Action preserving (weak) topologies on the category of presheaves

TL;DR

This paper constructs and analyzes action-preserving weak topologies on the category of presheaves over a finitely complete small category, introducing new methods using subfunctors and admissible classes.

Contribution

It introduces two new weak Lawvere-Tierney topologies on presheaf categories and studies their action-preserving properties, including conditions related to the double negation topology.

Findings

Constructed two weak topologies using subfunctors and admissible classes.

Identified conditions for these topologies to be action preserving.

Established an action preserving weak topology on presheaf categories.

Abstract

Let $\mathcal{C}$ be a finitely complete small category. In this paper, first we construct two weak (Lawvere-Tierney) topologies on the category of presheaves. One of them is established by means of a subfunctor of the Yoneda functor and the other one, is constructed by an admissible class on $\mathcal{C}$ and the internal existential quantifier in the presheaf topos $\widehat{\mathcal{C}}$. Moreover, by using an admissible class on $\mathcal{C},$ we are able to define an action on the subobject classifier $\Omega$ of $\widehat{\mathcal{C}}$. Then we find some necessary conditions for that the two weak topologies and also the double negation topology $\neg\neg$ on $\widehat{\mathcal{C}}$ to be action preserving maps. Finally, among other things, we constitute an action preserving weak topology on $\widehat{\mathcal{C}}$.

Equations

$$D_g{\times }_C{D}_f\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$[(n,g)]\circ [(m,f)]=[(m{f}^{-1}(n),g{n}^{-1}(f))].$$

$$f_{\ast }:\mathbf{y}(D)⟶\mathbf{y}(C);(f_{\ast }{)}_B(g)=fg,$$

$$I^2(C)=\{fg|f\in I_C,g\in I_{D_f}\},$$

$$C_F^I:{\mathrm{Sub}}_{\hat{\mathcal{C}}}(F)⟶{\mathrm{Sub}}_{\hat{\mathcal{C}}}(F);C_F^I(G)=\overline{G},$$

$$\overline{G}(C)=\{x\in F(C)|\forall f\in I_C,F(f)(x)\in G(D_f)\},$$

$${\mathrm{Sub}}_{\hat{\mathcal{C}}}(H)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$({\alpha }^{-1}\circ C_H^I(G))(C)$$

$$(C_F^I\circ {\alpha }^{-1}(G))(C)$$

$$\overline{\overline{I^2(D)}}(C)$$

$$\begin{array}[]{rcl}\overline{\overline{G}}(C)&=&\{x\in F(C)|~{}\forall f\in I_{C},F(f)(x)\in\overline{G}(D_{f})\}\\ &=&\{x\in F(C)|~{}\forall f\in I_{C},\forall g\in I_{D_{f}},F(fg)(x)\in G(D_{g})\}~{}(\textrm{by}~{}(\ref{def. of barG}))\\ &=&\{x\in F(C)|~{}\forall fg\in I^{2}(C),F(fg)(x)\in G(D_{g})\}~{}~{}~{}~{}~{}~{}~{}~{}(\textrm{by}~{}(\ref{I2}))\end{array}$$

$${\mathbf{J}}^I(C)=\{T_C\in \Omega (C)|I_C\subseteq T_C\}$$

$$j_C^I(S_C)=\{f|\forall g\in I_{D_f},fg\in S_C\},$$

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\neg }_{j^I}G{\vee }_{j^I}{\neg }_{j^I}{\neg }_{j^I}G=F.$$

$$\neg G(C)=\{x\in F(C)|\forall f\in t(C),F(f)(x)\notin G(D_f)\},$$

$$(\forall h\in t(C),\exists g\in I_{D_h},\exists k\in t(D_g);F(hgk)(x)\in G(D_k))$$

$$⟺(\exists w\in t(C);F(w)(x)\in G(D_w)).$$

$$({\neg }_{j^I}G{\vee }_{j^I}{\neg }_{j^I}{\neg }_{j^I}G)(C)$$

$$\neg {\neg }_C(T_C)=\{f|\forall g\in t(D_f),\exists h\in t(D_g);fgh\in T_C\},$$

$$(\neg {\neg }_I{)}_C(T_C)=\{f|\forall g\in I_{D_f},\exists h\in I_{D_g};fgh\in T_C\},$$

$${\Omega }_{j^I}(C)=\{T_C\in \Omega (C)|\forall h:D_h\to C,(\forall k\in I_{D_h},hk\in T_C)\iff h\in T_C\},$$

$$h\in I_{D_{fg}}⟺gh\in I_{D_f}.$$

$$({\sigma }_{{}_X}{)}_C(S_C)=S_C\times X\subseteq \mathbf{y}(C)\times X.$$

$$({\exists }_X{)}_C(U)=\{f|\exists x\in X(D_f);(f,x)\in U(D_f)\},$$

$${\Omega }^X\cong {\mathrm{Hom}}_{\hat{\mathcal{C}}}(\mathbf{y}(-)\times X,\Omega )$$

$$(({\sigma }_{{}_X}{)}_C(S_C){)}_D(f,x)=\Omega (f)(S_C)=f^{\ast }(S_C),$$

$$({\exists }_X{)}_C(\phi )=\{f|\exists x\in X(D_f);{\phi }_{D_f}(f,x)=t(D_f)\},$$

$$({\exists }_X\circ {\sigma }_{{}_X}{)}_C(S_C)=\{f\in S_C|X(D_f)\ne \varnothing \}.$$

$${\mathrm{true}}^X:1=1^X\rightarrowtail {\Omega }^X$$

Full text

AMS subject classification: 18D35, 18F10, 18F20, 18B25.

key words: (Weak) Lawvere-Tierney topology; (universal, modal) closure operator; admissible class; double negation topology; action.

1 Introduction and Preliminaries

One of the basic tools to construct new topoi from old ones is the notion of Lawvere-Tierney topology. In mathematics, a Lawvere-Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. Recently, applications of Lawvere-Tierney topologies on the category of presheaves in broad topics such as measure theory [13] and quantum physics [26, 27] are observed. In usual topology, closure operators without idempotency are so valuable [5]; such as $\check{{\rm C}}$ech closure operators which are just closure operators without idempotency. Considering (Lawvere-Tierney) topologies in the framework of [2], a weak Lawvere-Tierney topology (or weak topology, for short) is exactly a topology without idempotency. Originally, Hosseini and Mousavi in [11] applied the notion of weak Lawvere-Tierney topology in the category of presheaves on a small category. They proved that on a presheaf category, weak Lawver-Tierney topologies are in one-to-one correspondence with modal closure operators ‘(-)’. Recently, the authors studied this notion on an elementary topos in [18] and found some results in this respect.

On the other hand, action of a monoid over a set or an algebra is of interest to some mathematicians, see [1, 7, 10]. Some applications of these structures are in computer science, geometry and robotic manipulation; for example see [9, 12, 25].

This paper attempts to reconcile two abstract notions in a presheaf category which are ‘(weak) Lawvere-Tierney topology’ and ‘internal action of a monoid over a Heyting algebra’.

Let $\mathcal{C}$ be a finitely complete small category equipped with an admissible class $\mathcal{M}$ on $\mathcal{C}$. In this paper we first introduce the notion of an ‘ideal’ $I$ in the presheaf topos $\widehat{\mathcal{C}}$ and then find a weak Lawvere-Tierney topology with respect to an any ideal $I$ in $\widehat{\mathcal{C}},j^{I}$, which we shall call it ‘the associated weak Lawvere-Tierney topology’ given by $I$. Then we introduce the notion of action preserving weak Lawvere-Tierney topology on the presheaf topos $\widehat{\mathcal{C}}$. This is done first by constructing a presheaf $M$ on $\mathcal{C}$ (introduced in [11]) and then equipped $M$ with a monoid structure given by the structure of the slice categories $\mathcal{C}/B$, for any object $B$ of $\mathcal{C}/B$. Furthermore, using $M$ we construct a weak topology $j_{M}:\Omega{\longrightarrow}\Omega$ on $\widehat{\mathcal{C}}$. Then we provide some necessary conditions for that the two weak Lawvere-Tierney topologies $j^{I},j_{M}$ and also the double negation topology $\neg\neg$ on $\widehat{\mathcal{C}}$ to be action preserving. Meanwhile, a few basic examples throughout the paper are provided and are analysed. Finally, among other things, we constitute an action preserving weak Lawvere-Tierney topology on $\widehat{\mathcal{C}}$.

For general notions and results concerning topos theory we refer the reader to [20], [2] or [17]. In this manuscript, we use the following notations and notions: Let $\mathcal{C}$ be a small category and $\widehat{\mathcal{C}}$ its category of presheaves.

For any arrow $f$ in $\mathcal{C},D_{f}$ represents the ‘domain’ of $f$ and $C_{f}$ the ‘codomain’ of $f.$ Meanwhile, a pullback of a map $g$ along a map $f$ is denoted by $f^{-1}(g)$. 2. 2.

Let $C$ be an arbitrary object of $\mathcal{C}.$ Recall [20] that for objects $f:D_{f}\rightarrow C$ and $g:D_{g}\rightarrow C$ in the slice category $\mathcal{C}/C$, the product $f\times g$ in $\mathcal{C}/C$ is the diagonal of the following pullback square (if exist) in $\mathcal{C},$

[TABLE]

i.e., $f\times g=ff^{-1}(g)=gg^{-1}(f)(=g\times f).$ 3. 3.

An admissible class $\mathcal{M}$ on $\mathcal{C}$ (so-called a domain structure or a dominion in [14, 24]) is a family of subobjects of $\mathcal{C}$ for which:

(1) $\mathcal{M}$ contains all identities;

(2) $\mathcal{M}$ is closed under composition;

(3) $\mathcal{M}$ is closed under pullback (see, [29]).

