where $\phi \in F_{EL}::= p({e}_i|i:n)\ |\ \top \ |\ \phi _1\land \phi _2\ |\ \ldots \ |\ [x{\Leftarrow }{e}]_{}\phi$ and the formation rule for necessity $[x{\Leftarrow }{e}]_{}\phi$ is

  

  If $1\xrightarrow {\,\,\,\hspace{6pt}}\Omega$ is a dominance for ${\mathscr{M}}$ , then $\Omega$ is like a type of truth values; thus, formulas can be viewed as terms of type $\Omega$ and interpreted as arrows from $\Gamma$ to $\Omega$ .

The valid inference rules for Evaluation Logic depend on the assumptions on the category ${\mathscr{C}}$ , the dominion ${\mathscr{M}}$ , and the strong ${\mathscr{M}}$ -monad ${\mathbb{M}}$ used for interpreting the logic. We consider four sets of inference rules (from weaker to stronger):

1. (1) general rules, valid in any lex category ${\mathscr{C}}{\kern.8pt}$ with a dominion ${\mathscr{M}}$ and a strong ${\mathscr{M}}$ -monad ${\mathbb{M}}$

2. (2) rules valid when ${\mathscr{M}}$ includes regular monos; thus, one can interpret equality as ${e}_0={e}_1$

3. (3) rules valid when ${\mathscr{C}}{\kern.8pt}$ is lextensive and ${\mathscr{M}}$ is the dominion of decidable monos, and they coincide with regular monos

4. (4) additional rules valid for the powerset monad on ${\mathscr{S}}{\kern.8pt}$ and its sub-monads.

The third set of rules is particularly relevant, because it is possible to define a decidable evaluation predicate that can be interpreted as membership $c\ni x$ , that is, “element $x$ belongs to collection $c$ ,” and a limited form of existential quantification $\langle x{\Leftarrow }{e} \rangle _{}\phi$ , defined as $\neg [x{\Leftarrow }{e}]_{}\neg \phi$ . Moreover, in combination with the fourth set of rules, it allows establishing a link with Algebraic Set Theory (Joyal and Moerdijk, Reference Joyal and Moerdijk1995; Simpson, Reference Simpson1999; Awodey et al., Reference Awodey, Butz, Simpson and Streicher2014).

5.1 General rules

The following logical equivalences ( $\Gamma \vdash _{} \phi _1 \Longleftrightarrow \phi _2$ ) and inference rules (for deriving sequents $\Gamma \vdash _{} \Phi \Longrightarrow \phi$ ) involving necessity are sound w.r.t. any interpretation of Evaluation Logic.

5.2 Rules for equality

If the dominion ${\mathscr{M}}$ includes all regular monos, then the equality predicate ${e}_0={e}_1$ can be added to the set $F_{EL}$ of formulas, and the following inference rule involving necessity and equality is sound.

5.3 Rules for lextensive categories

If ${\mathscr{C}}{\kern.8pt}$ is lextensive and every object in ${\mathscr{C}}{\kern.8pt}$ has a decidable equality, then we may take for ${\mathscr{M}}$ the dominion $\mathscr{M}_2$ of decidable monos, which is classified by the dominance $1\xrightarrow {\,\,\,\hspace{6pt}} 2$ ) and coincides with the dominion of regular monos. In this case, one can extend the set $T_{EL}$ of types with empty $0$ and sum $\tau _0+\tau _1$ types (extend the set $E_{EL}$ of terms accordingly) and add negation $\neg \phi$ to the set $F_{EL}$ of formulas, since $2$ supports classical boolean logic. Therefore, one can define

• • disjunction $\phi _1\vee \phi _2\stackrel {\Delta }{\Longleftrightarrow }\neg (\neg \phi _1\wedge \neg \phi _2)$ ,

• • implication $\phi _1\supset \phi _2\stackrel {\Delta }{\Longleftrightarrow }\neg \phi _1\vee \phi _2$ ,

• • possibility $\langle x{\Leftarrow }{e} \rangle _{}\phi \stackrel {\Delta }{\Longleftrightarrow }\neg [x{\Leftarrow }{e}]_{}\neg \phi$ , and

• • the evaluation predicate $c:{\mathbb{M}}\tau ,x:\tau \vdash _{} c\mbox{$\Downarrow _{}$}x\stackrel {\Delta }{\Longleftrightarrow }\langle x'{\Leftarrow }c \rangle _{}x'=x$ .