For example, the class of all monomorphisms in $\mathcal{C}$ is an admissible class on the category $\mathcal{C}.$ The class $\mathcal{M}$ yields a subpresheaf $M$ of the presheaf ${\rm Sub}_{\mathcal{C}}(-):\mathcal{C}^{\rm op}\rightarrow{\bf Sets},$ given by $M(C)=\mathcal{M}/C$ (see also, [11]). 4. 4.

Consider the category ${\bf Ptl}(\mathcal{C})$ of partial maps on $\mathcal{C}$ and $\mathcal{M}$-Ptl$(\mathcal{C})$ the subcategory of ${\bf Ptl}(\mathcal{C})$ consisting of the partial maps in ${\bf Ptl}(\mathcal{C})$ whose domain of definition is in $\mathcal{M}$ (see also, [29]). In this paper, we consider (up to isomorphism) any partial map $[(n,f)]$ only as the pair $(n,f)$ where $f:A{\longrightarrow}B$ is a map in $\mathcal{C}$ and $n:A\rightarrowtail C$ is a monic in $\mathcal{C}$. Consider two composable partial maps $[(m,f)]$ and $[(n,g)],$ i.e., $C=C_{f}=C_{n}.$ By [29], we have

[TABLE] 5. 5.

Recall [11] that a weak Lawvere-Tierney topology on a topos $\mathcal{E}$ is a morphism $j:\Omega\rightarrow\Omega$ such that:

(i) $j\circ{\rm true}={\rm true}$;

(ii) $j\circ\wedge\leq\wedge\circ(j\times j)$,

in which $\leq$ stands for the internal order on $\Omega.$ Henceforth, by a weak topology on $\mathcal{E}$ we mean a weak Lawvere-Tierney topology on $\mathcal{E}$. Moreover, a weak topology $j$ on $\mathcal{E}$ is said to be productive, introduced in [18], if $j\circ\wedge=\wedge\circ(j\times j)$. The correspondence between modal closure operators on $\mathcal{C}$, weak topologies on $\widehat{\mathcal{C}}$ and weak Grothendieck topologies on $\mathcal{C}$ are given in [11], for the definition of a modal closure operator and a weak Grothendieck topologyon $\mathcal{C}$ we refer the reader to [11].

Furthermore, recall [3] that a pomonoid $S$ is a monoid $S$ together with a partially ordered such that partial order is compatible with the binary operation. A (right) $S$-poset is a (possibly empty) poset $A$ together with a monotone map $\lambda:A\times S\rightarrow A$, called the action of $S$ on $A$, such that, for all $a\in A$ and $s,t\in S$, we have $a1=a,$ and $a(st)=(as)t$ where $A\times S$ is considered as a poset with componentwise order and we denote $\lambda(a,s)$ by $as$.

2 (Weak) ideal topology on $\widehat{\mathcal{C}}$

In this section our aim is to introduce, for a small category $\mathcal{C},$ the notion of an ideal $I$ of $\widehat{\mathcal{C}}$ and next, to constitute a weak topology so-called weak ideal topology $j^{I}$ associated to $I$ on $\widehat{\mathcal{C}}.$ Then, we get some results in this respect.

Let us first assume that $\mathcal{C}$ is a small category (not necessarily with finite limits). Recall that the Yoneda functor ${\bf y}:\mathcal{C}\rightarrow\widehat{\mathcal{C}}$ assigns to any object $C\in\mathcal{C}$ the presheaf ${\rm Hom}_{\mathcal{C}}(-,C)$ and to any arrow $f:D\rightarrow C$ the map

[TABLE]

for any object $B\in\mathcal{C}$ and any map $g:B\rightarrow D$ of $\mathcal{C}.$ We will denote the category of all functors from $\mathcal{C}$ to $\widehat{\mathcal{C}}$ (so-called diagrams of type $\mathcal{C}$ in $\widehat{\mathcal{C}}$) and all natural transformations between them by ${\rm Fun}(\mathcal{C},\widehat{\mathcal{C}}).$ Let $F:\mathcal{C}\rightarrow\widehat{\mathcal{C}}$ be a functor. By a subfunctor $G:\mathcal{C}\rightarrow\widehat{\mathcal{C}}$ of $F$ we mean a subobject $i:G\rightarrow F$ in the functor category ${\rm Fun}(\mathcal{C},\widehat{\mathcal{C}}),$ such that its components $i_{C}:G(C)\rightarrow F(C),$ $C\in\mathcal{C},$ are monic in $\widehat{\mathcal{C}},$ indeed, $G(C)$ is a subfunctor of $F(C)$ in $\widehat{\mathcal{C}}$ for any $C\in\mathcal{C}.$ It is well known [20] that, for any $C\in\mathcal{C},$ sieves on $C$ are in one to one correspondence to subfunctors of ${\bf y}(C).$

Now we define

Definition 2.1**.**

By an ideal $I$ of $\widehat{\mathcal{C}}$ we mean a subfunctor $I:\mathcal{C}\rightarrow\widehat{\mathcal{C}}$ of the Yoneda functor ${\bf y}$ in ${\rm Fun}(\mathcal{C},\widehat{\mathcal{C}}).$ Indeed, an ideal $I$ of $\widehat{\mathcal{C}}$ is a family $\{I(C)\}_{C\in\mathcal{C}}$ for which any $I(C)$ is a sieve on $C$ and for any arrow $f:C\rightarrow D$ and any $g\in I(C)$ one has $fg\in I(D).$**

Note that the Yoneda functor is itself an ideal $\{{\bf y}(C)\}_{C\in\mathcal{C}}$ of $\widehat{\mathcal{C}}.$ From now on to the end of this section, for an ideal $I$ of $\widehat{\mathcal{C}}$ we denote the sieve $I(C)$ on $C$ by $I_{C},$ for any $C\in\mathcal{C},$ unless otherwise stated.

In the following we provide some ideals on some presheaf topoi.

Example 2.2**.**

(a) Let $(P,\leq)$ be a poset which we may realize it as a category. Then an ideal $I$ of $\widehat{P}$ is a family $\{I_{a}\}_{a\in P}$ in which any $I_{a}$ is a downward closed subset of $\downarrow a(=\{x\in P|x\leq a\})$ together with the property that $I_{a}\subseteq I_{b}$ whenever $a\leq b.$

(b) Let $S$ be a monoid. Then we can make it as a category (denoted by $S$ again) with just one object denoted by $*.$ It is well known that the category $\widehat{S}$ is isomorphic to $S$-Sets, the category of all (right) representations of $S.$ An ideal $I$ of $\widehat{S}$ is just a two sided ideal $I$ of the monoid $S$. Because $I_{*},$ again denoted by $I,$ is a sieve on $S$ and so indeed a right ideal of $S$. Meanwhile, by Definition 2.1, for any $m\in S$ and any $n\in I$ we must have $mn\in I$ and so, $I$ is also a left ideal of $S$.

(c) Let $\Gamma$ be the category with two objects $N$ and $A$, and two non-identity arrows $s,t:N\rightarrow A.$ It is well known that $\widehat{\Gamma}$ is the category of (directed) graphs and graph morphisms. Indeed, each presheaf $G$ in $\widehat{\Gamma}$ is given by a set $G(N)$, the set of nodes, and a set $G(A)$ of arcs. The arrows $s,t$ are mapped to functions $G(s),G(t):G(A)\rightarrow G(N)$ which assign to each arc its source and target, respectively. In [30] subfunctors of ${\bf y}(N)$ and ${\bf y}(A)$ are exactly determined. Then, by [20] we may find sieves on $N$ and $A$ as well. One can easily checked that the category $\widehat{\Gamma}$ has exactly two ideals $I:\Gamma\rightarrow\widehat{\Gamma},$ given by $I_{N}=\emptyset$ and $I_{A}=\emptyset$, and $I^{\prime}:\Gamma\rightarrow\widehat{\Gamma},$ given by $I^{\prime}_{N}=\{{\rm id}_{N}\}$ and $I^{\prime}_{A}=\{s,t\}.$

(d) Let $\mathbb{L}$ be the dual (or opposite) of the category of finitely generated $C^{\infty}$-rings. As usual, for a given such $C^{\infty}$-ring $A$, the corresponding locus-an object of $\mathbb{L}$-is denoted by $\ell(A)$. Notice that, for any $\ell A\in{\mathbb{L}},\ {\bf y}(\ell A)$ and $\ell A$ are identified in [28]. Moreover, let ${\mathbb{G}}$ and ${\mathbb{F}}$ be full subcategories of ${\mathbb{L}}$ consisting of opposites of germ determined finitely generated $C^{\infty}$-rings and closed finitely generated $C^{\infty}$-rings, respectively. A Grothendieck topology on ${\mathbb{L}}$ (on ${\mathbb{G}}$ and on ${\mathbb{F}}$) is introduced in [28, p. 241] by constructing a basis on ${\mathbb{L}}$. This basis, by [20, p. 112], generates a Grothendieck topology ${\bf J}_{{}_{\mathbb{L}}}$ on ${\mathbb{L}}.$ One can easily check that ${\bf J}_{{}_{\mathbb{L}}}$ is defined by a cosieve $S_{B}$ on $B$ (a cosieve on $B$ is the dual of a sieve on $B$) belongs to ${\bf J}_{{}_{\mathbb{L}}}(B),$ for any dual $\ell B\in{\mathbb{L}},$ iff there is a cover (in dual form) $R_{B}=\{\eta_{i}:B\rightarrow B[s_{i}^{-1}]|~{}i=1,\ldots,n\}$ with $R_{B}\subseteq S_{B}$. Using [20, Therorem V.4.1], ${\bf J}_{{}_{\mathbb{L}}}$ corresponds to a (Lawvere-Tierney) topology $j_{{}_{\widehat{{\mathbb{L}}}}}:\Omega\rightarrow\Omega$ on the presheaf topos $\widehat{{\mathbb{L}}}$ given by, for each object $A\in{\mathbb{L}}^{\rm op}$ and any cosieve $S_{A}$ on $A,$ the cosieve $(j_{{}_{\widehat{{\mathbb{L}}}}})_{{}_{A}}(S_{A})$ is the set of all those $C^{\infty}$-homomorphisms $g:A\rightarrow B$ for which there exists a cover (in dual form) $\{\eta_{i}:B\rightarrow B[s_{i}^{-1}]|~{}i=1,\ldots,n\}$ with $\eta_{i}g\in S_{A}$ for all indices $i$. Recall that [28], the smooth Zariski topos, denoted by $\mathcal{Z},$ is the category of all ${\bf J}_{{}_{\mathbb{L}}}$-sheaves on ${\mathbb{L}}$. Also, the Dubuc topos ([19]), denoted by $\mathcal{G},$ is the category of all ${\bf J}_{{}_{\mathbb{G}}}$-sheaves on ${\mathbb{G}}$. It is well known that ${\bf J}_{{}_{\mathbb{L}}}$-sheaves are exactly $j_{{}_{\widehat{{\mathbb{L}}}}}$-sheaves on $\widehat{{\mathbb{L}}}$ (for details, see [20, Theorem V.4.2]). This means that the topos $\mathcal{Z}$ is exactly the topos of all $j_{{}_{\widehat{{\mathbb{L}}}}}$-sheaves on $\widehat{{\mathbb{L}}}$ and also the topos $\mathcal{G}$ is the topos of all $j_{{}_{\widehat{{\mathbb{G}}}}}$-sheaves on $\widehat{{\mathbb{G}}}$. Note that $j_{{}_{\widehat{{\mathbb{G}}}}}$ and $j_{{}_{\widehat{{\mathbb{F}}}}}$ are defined exactly similar to $j_{{}_{\widehat{{\mathbb{L}}}}}$ only with different covers. Since the Grothendieck topology ${\bf J}_{{}_{\mathbb{L}}}$ on ${\mathbb{L}}$ is subcanonical it follows that for any ideal $I$ of $\widehat{{\mathbb{L}}}$ the sieve $I_{\ell A}$ is $j_{{}_{\widehat{{\mathbb{L}}}}}$-separated in $\widehat{{\mathbb{L}}},$ for any $\ell A\in{\mathbb{L}}.$ This also holds for the topologies $j_{{}_{\widehat{{\mathbb{G}}}}}$ on ${\mathbb{G}}$ and $j_{{}_{\widehat{{\mathbb{F}}}}}$ on ${\mathbb{F}},$ respectively. **

The following gives another ideal of $\widehat{\mathcal{C}}$ made of an ideal of $\widehat{\mathcal{C}}.$

Lemma 2.3**.**

Let $I$ be an ideal of $\widehat{\mathcal{C}}$. Then the family $I^{2}=\{I^{2}(C)\}_{C\in\mathcal{C}}$ in which

[TABLE]

for any object $C\in\mathcal{C}$, is an ideal of $\widehat{\mathcal{C}}.$

Proof. First we investigate that $I^{2}(C)$, for any $C\in\mathcal{C},$ is a sieve on $C.$ To achieve this, consider an element $fg\in I^{2}(C)$ and an arrow $h:D_{h}\rightarrow D_{fg}.$ We prove that $fgh\in I^{2}(C)$. By the definition of $I^{2}(C)$, one has $f\in I_{C}$ and $g\in I_{D_{f}}.$ Since $I_{D_{f}}$ is a sieve on $D_{f}$ we deduce that $gh\in I_{D_{f}}$ and then, $fgh\in I^{2}(C)$ by (2).

Next, choose an arrow $k:C\rightarrow D$ of $\mathcal{C}.$ We must show that if $fg\in I^{2}(C)$ then $kfg\in I^{2}(D)$. Since $I$ is an ideal of $\widehat{\mathcal{C}}$ and $f\in I_{C},$ thus $kf\in I_{D}$ for any $fg\in I^{2}(C).$ Then, by (2), we get $kfg\in I^{2}(D),$ as required.$\quad\quad\square$

Let $I$ be an ideal of $\widehat{\mathcal{C}}.$ For any presheaf $F,$ we define an assignment

[TABLE]

for any subpresheaf $G$ of $F,$ in which the subpresheaf $\overline{G}$ of $F,$ is defined by

[TABLE]

for any $C\in\mathcal{C}.$

This gives us a modal closure operator on $\widehat{\mathcal{C}}$.

Theorem 2.4**.**

The assignment $C^{I}$ defined as in (3) is a modal closure operator on $\widehat{\mathcal{C}}.$

Proof. First let us show that the functor $\overline{G}$ as in (4) is a subpresheaf of $F.$ To do this, we need to prove that $\overline{G}$ is well defined on arrows in $\mathcal{C}.$ Let $h:C\rightarrow D$ be an arrow in $\mathcal{C}.$ We must show that for any $x\in\overline{G}(D),$ $F(h)(x)\in\overline{G}(C).$ Since $I$ is a subfunctor of ${\bf y},$ it concludes that for any $g\in I_{C},$ $hg=h_{*}(g)\in I_{D}.$ That by assumption $x\in\overline{G}(D)$ and $hg\in I_{D},$ it follows that, by (4), $F(g)(F(h)(x))=F(hg)(x)\in G(D_{g}).$ Then, $F(h)(x)\in\overline{G}(C).$

Next, we show that $C^{I}$ defined as in (3) is a modal closure operator on $\widehat{\mathcal{C}}$ in the following steps:

(i) $C^{I}$ is extensive, i.e. for any presheaf $F$ and any subpresheaf $G$ of $F$, $G$ is a subpresheaf of $\overline{G}.$ For any $C\in\mathcal{C},$ $f\in I_{C}$ and $x\in G(C)$, since $G$ is a subfunctor of $F$ we obtain $F(f)(x)=G(f)(x)\in G(D_{f}).$ This means that $x\in\overline{G}(C)$, by (4).

(ii) $C^{I}$ is monotone. Let $G$ and $H$ be two subpresheaves of $F$ such that $G$ is a subpresheaf of $H.$ We must show that $\overline{G}$ is a subpresheaf of $\overline{H}$. By (4), it is evident $\overline{G}(C)\subseteq\overline{H}(C)$, for any $C\in\mathcal{C}.$

(iii) $C^{I}$ is modal, i.e. for any arrow $\alpha:F\rightarrow H$ in $\widehat{\mathcal{C}},$ the following diagram commutes,

[TABLE]

For any subfunctor $G$ of $H$ and any $C\in\mathcal{C},$ we have

[TABLE]

On the other hand, by (4), we have

[TABLE]

Now let $x\in(\alpha^{-1}\circ C^{I}_{H}(G))(C).$ For any $f\in I_{C},$ by (5) and (4), one has $H(f)(\alpha_{C}(x))\in G(D_{f}).$ Naturality of $\alpha$ implies that $\alpha_{D_{f}}(F(f)(x))=H(f)(\alpha_{C}(x))\in G(D_{f}),$ and hence, by (2), $x\in(C^{I}_{F}\circ\alpha^{-1}(G))(C).$ Conversely, choose an $x\in(C^{I}_{F}\circ\alpha^{-1}(G))(C).$ By (2) and naturality of $\alpha$, we deduce that $H(f)(\alpha_{C}(x))=\alpha_{D_{f}}(F(f)(x))\in G(D_{f})$ for any $f\in I_{C}.$ Then, by (4), we get $\alpha_{C}(x)\in\overline{G}(C)$ and so by (5) we get $x\in(\alpha^{-1}\circ C^{I}_{H}(G))(C).$ Therefore, $(\alpha^{-1}\circ C^{I}_{H}(G))(C)=(C^{I}_{F}\circ\alpha^{-1}(G))(C).$ $\quad\quad\square$

We say to the modal closure operator $C^{I}$ on $\widehat{\mathcal{C}},$ defined as in (3), the ideal closure operator.

The following provides a necessary and sufficient condition for that an ideal closure operator to be idempotent.

Lemma 2.5**.**

Let $I$ be an ideal of $\widehat{\mathcal{C}}.$ Then the ideal closure operator $C^{I}$ is idempotent iff $I$ is idempotent, i.e., $I^{2}=I.$

Proof. Necessity. We investigate $I^{2}=I.$ Since, for any $C\in\mathcal{C},$ $I_{C}$ is a sieve on $C,$ by (2), we conclude that $I^{2}(C)\subseteq I_{C}$ and thus, $I^{2}\subseteq I.$ We prove that $I\subseteq I^{2}$ or equivalently, $I_{C}\subseteq I^{2}(C)$ for any $C\in\mathcal{C}.$ At the beginning, we remark that for any $D,C\in\mathcal{C},$ by the definition of $C^{I}_{{\bf y}(D)}$ as in (3), for the subfunctor $I^{2}(D)$ of ${\bf y}(D)$ we achieve

[TABLE]

On the other hand, by (4) and (2), we have

[TABLE]

for any $D,C\in\mathcal{C}.$ By (8) we deduce that ${\rm id}_{C}\in\overline{\overline{I^{2}(C)}}(C).$

Now, let $C\in\mathcal{C}$ and $g\in I_{C}.$ Then, since ${\rm id}_{C}\in\overline{\overline{I^{2}(C)}}(C),$ by (7), it follows that $g\in I^{2}(C)$ and so, $g$ belongs to the sieve $I^{2}(C),$ as required.