Moreover, the following inference rules (entailments $\phi _1\implies \phi _2$ ) and equivalences are derivableFootnote ³

1. (2) if $x\notin \textrm {FV}(\phi )$ , then ${\langle x{\Leftarrow }{e} \rangle _{}\phi }\implies \phi$ and $\phi \implies {[x{\Leftarrow }{e}]_{}\phi }$

2. (3) $\bot \Longleftrightarrow \langle x{\Leftarrow }{e} \rangle _{}\bot$ and ${\langle x{\Leftarrow }{e} \rangle _{}(\phi _1\vee \phi _2)}\Longleftrightarrow {(\langle x{\Leftarrow }{e} \rangle _{}\phi _1)\vee (\langle x{\Leftarrow }{e} \rangle _{}\phi _2)}$

3. (4) if $x_1\notin \textrm {FV}(\phi )$ , then ${\langle x_1{\Leftarrow }{e}_1 \rangle _{}\langle x_2{\Leftarrow }{e}_2 \rangle _{}\phi }\Longleftrightarrow {\langle x_2{\Leftarrow }(\textrm {let}_{}\,x_1{\Leftarrow }{e}_1\,\textrm {in}\,{e}_2) \rangle _{}\phi }$

4. (5) ${[x{\Leftarrow }{e}]_{}\phi }\Longleftrightarrow {(\textrm {let}_{}\,x{\Leftarrow }{e}\,\textrm {in}\,[\phi ]_{})=(\textrm {let}_{}\,x{\Leftarrow }{e}\,\textrm {in}\,[\top ]_{})}$ and ${\langle x{\Leftarrow }{e} \rangle _{}\phi }\Longleftrightarrow {(\textrm {let}_{}\,x{\Leftarrow }{e}\,\textrm {in}\,[\neg \phi ]_{})\neq (\textrm {let}_{}\,x{\Leftarrow }{e}\,\textrm {in}\,[\top ]_{})}$

5. (6) if $x\notin \textrm {FV}(\phi _1)$ , then ${\phi _1\supset [x{\Leftarrow }{e}]_{}\phi _2}\Longleftrightarrow {[x{\Leftarrow }{e}]_{}(\phi _1\supset \phi _2)}$ and ${\phi _1\wedge \langle x{\Leftarrow }{e} \rangle _{}\phi _2}\Longleftrightarrow {\langle x{\Leftarrow }{e} \rangle _{}(\phi _1\wedge \phi _2)}$

6. (7) if $x\notin \textrm {FV}({e}_0)$ , then ${(\textrm {let}_{}\,x{\Leftarrow }{e}_1\,\textrm {in}\,{e}_2)\mbox{$\Downarrow _{}$}{e}_0}\Longleftrightarrow {\langle x{\Leftarrow }{e}_1 \rangle _{}{e}_2\mbox{$\Downarrow _{}$}{e}_0}$ and ${[{e}_1]_{}\mbox{$\Downarrow _{}$}{e}_0}\Longleftrightarrow {e}_1={e}_0$ .

6. Assemblies & co.

We introduce the category of assemblies $\mathsf{A}[C]$ for a sub-monoid $C$ of the monoid of partial maps on a set $D$ , and several full sub-categories of $\mathsf{A}[C]$ . The aim is to provide realizability-like models for low-complexity domain-specific languages (DSL), where $D$ is the set of data manipulated by programs in the DSL, and $C$ are the maps computed by such programs. The general idea, following Scott (Reference Scott1980), is to replace application with composition in order to achieve greater flexibility over realizability models based on partial Combinatory Algebras (pCA), such as the Effective topos in Hyland (Reference Hyland1982). Our definition of $\mathsf{A}[C]$ is an instance of a construction in Longley (Reference Longley2014) based on computability structures.

If $D$ is trivial, that is, it has at most one element, then $C$ is irrelevant. More precisely:

• • if $D$ is the empty set, then $\mathsf{A}[C]$ is the discrete category with one object (the empty set);

• • if $D$ is a singleton, then $\mathsf{A}[C]$ is (isomorphic to) the category ${\mathscr{S}}{\kern.8pt}$ of sets.