Sufficiency. Let $F$ be a presheaf and $G$ a subfunctor of $F.$ For any $C\in\mathcal{C},$ we get

[TABLE]

Since by assumption, $I^{2}(C)=I_{C}$, by (4) it follows that $\overline{\overline{G}}(C)\subseteq\overline{G}(C).$ Therefore $C^{I}_{F}C^{I}_{F}(G)\subseteq C^{I}_{F}(G).$ The converse easily holds by the extension property of $C^{I}_{F}.\quad\quad\square$

Clearly, the ideal ${\bf y}$ of $\widehat{\mathcal{C}}$ is idempotent. It is straightforward to see that the two ideals $I$ and $I^{\prime}$ of $\widehat{\Gamma}$ in Example 2.2(c) are idempotent. Meanwhile, in Example 2.2(b), idempotent ideals of $\widehat{S}$ are exactly idempotent two sided ideals of the monoid $S.$

The following gives us a weak topology associated to any ideal closure operator on $\widehat{\mathcal{C}}.$

Corollary 2.6**.**

Let $I$ be an ideal of $\widehat{\mathcal{C}}.$ Then the weak Grothendieck topology ${\bf J}^{I}$ on $\mathcal{C}$ associated to $C^{I}$, as in (3), is given by

[TABLE]

for any $C\in\mathcal{C}.$ Furthermore, the weak topology on $\widehat{\mathcal{C}}$ associated to $C^{I},$ denoted by $j^{I}:\Omega\rightarrow\Omega,$ is given by

[TABLE]

for any $C\in\mathcal{C}$ and any $S_{C}\in\Omega(C).$ Moreover, $j^{I}$ is a topology on $\widehat{\mathcal{C}}$ iff $I$ is idempotent.

Proof. It is easy to check by [11].$\quad\quad\square$

We say to the weak topology $j^{I}$ on $\widehat{\mathcal{C}},$ defined as in (10), the weak ideal topology on $\widehat{\mathcal{C}}$ associated to an ideal $I$ of $\widehat{\mathcal{C}}.$ By (10), we may deduce that $j^{I}$ is a productive weak topology on $\widehat{\mathcal{C}}$, i.e., $j^{I}_{C}(S_{C}\cap T_{C})=j^{I}_{C}(S_{C})\cap j^{I}_{C}(T_{C})$ for any $C\in\mathcal{C}$ and any $S_{C},T_{C}\in\Omega(C).$ We point out that, by (10), $j^{\bf y}={\rm id}_{\Omega}$ and then, ${\bf Sh}_{j^{\bf y}}(\widehat{\mathcal{C}})=\widehat{\mathcal{C}}.$ Meanwhile, $j^{0}={\rm true}\circ!_{\Omega}$ on $\widehat{\mathcal{C}}$ which is the topology associated to the chaotic or indiscrete Grothendieck topology on $\mathcal{C}$ where only cover sieve on an object $C\in\mathcal{C}$ is the maximal sieve $t(C)$ on $C.$

Take an ideal $I$ of $\widehat{\mathcal{C}}.$ By (4), one can easily observe that a subpresheaf $G$ of any presheaf $F$ is $j^{I}$-dense iff for any $C\in\mathcal{C},$ $x\in F(C)$ and $f\in I_{C}$ one has $F(f)(x)\in G(D_{f}).$

The succeeding example gives us some weak ideal topologies, defined as in (10), on some presheaf topoi.

Example 2.7**.**

(1) For a (left) two sided ideal $I$ of a monoid $S$, the weak ideal topology on $\widehat{S}$ is introduced in [18].

(2) Consider the idempotent ideals $I$ and $I^{\prime}$ of $\widehat{\Gamma}$ as in Example 2.2(c). It is easy to see that the ideal topologies $j^{I}$ and $j^{I^{\prime}}$ on $\widehat{\Gamma}$ coincide with the two topologies ${\rm true}\circ!_{\Omega}$ and ${\rm id}_{\Omega},$ respectively.**

In what follows, we give a significant property of the ideal closure operator on $\widehat{\mathcal{C}}.$ First recall [17] that a category $\mathcal{C}$ satisfies the right Ore condition whenever any two morphisms $f:A\rightarrow C$ and $g:B\rightarrow C$ of $\mathcal{C}$ with common codomain $C$ can be completed to a commutative square as follows:

[TABLE]

Theorem 2.8**.**

Let $\mathcal{C}$ be a category with the right Ore condition and $I$ an idempotent ideal of $\widehat{\mathcal{C}}$ for which $I_{C}\not=\emptyset$, for any $C\in\mathcal{C}.$ Then the topos ${\bf Sh}_{j^{I}}(\widehat{\mathcal{C}})$ is a De Morgan topos.

Proof. To check the claim, we need to show that for any $j^{I}$-sheaf $F$, the Heyting algebra ${\rm Sub}_{{\bf Sh}_{j^{I}}(\widehat{\mathcal{C}})}(F)$, which its structure can be found in Lemma VI.1.2 of [20], satisfies De Morgan’s law. Following [15], it is sufficient to show that for any $j^{I}$-sheaf $F$ and any subpresheaf $G$ of $F$ the following equality holds

[TABLE]

We know that the join of any two closed subpresheaves of a common presheaf, is the closure of their join (see also [20, Lemma VI.1.2]). On the other hand, using [20, p. 272], for any $C\in\mathcal{C},$ we have

[TABLE]

where $t(C)$ is the maximal sieve on $C$. First of all, we prove that for any $C\in\mathcal{C}$ and any $x\in F(C),$ the equivalence below holds:

[TABLE]

By putting $h={\rm id}_{C}$, the ‘only if’ part of (2) is clear. For establishing the ‘if’ part, take $h\in t(C).$ Since $\mathcal{C}$ satisfies the right Ore condition, there are $u,v$ such that $wu=hv.$ One has $F(hv)(x)=F(wu)(x)\in G(D_{u})$ and so, $F(hvl)(x)\in G(D_{l})$ for any $l\in I_{D_{v}},$ as $G\subseteq F$. Note that by the assumption one has $I_{D_{v}}\not=\emptyset$ and by Definition 2.1, $vl\in I_{D_{l}}$. Setting $g=vl$ and $k={\rm id}_{D_{g}},$ the ‘if’ part of (2) holds.

Next, notice that we have $x\in\neg(\overline{\neg G})(C)$ iff $x$ satisfies in the left side of (2). Finally, by (2), Lemma VI.1.2 of [20] and the definition of closure as in (4), we can deduce that

[TABLE]

for any $C\in\mathcal{C}.$ This is the required result.$\qquad\square$

According to [20, Lemma VI.1.4] one can easily check that the double negation topology (or dense topology) $\neg\neg:\Omega\rightarrow\Omega$ on $\widehat{\mathcal{C}}$ is defined by

[TABLE]

for any $C\in\mathcal{C}$ and $T_{C}\in\Omega(C).$

The following presents an explicit description of the double negation topology on $\widehat{\mathcal{C}}$ associated to an ideal.

Proposition 2.9**.**

Let $I$ be an ideal of $\widehat{\mathcal{C}}$ for which $I_{C}\not=\emptyset$ for any $C\in\mathcal{C}.$ Then, the double negation topology $\neg\neg$ on $\widehat{\mathcal{C}}$ coincides with $I$-double negation topology $\neg\neg_{I}$ which is defined by

[TABLE]

for any $C\in\mathcal{C}$ and $T_{C}\in\Omega(C).$

Proof. Let $C\in\mathcal{C}$ and $T_{C}\in\Omega(C).$ We show that, $\neg\neg_{C}(T_{C})=(\neg\neg_{I})_{C}(T_{C})$. To check this, fix an element $k\in I_{C}$. Take $f\in(\neg\neg_{I})_{C}(T_{C})$ and $g\in t(D_{f})$. Since $I$ is an ideal of $\widehat{\mathcal{C}}$, $gk$ lies in $I_{D_{f}}$. Thus, by (14), there is an element $h\in I_{D_{gk}}$ such that $fgkh\in T_{C}$. Putting $\bar{h}=kh\in I_{D_{g}}$, by (13), one has $f\in\neg\neg_{C}(T_{C})$.

Conversely, let $f\in\neg\neg_{C}(T_{C})$ and $g\in I_{D_{f}}$. Then, there exists some $h\in t(D_{g})$ for which $fgh$ lies in $T_{C}$. Take $l\in I_{D_{h}}.$ Since $T_{C}$ is a sieve on $C$ it follows that $fghl\in T_{C}.$ Also, as $I$ is an ideal of $\widehat{\mathcal{C}}$, we deduce that $hl\in I_{D_{g}}$ and then, by (14), $f\in(\neg\neg_{I})_{C}(T_{C})$.$\qquad\square$

Note that by (10) and Proposition 2.9, we achieve ${\bf Sh}_{\neg\neg}(\widehat{\mathcal{C}})\subseteq{\bf Sh}_{j^{I}}(\widehat{\mathcal{C}}),$ for an ideal $I$ of $\widehat{\mathcal{C}}$ for which $I_{C}\not=\emptyset$ for any $C\in\mathcal{C}.$ Further, if such an ideal $I$ is idempotent also, then one has $j^{I}\leq\neg\neg.$

Let $I$ be an ideal of $\widehat{\mathcal{C}}.$ It is straightforward to see that the subobject classifier of the topos ${\bf Sh}_{j^{I}}(\widehat{\mathcal{C}}),$ denoted by $\Omega_{j^{I}}$ as stated in [20, p. 224] stands for the sheaf given by

[TABLE]

for any $C\in\mathcal{C}.$ By (15), we achieve that $I_{C}\in\Omega_{j^{I}}(C)$ iff $I_{C}=t(C),$ for any $C\in\mathcal{C}.$ It is convenient to see that $\overline{I_{C}}=t(C)$ lies in $\Omega_{j^{I}}(C),$ for any $C\in\mathcal{C}.$

Also, by (9), for any $C\in\mathcal{C},\ I_{C}$ lies in ${\bf J}^{I}(C),$ i.e., $I_{C}$ is a covering sieve. Fix $C\in\mathcal{C}.$ It is well known [20] that the family $\{I_{D_{f}}\}_{f\in I_{C}}$ is a matching family whenever for any $f\in I_{C}$ and any arrow $g:D_{g}\rightarrow D_{f}$ one has

[TABLE]

Note that, using Definition 2.1, the implication ($\Rightarrow$) as in (16) always holds. If the family $\{I_{D_{f}}\}_{f\in I_{C}}$ is a matching family, then $\overline{I^{2}(C)}$ is just the amalgamation of the $I_{D_{f}}$’s which is defined as in [20, p.142] while $I_{D_{f}}$’s and $\overline{I^{2}(C)}$ does not necessarily lie in $\Omega_{j^{I}}$.

3 An admissible class on $\mathcal{C}$ and a topology on $\widehat{\mathcal{C}}$

Let $\mathcal{C}$ be a small category with finite limits. This section is devoted to establish a topology on $\widehat{\mathcal{C}}$ by means of the internal existential quantifier and an admissible class on $\mathcal{C}.$

For the beginning, select a presheaf $X\in\widehat{\mathcal{C}}.$ One has an arrow $\sigma_{{}_{X}}:\Omega\rightarrow\Omega^{X}$ as the cartesian transpose of $\Omega^{-1}(X)=\pi_{2}:X\times\Omega\rightarrow\Omega$ for the pullback functor $\Omega^{-1}:\widehat{\mathcal{C}}\rightarrow\widehat{\mathcal{C}}/\Omega.$ Indeed, for any $C\in\mathcal{C}$ and $S_{C}\in\Omega(C)$, one has

[TABLE]

It is well known that $\sigma_{{}_{X}}$ has an internal left adjunction, so-called the internal existential quantifier and denoted by $\exists_{X}:\Omega^{X}\rightarrow\Omega$. In this route, $\exists_{X}$ is monotone and join preserving map given by

[TABLE]

for any $C\in\mathcal{C}$ and $U\subseteq{\bf y}(C)\times X$ of $\Omega^{X}(C).$ The pair $(\exists_{X},\sigma_{{}_{X}})$ as in (18) and (17), establishes a Galois connection between two locales $(\Omega,\subseteq)$ and $(\Omega^{X},\subseteq)$.

Remark 3.1**.**

Using the isomorphism

[TABLE]

and by (17), we achieve another description of $\sigma_{{}_{X}}$ as follows

[TABLE]

for any $C,D\in\mathcal{C},$ $f\in{\bf y}(C)(D),$ $S_{C}\in\Omega(C)$ and $x\in X(D).$ Furthermore, by the isomorphism (19), one may rewrite (18) by

[TABLE]

for any $C\in\mathcal{C}$ and $\varphi:{\bf y}(C)\times X\rightarrow\Omega$ of $\Omega^{X}(C).$**

Now assume that $\mathcal{C}$ has an admissible class $\mathcal{M}.$ The class $\mathcal{M}$ yields a subpresheaf $M$ of the presheaf ${\rm Sub}_{\mathcal{C}}(-):\mathcal{C}^{\rm op}\rightarrow{\bf Sets},$ given by $M(C)=\mathcal{M}/C$ (see also, [11]).

Actually:

Remark 3.2**.**

Let $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$ By (17) and (18), it is easy to see that

[TABLE]

In particular, since ${\rm id}_{C}\in\mathcal{M}/C$, $\mathcal{M}/C\not=\emptyset,$ so one has $(\exists_{M}\circ\sigma_{{}_{M}})_{C}(S_{C})=S_{C}$ and then, $\exists_{M}\circ\sigma_{{}_{M}}={\rm id}_{\Omega}.$ On the other hand, the resulting monad of the Galois connection $(\exists_{X},\sigma_{{}_{X}})$, which is a closure operator on $\Omega^{X},$ is the arrow $T_{X}=\sigma_{{}_{X}}\circ\exists_{X}:\Omega^{X}\rightarrow\Omega^{X}$ in $\widehat{\mathcal{C}}$ given by $(T_{X})_{C}(U)=(\exists_{X})_{C}(U)\times X$ for any $C\in\mathcal{C}$ and $U\in\Omega^{X}(C).$ Meanwhile, for the exponential arrow

[TABLE]

which is given by

[TABLE]

one has

[TABLE]

for any $C\in\mathcal{C}.$ In particular, we get $T_{M}^{2}=T_{M}$ and $T_{M}\circ{\rm true}^{M}={\rm true}^{M}$.**

It is easy to check that any Galois connection $v\dashv u:\Omega^{X}\rightarrow\Omega$ on $\widehat{\mathcal{C}}$ gives a topology $u\circ v$ on $\widehat{\mathcal{C}}.$ In particular, the Galois connection $\sigma_{{}_{X}}\dashv\forall_{X}:\Omega^{X}\rightarrow\Omega,$ for the internal universal quantifier $\forall_{X}$ introduced in [20], produces a topology $\forall_{X}\circ\sigma_{{}_{X}}:\Omega\rightarrow\Omega$ on $\widehat{\mathcal{C}}.$

Note that some results of the rest of this manuscript hold when $\mathcal{C}$ is $\mathcal{M}$-complete or it has inverse images (= pullbacks of monics).

Now we proceed to construct a natural transformation in terms of the presheaf $M$ which we are interested in.

Lemma 3.3**.**

The assignment $\mu_{{}_{M}}:\Omega\rightarrow\Omega^{M},$ which for any objects $C,D\in\mathcal{C}$ and any sieve $S_{C}\in\Omega(C)$ given by

[TABLE]

and $(\mu_{{}_{M}})_{C}(S_{C})$ assigns to any arrow $h:D\rightarrow D^{\prime}$ the map

[TABLE]

is a natural transformation. It is also a map between internal posets.

Proof. First of all we show the second assertion. That is $\mu_{{}_{M}}$ is a monotone map. But by (22), this is clear. Next, let $C\in\mathcal{C},$ $S_{C}\in\Omega(C)$ and $h:D\rightarrow D^{\prime}$ be an arrow in $\mathcal{C}.$ For any $(l,t)\in(\mu_{{}_{M}})_{C}(S_{C})(D^{\prime})$ one has

[TABLE]

because $\mathcal{M}$ is closed under pullback. Meanwhile, that $S_{C}$ is a sieve on $C$ implies that $lhh^{-1}(t)=ltt^{-1}(h)\in S_{C}.$ These show that $(\mu_{{}_{M}})_{C}(S_{C})$ is well defined on morphisms as a subpresheaf of ${\bf y}(C)\times M,$ i.e., $(\mu_{{}_{M}})_{C}(S_{C})\in\Omega^{M}(C).$

Now we prove that $\mu_{{}_{M}}$ is natural. To this end , for any $h:C\rightarrow D$ we show that

[TABLE]

For any $S_{D}\in\Omega(D)$ and $D^{\prime}\in\mathcal{C},$ one has

[TABLE]

This completes the proof. $\quad\quad\square$

In the other words, by the isomorphism (19), by (22) we can achieve

[TABLE]

for any $C,D\in\mathcal{C},$ $(f,g)\in{\bf y}(C)(D)\times M(D)$ and $S_{C}\in\Omega(C).$

To have a better intuition of $\mu_{M},$ let $\mathcal{C}$ be a monoid $S$ (of course, as a category with just one object it is not necessary finitely complete) and $M$ be the $S$-set of all left cancellable elements of $S$ endowed with the trivial action. Indeed, $M$ is the class of all monics on $S$ which is an admissible class. Then, the action preserving map $\mu_{{}_{M}}:\Omega\rightarrow\Omega^{M}$ is given by $\mu_{{}_{M}}(R)(s,m)=R\cdot(sm)$ in which $\cdot$ is the action of $S$ on $\Omega,$ for any right ideal $R$ of $S$ and any $(s,m)\in S\times M.$ Note that in [21] it is proved that, as a category, $S$ has products iff $S\times S$ is isomorphic to $S$ as $S$-sets.

The arrow $\mu_{{}_{M}}$ defined as in (22) gives us actually a copy of $\Omega$ in $\Omega^{M}$ as the following proposition shows.

Proposition 3.4**.**

The arrow $\mu_{{}_{M}}:\Omega\rightarrow\Omega^{M},$ defined as in (22), is a monic arrow in $\widehat{\mathcal{C}}.$ Indeed, for any $C\in\mathcal{C},$ one has

[TABLE]

Proof. We show that for any $C\in\mathcal{C},$ the function $(\mu_{{}_{M}})_{C}:\Omega(C)\rightarrow\Omega^{M}(C)$ is one to one. Let $S_{C},T_{C}$ be two sieves on $C$ such that $(\mu_{{}_{M}})_{C}(S_{C})=(\mu_{{}_{M}})_{C}(T_{C})$ and so, $(\mu_{{}_{M}})_{C}(S_{C})(D)=(\mu_{{}_{M}})_{C}(T_{C})(D)$ for any $D\in\mathcal{C}.$ For any $f\in S_{C},$ since by (22) one has $(f,{\rm id}_{D})\in(\mu_{{}_{M}})_{C}(S_{C})(D)$ we deduce that $(f,{\rm id}_{D})\in(\mu_{{}_{M}})_{C}(T_{C})(D)$ and then, by (22), $f\in T_{C}.$ Analogously, the converse also is true. Thus, $S_{C}=T_{C}.$ $\quad\quad\square$

Now we will consider an element of $\Omega^{M}(C)$ as an arrow $\varphi:{\bf y}(C)\times M\rightarrow\Omega$ in $\widehat{\mathcal{C}}$ for any $C\in\mathcal{C}.$ By Proposition 3.4 and (23), one can easily checked that the characteristic map ${\rm Char}(\mu_{{}_{M}}):\Omega^{M}\rightarrow\Omega$ is an arrow in $\widehat{\mathcal{C}}$ given by

[TABLE]

for any $C\in\mathcal{C}$ and any arrow $\varphi:{\bf y}(C)\times M\rightarrow\Omega$ of $\Omega^{M}(C).$

By (17) and (22), it is straightforward to see that $(\sigma_{{}_{M}})_{C}(S_{C})\subseteq(\mu_{{}_{M}})_{C}(S_{C})$ for any $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$ In the following we find a relationship between arrows $\mu_{{}_{M}}:\Omega\rightarrow\Omega^{M}$ and $\exists_{M}:\Omega^{M}\rightarrow\Omega$ which is close to be an adjoint.