We now give several examples of $D$ and $C$ that do not arise from a pCA.

We define two full sub-categories of $\mathsf{A}[C]$ : the category $\mathsf{M}[C]$ of modest sets and the category $\mathsf{P}[C]$ of predicates.

Without further assumptions on $C$ , $\mathsf{A}[C]$ and its sub-categories have limited properties.

7. Special monoids

In this section we consider additional properties of a sub-monoid $C\subseteq D\to D$ , which imply that $\mathsf{P}[C]$ is lextensive, has enough points, and all its objects have decidable equality.

8. Collection monads revised

Given a category ${\mathscr{C}}{\kern.8pt}$ with finite products, monadicity of the forgetful functor from the category of algebras in ${\mathscr{C}}{\kern.8pt}$ for a (balance) equational theory could be a property difficult to satisfy. The following definition replaces the equational theory with a monad ${\mathbb{M}}$ on ${\mathscr{S}}$ , and we look for (strong) monads ${\mathbb{M}}'$ on ${\mathscr{C}}{\kern.8pt}$ behaving like ${\mathbb{M}}$ .

We are mainly interested in the case of a lextensive category ${\mathscr{C}}{\kern.8pt}$ satisfying the properties of Theorem 3.1; thus, the global section functor is lextensive, and a (strong) $\mathscr{M}_2$ -monad ${\mathbb{P}}$ on ${\mathscr{C}}{\kern.8pt}$ (where $\mathscr{M}_2$ is the dominion of decidable monos) behaving like the finite powerset monad $\mathbb{P}_f$ on ${\mathscr{S}}$ . Since $|$ ${\mathbb{P}}$ $(n)|\cong \mathbb{P}_f(n)$ for every finite cardinal $n$ , the algebraic operations for ${\mathbb{P}}$ and $\mathbb{P}_f$ coincide.

If $C\subseteq D\to D$ is a sub-monoid satisfying properties (K,P,E) of Definition 7.1 and ${\mathscr{C}}{\kern.8pt}$ is the lextensive category $\mathsf{P}[C]$ of predicates (or the category $\mathsf{P}_d[C]$ of decidable predicates, see Remark 7.5), then $\mathscr{M}_2=\mathscr{M}_r$ and any (strong) monad ${\mathbb{P}}$ on ${\mathscr{C}}{\kern.8pt}$ behaving like $\mathbb{P}_f$ on ${\mathscr{S}}{\kern.8pt}$ restricts to the full sub-category ${\mathscr{C}}_f$ of finite cardinals in ${\mathscr{C}}{\kern.8pt}$ (i.e., finite sums of terminal objects), which is equivalent to ${\mathscr{S}}_f$ , since $X\in \mathsf{P}[C]_f\Longleftrightarrow |X|\in \mathscr{S}_f$ . The monad $\mathbb{P}_f$ has additional properties; thus, one expects that a monad ${\mathbb{P}}$ behaving like $\mathbb{P}_f$ should have them too.

• • $\mathbb{P}_f$ is taut. In $\mathsf{P}[C]$ not all mono are decidable; thus, we require ${\mathbb{P}}$ to be a (strong) $\mathscr{M}_2$ -monad.

• • In ${\mathscr{S}}{\kern.8pt}$ there is a small relation classifier , that is, every small relation (i.e., for every $x$ there are finitely many $y$ such that $R(x,y)$ ) is the pullback of $\ni _Y$ along $r\times \textrm {id}_{Y}$ for a unique $r:X\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{P}_f Y$ . Moreover, it coincides with the interpretation of the predicate $c\mbox{$\Downarrow _{}$}v$ of Evaluation Logic; thus, we expect something similar for ${\mathbb{P}}$ by exploiting the interpretation of Evaluation Logic.

Another desirable property of a monad ${\mathbb{M}}$ on $\mathsf{P}[C]$ ( $\mathsf{M}[C]$ or $\mathsf{A}[C]$ ) is realizability.