Proposition 3.5**.**

*Let $C\in\mathcal{C},$ $S_{C}\in\Omega(C)$ and $U\in\Omega^{M}(C).$ If $(\mu_{{}_{M}})_{C}(S_{C})\subseteq U$ then one has $S_{C}\subseteq(\exists_{M})_{C}(U).$ *

Proof. First we remark that the inclusion in the assumption, i.e., the order on $\Omega^{M},$ means that $((\mu_{{}_{M}})_{C}(S_{C}))(D)\subseteq U(D)$ for all $D\in\mathcal{C}.$ Let $D\in\mathcal{C}$ and $f\in S_{C}$ with domain $D.$ Since $S_{C}$ is a sieve on $C$ we deduce that $fg\in S_{C}$ for any $g\in\mathcal{M}/D.$ Now by assumption, (22) and (18), we have $f\in(\exists_{M})_{C}(U)$ since $\mathcal{M}/D\not=\emptyset.$ $\quad\quad\square$

Since $\mathcal{C}$ is finitely complete it follows that $\mathcal{C}$ satisfies in the right Ore condition. Then, the double negation topology $\neg\neg:\Omega\rightarrow\Omega$ on $\widehat{\mathcal{C}},$ defined as in (13), coincides with the atomic topology on $\mathcal{C}$ which takes all nonempty sieves as covers, i.e., we have

[TABLE]

for any $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$

In what follows we are going to define a topology on $\widehat{\mathcal{C}}$ similar to $\neg\neg$ as in (25) in terms of the two arrows $\exists_{M}$ and $\mu_{M}$ in $\widehat{\mathcal{C}}$.

Theorem 3.6**.**

Consider the arrows $\mu_{{}_{M}}:\Omega\rightarrow\Omega^{M}$ and $\exists_{M}:\Omega^{M}\rightarrow\Omega$ in $\widehat{\mathcal{C}}$, defined as in (22) and (18), respectively. Then the compound arrow $\exists_{M}\circ\mu_{{}_{M}}$ is a topology on $\widehat{\mathcal{C}}$ which is given by

[TABLE]

for any $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$

Proof. We check that $\exists_{M}\circ\mu_{{}_{M}}$ satisfies in the axioms of a topology on $\widehat{\mathcal{C}}.$ First, it is easy to see that $(\exists_{M}\circ\mu_{{}_{M}})\circ{\rm true}={\rm true}.$

Next, let $C\in\mathcal{C}$ and $S_{C},T_{C}\in\Omega(C).$ By the definition of $\exists_{M}\circ\mu_{{}_{M}}$, it is clear

[TABLE]

To check the converse, let $f\in(\exists_{M}\circ\mu_{{}_{M}})_{C}(S_{C})\cap(\exists_{M}\circ\mu_{{}_{M}})_{C}(T_{C}).$ By (26) there are $g,h\in\mathcal{M}/D_{f}$ such that $fg\in S_{C}$ and $fh\in T_{C}.$ That $\mathcal{M}$ is closed under pullback implies $g^{-1}(h),h^{-1}(g)\in\mathcal{M}$ and then, $hh^{-1}(g)\in\mathcal{M}$ because $\mathcal{M}$ is closed under composition. Therefore, $fhh^{-1}(g)=fgg^{-1}(h)\in S_{C}$ and $fhh^{-1}(g)\in T_{C}$ as $S_{C}$ and $T_{C}$ are sieves. This shows that $fhh^{-1}(g)\in S_{C}\cap T_{C}$ and then, $f\in(\exists_{M}\circ\mu_{{}_{M}})_{C}(S_{C}\cap T_{C}).$

Finally, we prove that $\exists_{M}\circ\mu_{{}_{M}}$ is idempotent. It is sufficient to show that $(\exists_{M}\circ\mu_{{}_{M}})^{2}\leq(\exists_{M}\circ\mu_{{}_{M}}).$ To do so, let $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$ By (26), we get

[TABLE]

and then, $(\exists_{M}\circ\mu_{{}_{M}})_{C}^{2}(S_{C})\subseteq(\exists_{M}\circ\mu_{{}_{M}})_{C}(S_{C})$ since $\mathcal{M}$ is closed under composition. $\quad\quad\square$

It is easily to see that the idempotent modal closure operator associated to the topology $\exists_{M}\circ\mu_{{}_{M}}$ defined as in (26), denoted by

[TABLE]

for any presheaf $F,$ assigns to any subpresheaf $G$ of $F,$ the subpresheaf $\overline{G}$ of $F$ given by

[TABLE]

for any $C\in\mathcal{C}.$ Note that for the closure operator associated to $\neg\neg$ in place of $\mathcal{M}/D_{f}$ as in (28) one has $t(D_{f})$. Furthermore, setting $M={\rm Sub}_{\mathcal{C}}(-)$ in (26) and (28) we may achieve a topology and an idempotent modal closure operator on $\widehat{\mathcal{C}}.$ Henceforth, we denote the topologies $\exists_{M}\circ\mu_{{}_{M}}$ and $\Omega\stackrel{{\scriptstyle\mu_{{}_{{\rm Sub}_{\mathcal{C}}(-)}}}}{{\longrightarrow}}\Omega^{{\rm Sub}_{\mathcal{C}}(-)}\stackrel{{\scriptstyle\exists_{{\rm Sub}_{\mathcal{C}}(-)}}}{{\longrightarrow}}\Omega,$ on $\widehat{\mathcal{C}}$ by $j_{{}_{M}}$ and $j_{{}_{\rm Sub}},$ respectively. Now one has the chain $j_{{}_{M}}\leq j_{{}_{\rm Sub}}\leq\neg\neg$ and then, ${\bf Sh}_{\neg\neg}(\widehat{\mathcal{C}})\subseteq{\bf Sh}_{j_{{}_{\rm Sub}}}(\widehat{\mathcal{C}})\subseteq{\bf Sh}_{j_{{}_{M}}}(\widehat{\mathcal{C}}).$ Note that for any $C\in\mathcal{C}$ and $S_{C}\in\Omega(C),$ by (26) we have

[TABLE]

Also, it is easy to check that the Grothendieck topology associated to $j_{{}_{M}},$ for any $C\in\mathcal{C},$ is given by

[TABLE]

These are definable for $j_{{}_{\rm Sub}}$ by replacing ${\rm Sub}_{\mathcal{C}}(-)$ instead of $M.$

Let $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$ By (26), we may extract a class of partial maps on $\mathcal{C}$ as follows:

[TABLE]

Indeed, $\mathcal{P}_{S_{C}}$ is a class of arrows in the category $\mathcal{M}$-Ptl$(\mathcal{C}).$ Notice that for two arrows $f,g$ and any sieve $S_{C}$ on $C,$ one has $[(m,fm)]\in\mathcal{P}_{g^{*}(S_{C})}$ iff $[(m,gfm)]\in\mathcal{P}_{S_{C}}.$ It is well known that for two partial maps $[(n,f)]$ and $[(t,s)]$ one has $[(n,f)]\leq[(t,s)]$ iff there is an arrow (monic) $p:D_{f}\rightarrow D_{s}$ such that $sp=f$ and $tp=n.$ Here, any $[(m,fm)]\in\mathcal{P}_{S_{C}}$ as in (29) belongs to $\downarrow\![({\rm id}_{D_{f}},f)]=\{~{}[(m,fm)]~{}|~{}m\in{\rm Sub}_{\mathcal{C}}(D_{f})\}.$ Analogously, we can establish a class of partial maps in the category ${\bf Ptl}(\mathcal{C}),$ given by

[TABLE]

One has

[TABLE]

for any $C\in\mathcal{C}.$ It is clear that

[TABLE]

for any $C\in\mathcal{C}.$

In what follows we constitute a subcategory of $\mathcal{M}$-Ptl$(\mathcal{C}).$

Proposition 3.7**.**

The classes of partial maps as in (29) and (30) are closed under composition. More generally, the objects of $\mathcal{C}$ together with the set $\bigcup_{C\in\mathcal{C}}\mathcal{P}_{t(C)}$ as the arrows constitutes a subcategory of $\mathcal{M}$-Ptl$(\mathcal{C}),$ denoted by $\mathcal{M}\mathcal{P}(\mathcal{C}).$ Similarly, one can construct a subcategory of ${\bf Ptl}(\mathcal{C})$ via $\mathcal{P}^{\prime}_{t(C)}$’s, denoted by $\mathcal{P}^{\prime}(\mathcal{C}).$

Proof. We prove only the first assertion. The second assertion follows analogously. First we investigate that the partial maps as in (29) are closed under composition. To verify this, consider two composable partial maps $[(m,fm)]$ and $[(n,gn)],$ i.e., $C=C_{fm}=C_{n}.$ We have

[TABLE]

Since $n\in\mathcal{M}$ hence, $m(fm)^{-1}(n)\in\mathcal{M}$ because $\mathcal{M}$ is stable under pullback and is closed under composition. That $gn\in S_{C}$ by (29) and $S_{C}$ is a sieve on $C$ implies that $gnn^{-1}(fm)$ lies in $S_{C}$ and then, $gfm(fm)^{-1}(n)\in S_{C}$ as $gfm(fm)^{-1}(n)=gnn^{-1}(fm).$ Then,