In the rest of this section, we consider several examples of $C\subseteq D\to D$ satisfying properties (K,P,E) of Definition 7.1 and ask whether there exists a strong monad ${\mathbb{M}}'$ on $\mathsf{P}[C]$ behaving like a chosen monad ${\mathbb{M}}$ on ${\mathscr{S}}$ . For defining such a strong monad (and ensure realizability), it suffices to fix an injective encoding $M:{\mathbb{M}}(D)\succ \!\!\longrightarrow D$ s.t.

• • for each $X\in \mathsf{P}[C]$ (i.e., $X\subseteq D$ ) the object ${\mathbb{M}}'(X)$ is given by the subset $\{M(A)|A\in {\mathbb{M}}(X)\}$ ,

• • for every $f\in C$ for some $f'\in C$ ,

and have maps $\eta ',\mu ',\mathsf{t}'\in C$ s.t.

these maps are unique if the encodings $p:D^2\to D$ and $M:{\mathbb{M}}(D)\to D$ are bijective.

8.1 Examples of $C$ on ${\mathbb{N}}$

Given a subset $C\subseteq \mathbb{N}\to \mathbb{N}$ , the Skolem (elementary) expressions on $C$ are defined by the BNF

(1) \begin{eqnarray} {e}\in E_{S}[C]&::=& x\ |\ f({e})\ |\ 0\ |\ 1\ |\ {e}_0+{e}_1\ |\ {e}_0\doteq {e}_1\ |\ \Sigma x\lt {e}_0.{e}_1 \end{eqnarray}

where $x$ ranges over a countable set of variables and $f$ ranges over $C$ . There is a standard interpretation of Skolem expressions in ${\mathbb{N}}$ , in particular $n_0\doteq n_1\stackrel {\Delta }{=}\mbox{$1$ if $n_0=n_1$ else $0$}$ .

We recall two standard bijective encodings for pairs and finite subsets of natural numbers.

Definition 8.4. The encodings $p:\mathbb{N}^2\to \mathbb{N}$ (aka Cantor pairing) and $P:\mathbb{P}_f(\mathbb{N})\to \mathbb{N}$ are

\begin{equation*}p(n_0,n_1)\stackrel {\Delta }{=}\frac {(n_0+n_1)(n_0+n_1+1)}{2}+n_1\quad \textit{and}\quad P(A)\stackrel {\Delta }{=}\Sigma _{n\in A}2^n.\end{equation*}

The map $p$ is Skolem (i.e., it is definable by an expression in $E_S[\emptyset ]$ ), but $P$ is not. More precisely, $f(n)=2^n=P(\{n\})$ is not Skolem, because Skolem maps are always sub-polynomial, that is, $f\leq p_f$ for some polynomial function $p_f:\mathbb{N}\to \mathbb{N}$ .

Proposition 8.5. If $C$ is closed w.r.t. Skolem-definable maps, that is, the interpretation of every map $\lambda x.{e}$ with ${e}\in E_{S}[C]$ is in $C$ , then $C$ satisfies properties (K,P,E).

Proof. We introduce some derived Skolem expressions (i.e., shorthand for Skolem expressions), we write $\phi$ for Skolem expressions that can only denote $0$ or $1$ (e.g., ${e}_0\doteq {e}_1$ ):

• • ${e}_1*{e}_2\stackrel {\Delta }{=}\Sigma x\lt {e}_1.{e}_2$ product and ${e}_1\ominus {e}_2\stackrel {\Delta }{=}\mu x\leq {e}_1.{e}_1\dot {\leq }({e}_2+x)$ truncated subtraction;

• • ${e}_1\dot {\lt }{e}_2\stackrel {\Delta }{=}\Sigma x\lt {e}_2.({e}_1\doteq x)$ , $\neg \phi \stackrel {\Delta }{=}(\phi \doteq 0)$ , $\exists x\lt {e}.\phi \stackrel {\Delta }{=}0\dot {\lt }(\Sigma x\lt {e}.\phi )$ ;

• • $\mu x\leq {e}.\phi \stackrel {\Delta }{=}\Sigma x\leq {e}.(0\doteq \Sigma x\lt x.\phi )*(0\dot {\lt }\phi )*x$ bounded minimization (it is $0$ if $\neg \exists x\leq {e}.\phi$ );

• • $p({e}_0,{e}_1)\stackrel {\Delta }{=}(\Sigma x\leq {e}_0+{e}_1.x)+{e}_1$ pairing, and $\pi _i({e})\stackrel {\Delta }{=}\mu x_i\leq {e}.(\exists x_{1-i}\leq {e}.\langle x_0,x_1 \rangle \doteq {e})$ projections for $i=0,1$ ;

• • $if(\phi ,{e}_t,{e}_f)\stackrel {\Delta }{=}(0\doteq \phi )*{e}_f+(0\dot {\lt }\phi )*{e}_t$ conditional.

The map $t:\mathbb{N}^4\to \mathbb{N}$ in property (E) is given by $t(n_0,n_1,n_2,n_3)\stackrel {\Delta }{=} if(n_0\doteq n_1,n_2,n_3)$ . In property (P) we use Cantor pairing $p$ , but any other encoding $p'$ of pairs is $C$ -reducible to $p$ (i.e., $p'=f\circ p$ for some $f\in C$ ) and conversely.

To prove that the bijective encoding $P:\mathbb{P}_f(\mathbb{N})\to \mathbb{N}$ determines a strong monad on $\mathsf{P}[C]$ behaving like $\mathbb{P}_f$ , it is not sufficient to assume that $C$ is closed w.r.t. Skolem-definable maps, since $\eta '$ is not a Skolem map. We have to take Kalmar (elementary) expressions, whose BNF has an extra clause for bounded products, namely

(2) \begin{eqnarray} {e}\in E_{K}[C]&::=& x\ |\ f({e})\ |\ 0\ |\ 1\ |\ {e}_0+{e}_1\ |\ {e}_0\doteq {e}_1\ |\ \Sigma x\lt {e}_0.{e}_1\ |\ \Pi x\lt {e}_0.{e}_1 \end{eqnarray}

Proposition 8.6. If $C$ is closed w.r.t. Kalmar-definable maps, then $C$ satisfies properties (K,P,E) and the bijective encoding $P:\mathbb{P}_f(\mathbb{N})\to \mathbb{N}$ determines a strong monad ${\mathbb{P}}$ on $\mathsf{P}[C]$ behaving like $\mathbb{P}_f$ .

Proof. By Proposition 8.5 we already know that $C$ satisfies properties (K,P,E), and one can assume that the map $p$ in (P) is Cantor pairing. Therefore, we are left to prove that $P$ given in Definition 8.4 satisfies the properties for determining a strong monad on $\mathsf{P}[C]$ behaving like $\mathbb{P}_f$ . To do this we introduce some derived Kalmar (and Skolem) expressions needed to prove that the required maps are in $C$ (in the following $P_*$ denotes the inverse of $P$ ):

• • ${e}_2^{{e}_1}\stackrel {\Delta }{=}\Pi x\lt {e}_1.{e}_2$ exponentiation and $\eta '({e})\stackrel {\Delta }{=} 2^{{e}}$ , thus $\eta (n)=P_*(\eta '(n))$ for $n\in \mathbb{N}$ ;

• • ${e}_2//({e}_1+1)\stackrel {\Delta }{=}(\mu x\leq {e}_2.{e}_2\dot {\lt }({e}_1+1)*x)\ominus 1$ quotient and ${e}_2\%({e}_1+1)\stackrel {\Delta }{=}{e}_2\ominus ({e}_2//({e}_1+1))$ reminder;

• • ${e}_1\dot {\ni }{e}_2\stackrel {\Delta }{=}({e}_1//2^{{e}_2})\%2$ , thus $n_1\dot {\ni }n_2=\mbox{$1$ if $P_*(n_1)\ni n_2$ else $0$}$ ;

• • $\mu '({e})\stackrel {\Delta }{=}\Sigma x\lt {e}.if(\exists x'\lt {e}.{e}\dot {\ni }x'*x'\dot {\ni }x,2^x,0)$ , thus $\mu (B)=\cup B=P_*(\mu '(P(\mathbb{P}_f(P)(B))))$ for $B\in \mathbb{P}_f^2(\mathbb{N})$ ;

• • $\mathsf{t}'({e}_0,{e}_1)\stackrel {\Delta }{=}\Sigma x\lt {e}_1.if({e}_1\dot {\ni }x,2^{p({e}_0,x)},0)$ , thus $p(\mathsf{t}(n,A))=\{p(n,n')|n'\in A\}=P_*(\mathsf{t}'(n,P(A)))$ for $n\in \mathbb{N}$ and $A\in \mathbb{P}_f(\mathbb{N})$ .