[TABLE]

is a partial map in $\mathcal{P}_{S_{C}}$. More generally, the elements of $\bigcup_{C\in\mathcal{C}}\mathcal{P}_{t(C)}$ are closed under composition. Meanwhile, from (31), we can deduce that the partial map $[({\rm id}_{C},{\rm id}_{C})]$ lies in $\mathcal{P}_{t(C)}$ for any $C\in\mathcal{C}.$ $\quad\quad\square$

Note that the two categories $\mathcal{M}\mathcal{P}(\mathcal{C})$ and $\mathcal{P}^{\prime}(\mathcal{C})$ contain all whole maps, i.e., all partial maps of the form $[({\rm id}_{C},f)]$ where $f:C\rightarrow D$ for $C\in\mathcal{C}.$

Next we define a functor $A_{{}_{M}}:\mathcal{C}\rightarrow{\bf Sets}$ which assigns to any $C\in\mathcal{C}$ the set $\mathcal{P}_{t(C)}$ defined as in (31), and to any arrow $g:C\rightarrow D$ the map $A_{{}_{M}}(g):A_{{}_{M}}(C)\rightarrow A_{{}_{M}}(D)$ given by $A_{{}_{M}}(g)([(m,fm)])=([(m,gfm)]).$ In this route, the functor $-\otimes_{\mathcal{C}}A_{{}_{M}}:\widehat{\mathcal{C}}\rightarrow{\bf Sets},$ which is just the (left) Kan extension of $A_{{}_{M}}$ along y, is not filtering and then, nor left exact [20, Theorem VII.6.3]. Because, if $-\otimes_{\mathcal{C}}A_{{}_{M}}$ satisfies in [20, Definition VII.6.2], then for given elements $[(m,fm)]\in A_{{}_{M}}(C)$ and $[(n,gn)]\in A_{{}_{M}}(D),$ there must exist an object $B\in\mathcal{C}$, morphisms $C\stackrel{{\scriptstyle u}}{{\rightarrow}}B\stackrel{{\scriptstyle v}}{{\rightarrow}}D$ in $\mathcal{C},$ and an element $[(t,ht)]\in A_{{}_{M}}(B)$ such that $[(t,uht)]=[(m,fm)]$ and $[(t,vht)]=[(n,gn)].$ This holds only if $t=m=n$ but this is not the case since we choose $[(m,fm)]$ and $[(n,gn)]$ arbitrary.

Analogously, we may obtain a functor $A_{{}_{\rm Sub}}:\mathcal{C}\rightarrow{\bf Sets}$ via the sets $\mathcal{P}^{\prime}_{t(C)}$, where $C\in\mathcal{C}.$ Note that the functor $-\otimes_{\mathcal{C}}A_{{}_{\rm Sub}}$ is not necessary left exact.

4 An admissible class on $\mathcal{C}$ and an action on $\Omega$

In this section, for a small category $\mathcal{C}$ with finite limits, among other things, we define an action on the subobject classifier $\Omega$ of $\widehat{\mathcal{C}}$ by means of an admissible class on the category $\mathcal{C}.$

Let $\mathcal{C}$ be a finitely complete small category equipped with an admissible class $\mathcal{M}.$ As we have already mentioned the class $\mathcal{M}$ yields a subpresheaf $M$ of the presheaf ${\rm Sub}_{\mathcal{C}}(-):\mathcal{C}^{\rm op}\rightarrow{\bf Sets},$ given by $M(C)=\mathcal{M}/C$. By (23), the cartesian transpose of $\mu_{{}_{M}}:\Omega\rightarrow\Omega^{M},$ denoted by $\widehat{\mu_{{}_{M}}}:\Omega\times M\rightarrow\Omega$ is given by

[TABLE]

for any $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega(C).$ Recall [20] that for any arrow $f:D\rightarrow C,$ the pullback (or change of base) functor $f^{-1}:\mathcal{C}/C\rightarrow\mathcal{C}/D$ has a left adjoint $\Sigma_{f},$ given by composition with $f.$ By (32), indeed for any $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega(C)$ one has $(\widehat{\mu_{{}_{M}}})_{C}(S_{C},f)=(\Sigma_{f}\circ f^{-1})^{-1}(S_{C})$ in which $\Sigma_{f}\circ f^{-1}$ stands for the object function of the product functor $f\times-:\mathcal{C}/C\rightarrow\mathcal{C}/C.$

It is easy to check that the subobject $W_{\mu_{{}_{M}}}$ of $\Omega\times M$ which has the characteristic map $\widehat{\mu_{{}_{M}}},$ is the presheaf given by

[TABLE]

for any $C\in\mathcal{C}.$

In the next lemma we turn $M$ into an ordered algebraic structure.

Lemma 4.1**.**

The triple $(M,\cdot,e)$ is a (internal) commutative monoid on $\widehat{\mathcal{C}}$ in which for any $C\in\mathcal{C},$ the arrow $\cdot_{C}:M(C)\times M(C)\rightarrow M(C)$ defined by the product in $\mathcal{C}/C,$ and the map $e_{C}:1(C)(=\{*\})\rightarrow M(C)$ is given by $e_{C}(*)={\rm id}_{C}.$ Moreover, the monoid structure of $M$ is compatible with the order on $M$ induced by ${\rm Sub}_{\mathcal{C}}(-).$ That is, $M$ is an ordered commutative monoid in $\widehat{\mathcal{C}}$.

Proof. It is well known that for any $k:C\rightarrow D$ the function $k^{-1}:M(D)\rightarrow M(C)$ is just the object function of the restriction of the pullback functor $k^{-1}:\mathcal{C}/D\rightarrow\mathcal{C}/C$ to $\mathcal{M}/D,$ i.e. $k^{-1}:\mathcal{M}/D\rightarrow\mathcal{M}/C$. That pullback functors preserve products and identities it follows that $\cdot:M\times M\rightarrow M$ and $e:1\rightarrow M$ are natural. The rest of the proof is clear. $\quad\quad\square$

We proceed to present an action of $M$ on $\Omega$ as follows.

Lemma 4.2**.**

The arrow $\widehat{\mu_{{}_{M}}}:\Omega\times M\rightarrow\Omega$, defined as in (32), is an (internal) action of the commutative monoid $M$ on $\Omega.$ This means that $\Omega$ is an (left and right) $M$-set in $\widehat{\mathcal{C}}.$

Proof. By the formula (32), one has $(\widehat{\mu_{{}_{M}}})_{C}(S_{C},{\rm id}_{C})=S_{C}$ for any $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$ Furthermore,

[TABLE]

for any $C\in\mathcal{C},$ $f,g\in M(C)$ and $S_{C}\in\Omega(C).$ $\quad\quad\square$

From now on, for simplicity we denote $(\widehat{\mu_{{}_{M}}})_{C}(S_{C},f)$, by $S_{C}\cdot f$ for any $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega(C).$

By the terminology provided in [20, p. 238], since $M$ is commutative, $\widehat{\mathcal{C}}^{M}$ is the category of objects of $\widehat{\mathcal{C}}$ equipped with a (left, right) $M$-action. In this way, by an $M$-frame in $\widehat{\mathcal{C}}$ we mean a frame $F$ in $\widehat{\mathcal{C}}^{M}$. That is the operations on the frame $F$ are equivariant. As further properties of $\Omega$ as an $M$-set in $\widehat{\mathcal{C}}$ we have:

Proposition 4.3**.**

The presheaf $\Omega$ is an $M$-frame in $\widehat{\mathcal{C}}.$

Proof. First recall [20, Proposition I.8.5] that the subobject classifier $\Omega$ of $\widehat{\mathcal{C}}$ is a frame in $\widehat{\mathcal{C}}$. By (32), it is straightforward to see that the binary operations $\vee$ and $\wedge$ on $\Omega$ are equivariant. Meanwhile, the terminal presheaf $1$ in $\widehat{\mathcal{C}}$ endowed with the trivial action is an $M$-set. Again by (32), one can easily checked that the arrows ‘true’ and ‘false’ $:1\rightarrow\Omega$ which are nullary operations of $\Omega$ are equivariant. $\quad\quad\square$

Now we find two sub $M$-sets of $\Omega.$

Proposition 4.4**.**

For the topology $j_{{}_{M}}$ ($j_{{}_{\rm Sub}}$) on $\widehat{\mathcal{C}},$ the presheaf $\Omega_{j_{{}_{M}}}{\rm(}\Omega_{j_{{}_{\rm Sub}}}$)* is a sub $M$-set of $\Omega.$*

Proof. First of all by (26), we obtain

[TABLE]

for any $C\in\mathcal{C},$ which is non-empty since $t(C)\in\Omega_{j_{{}_{M}}}(C).$ (Similarly, one can define $\Omega_{j_{{}_{\rm Sub}}}(C).$) Note that one has a chain of subobjects as in $\Omega_{j_{{}_{M}}}\rightarrowtail\Omega_{j_{{}_{\rm Sub}}}\rightarrowtail\Omega$ in $\widehat{\mathcal{C}}.$ Also, we point out that the implication $(\Leftarrow)$ as in (33) always holds since for any $f\in S_{C}$ one has ${\rm id}_{D_{f}}\in\mathcal{M}/D_{f}.$ To check the assertion, let $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega_{j_{{}_{M}}}(C).$ We show that the sieve $S_{C}\cdot f$ lies in $\Omega_{j_{{}_{M}}}(C).$ Let $h:D_{h}\rightarrow C$ be an arrow for which there exists $m\in\mathcal{M}/D_{h}$ such that $hm\in S_{C}\cdot f.$ We show that $h\in S_{C}\cdot f$. We have $f\times hm\in S_{C}$ by (32). We also have

[TABLE]