Finally, if $f\in C$ , then $f'({e})\stackrel {\Delta }{=}\Sigma x\lt {e}.if({e}\dot {\ni }x,2^{f(x)},0)\in E_{K}[C]$ and $\mathbb{P}_f(f)(A)=P_*(f'(P(A)))$ for $A\in \mathbb{P}_f(\mathbb{N})$ .

Remark 8.7. Among the $C\subset \mathbb{N}\to \mathbb{N}$ in Example 6.3 only $\mathsf{TR}$ , $\mathsf{PR}$ and $\mathsf{E}_{n}$ for $n\geq 3$ are closed w.r.t. Kalmar-definable maps. Under the assumptions of Proposition 8.6 one can prove that there is a natural transformation $\cup _X:{\mathbb{P}}(X)^2\xrightarrow {\,\,\,\hspace{6pt}}{\mathbb{P}}(X)$ behaving like $\cup _X:\mathbb{P}_f(X)^2\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{P}_f(X)$ realized by $\cup '({e}_1,{e}_2)\stackrel {\Delta }{=}\Sigma x\lt {e}_1+{e}_2. if(0\dot {\lt }({e}_1\dot {\ni }x+{e}_2\dot {\ni }x),2^{x},0)$ . Following Remark 5.1, another way to get this natural transformation is to observe that we have element $none$ and $two$ , namely $none$ is the only element $0\in {\mathbb{P}}(\emptyset )$ and $two$ is the element $\{0,1\}\in {\mathbb{P}}(\{0,1\})$ .

If the sub-monoid $C$ has the closure properties of primitive recursion maps, then ${\mathbb{N}}$ (with the obvious structure) is a pNNO (parameterized NNO) in $\mathsf{P}[C]$ , thus $\mathsf{P}[C]$ is a Skolem category in the sense of Maietti (Reference Maietti2010, Definition 2.2), and every commutative idempotent monoid $(X,e,m)$ in $\mathsf{P}[C]$ corresponds to an EM-algebra $(X,\alpha )$ for ${\mathbb{P}}$ .

However, if $C\subseteq O(T)\subset {\mathsf{PR}}$ for some $T\in {\mathsf{PR}}$ and $C$ satisfies properties (K,P,E), then $\mathsf{P}[C]$ cannot have a pNNO. In fact, if were a pNNO in $\mathsf{P}[C]$ , then there is a unique map $e:\mathbb{N}\xrightarrow {\,\,\,\hspace{6pt}} X$ in $\mathsf{PR}$ s.t. $e(n)=x_n\stackrel {\Delta }{=} f^n(a)$ , because is a pNNO in $\mathsf{P}[{\mathsf{PR}}]$ , and conversely there is a unique map $e':X\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{N}$ (realized by a map) in $C$ s.t. $e'(x_n)=n$ , because $X$ is a pNNO in $\mathsf{P}[C]$ . In a Skolem category every (unary) primitive recursive map $g$ is representable by an arrow $g':\mathbb{N}\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{N}$ on the pNNO. In $\mathsf{P}[C]$ this means that for every $g\in {\mathsf{PR}}$ the map $g':X\xrightarrow {\,\,\,\hspace{6pt}} X$ s.t. $g'(x_n)=x_{g(n)}$ is in $\mathsf{P}[C]$ . Therefore, the map $e'\circ g':X\xrightarrow {\,\,\,\hspace{6pt}}\mathbb{N}$ , that is, $x_n\mapsto g(n)$ , must be (realized by a map) in $C$ . Choose $T'\in {\mathsf{PR}}-O(T)$ and take $g=T'\circ e\in {\mathsf{PR}}$ , then $e'\circ g'=g\circ e'$ maps $x_n\mapsto T'(x_n)$ . We get a contradiction, because $e'\circ g'$ is (realized by a map) in $C\subseteq O(T)$ , while $T'\not \in O(T)$ .