Since $m\in\mathcal{M}$ and $\mathcal{M}$ is stable under pullback, thus $(h^{-1}(f))^{-1}(m)$ belongs to $\mathcal{M}.$ This fact together with the assumption $f\times hm\in S_{C}$ and $S_{C}\in\Omega_{j_{{}_{M}}}(C),$ by (33), show that $hh^{-1}(f)\in S_{C},$ i.e., $f\times h\in S_{C}$ or $h\in S_{C}\cdot f.$ $\quad\quad\square$

Next we exhibit a ${\rm Sub}_{\mathcal{C}}(-)$-poset in $\widehat{\mathcal{C}}.$

Proposition 4.5**.**

For the topology $j_{{}_{\rm Sub}}$ on $\widehat{\mathcal{C}},$ the presheaf $\Omega_{j_{{}_{\rm Sub}}}$ is a ${\rm Sub}_{\mathcal{C}}(-)$-poset in $\widehat{\mathcal{C}}.$

Proof. By Proposition 4.4, $\Omega_{j_{{}_{\rm Sub}}}$ is an ${\rm Sub}_{\mathcal{C}}(-)$-set in $\widehat{\mathcal{C}}.$ To establish the aim, let $C\in\mathcal{C},$ $S_{C},T_{C}\in\Omega_{j_{{}_{\rm Sub}}}(C)$ and $m,n\in{\rm Sub}_{\mathcal{C}}(C)$ for which $S_{C}\subseteq T_{C}$ and $m\leq n.$ We show that $S_{C}\cdot m\subseteq T_{C}\cdot n$. At the beginning, by Proposition 4.3, we deduce $S_{C}\cdot m\subseteq T_{C}\cdot m.$ Therefore, it only remains to prove $T_{C}\cdot m\subseteq T_{C}\cdot n.$ To do so, let $h\in T_{C}\cdot m$. Then, by (32), $mm^{-1}(h)=m\times h\in T_{C}.$ Since $m\leq n,$ there is an arrow $k$ such that $nk=m.$ This concludes that corresponding to arrows $h^{-1}(m)$ and $km^{-1}(h)$ in the following pullback diagram there exists a unique arrow $w$ which commutes the resulting triangles,

[TABLE]

Since monics are stable under pullback, that $m$ is monic implies that $h^{-1}(m)$ is also and then, by the diagram (35), $w$ is monic too. On the other hand, by the diagram (35), one has $nn^{-1}(h)w=nkm^{-1}(h)=mm^{-1}(h)$ and then, $nn^{-1}(h)w$ lies in $T_{C}$ for $mm^{-1}(h)\in T_{C}.$ Finally, using this fact and that $w$ is monic, by the definition of $\Omega_{j_{{}_{\rm Sub}}}$ which is similar to (33), we get $n\times h=nn^{-1}(h)\in T_{C}$ and then, $h\in T_{C}\cdot n.$ $\quad\quad\square$

Replacing $\widehat{\mathcal{C}}$ by ${\bf Sh}_{j_{{}_{M}}}(\widehat{\mathcal{C}})$ in (20), one can observe that in ${\bf Sh}_{j_{{}_{M}}}(\widehat{\mathcal{C}}),$ we have the arrow $\sigma_{{}_{M}}:\Omega_{j_{{}_{M}}}\rightarrow\Omega_{j_{{}_{M}}}^{M}$ as the restriction of $\sigma_{{}_{M}}:\Omega\rightarrow\Omega^{M}$ to $\Omega_{j_{{}_{M}}}.$

We proceed to present a relationship between the restricted arrows $\sigma_{{}_{M}}:\Omega_{j_{{}_{M}}}\rightarrow\Omega_{j_{{}_{M}}}^{M}$ and ${\rm Char}(\mu_{{}_{M}}):\Omega_{j_{{}_{M}}}^{M}\rightarrow\Omega$ defined as in (20) and (3), respectively.

Proposition 4.6**.**

Consider the restricted arrows $\sigma_{{}_{M}}:\Omega_{j_{{}_{M}}}\rightarrow\Omega_{j_{{}_{M}}}^{M}$ and ${\rm Char}(\mu_{{}_{M}}):\Omega_{j_{{}_{M}}}^{M}\rightarrow\Omega$, defined by the formulas (20) and (3) before. Then, the compound arrow ${\rm Char}(\mu_{{}_{M}})\circ\sigma_{{}_{M}}:\Omega_{j_{{}_{M}}}\rightarrow\Omega$ is extensive, i.e., one has

[TABLE]

for any $C\in\mathcal{C}$ and $S_{C}\in\Omega_{j_{{}_{M}}}(C).$

Proof. First of all let us consider $C\in\mathcal{C}$ and $S_{C}\in\Omega_{j_{{}_{M}}}(C).$ By the definitions of $\sigma_{{}_{M}}$ and ${\rm Char}(\mu_{{}_{M}})$, we have

[TABLE]

To investigate (36), let $f\in S_{C}.$ We show $f\in{\rm Char}(\mu_{{}_{M}})_{C}\circ(\sigma_{{}_{M}})_{C}(S_{C}).$ To do this, in light of (4), for an arbitrary $B\in\mathcal{C}$ and $(h,g)\in{\bf y}(D_{f})(B)\times M(B)$ we prove that

[TABLE]

Note that one has $h\circ(k\times g)\in f^{*}(S_{C})$ iff $fh\circ(k\times g)\in S_{C}.$ The ‘only if’ part of (38) holds because if $fhk\in S_{C}$ and $S_{C}$ is a sieve on $C,$ then $fh\circ(k\times g)=fhkk^{-1}(g)\in S_{C}.$ To check the ‘if’ part of (38), suppose $h\circ(k\times g)\in f^{*}(S_{C})$. Equivalently, $fh\circ(k\times g)\in S_{C}$ and then, $fhkk^{-1}(g)\in S_{C}$. Since $g\in\mathcal{M}$ and $\mathcal{M}$ is stable under pullback, it yields that $k^{-1}(g)\in\mathcal{M}.$ Roughly, this fact together with the assumption and $S_{C}\in\Omega_{j_{{}_{M}}}(C)$, in view of (33), imply that $fhk\in S_{C}.$ $\quad\quad\square$

The following provides a necessary condition so that $\neg\neg$ is action preserving.

Theorem 4.7**.**

Let $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega(C).$ Then one always has $\neg\neg_{C}(S_{C}\cdot f)\subseteq\neg\neg_{C}(S_{C})\cdot f.$ The converse holds, i.e., $\neg\neg:\Omega\rightarrow\Omega$ is an $M$-action preserving map in $\widehat{\mathcal{C}}$, whenever in the category $\mathcal{C}$ any composable pair $(s,t)$ admits a pullback as follows:

[TABLE]

In particular, $\neg\neg:\Omega\rightarrow\Omega$ is a ${\rm Sub}_{\mathcal{C}}(-)$-action preserving map in $\widehat{\mathcal{C}}$ if $\mathcal{C}$ is cartesian closed.

Proof. In view of (25) and (32), we have

[TABLE]

and

[TABLE]

To check the goal, first we show that $\neg\neg_{C}(S_{C}\cdot f)\subseteq\neg\neg_{C}(S_{C})\cdot f.$ Let $h\in\neg\neg_{C}(S_{C}\cdot f)$. Then, there is an arrow $k:B\rightarrow D_{h}$ for which $f\times(hk)\in S_{C}.$ Since

[TABLE]

setting $g=(h^{-1}(f))^{-1}(k),$ by (40), we get $h\in\neg\neg_{C}(S_{C})\cdot f$.

To prove the converse, now let $h\in\neg\neg_{C}(S_{C})\cdot f.$ Then, there exists an arrow $g:A\rightarrow D_{f\times h}$ such that $(f\times h)\circ g\in S_{C}.$ Here, by assumption, the composable pair $(g,h^{-1}(f))$ form part of a pullback diagram as follows:

[TABLE]

Now, we get

[TABLE]

and thus, $f\times(hk)\in S_{C}$.

In the weaker case if $\mathcal{C}$ has pullback complements over all its morphisms, introduced in [6], then any composable pair in $\mathcal{C}$ admits a pullback. On the other hand, by [6, Theorem 4.4], $\mathcal{C}$ has pullback complements over all its monics whenever $\mathcal{C}$ is cartesian closed (of course, here it is sufficient that any monic in slices of $\mathcal{C}$ be exponentiable). By this fact and what mentioned in the two preceding paragraphs, since for any monic $f$ any arrow of the form $h^{-1}(f)$, is monic it follows that $\neg\neg$ is ${\rm Sub}_{\mathcal{C}}(-)$-action preserving. $\quad\quad\square$

Note that for a regular category $\mathcal{C},$ Meisen [22] has given a necessary and sufficient condition that a composable pair in $\mathcal{C}$ form part of a pullback diagram. Ehrig and Kreowski [8] discuss the same problem in the opposites situation, and have given solutions for the categories of sets and of graphs.

In a similar method of the proof of Theorem 4.7, we will give some necessary conditions so that two topologies $j_{{}_{M}},$ and $j_{{}_{\rm Sub}}$ are equivariant.

Corollary 4.8**.**

For any $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega(C),$ one has $(j_{{}_{M}})_{C}(S_{C}\cdot f)\subseteq(j_{{}_{M}})_{C}(S_{C})\cdot f.$ The converse holds, i.e., $j_{{}_{M}}$ is $M$-action preserving, whenever in the category $\mathcal{C}$ any composable pair $(s,t)$ such that $s\in\mathcal{M}$ admits a pullback as follows

[TABLE]

where $h\in\mathcal{M}.$ Similarly, $j_{{}_{\rm Sub}}$ is $M$-action preserving if the above condition holds by setting the class of monics in $\mathcal{C}$ instead of $\mathcal{M}.$

Proof. We only mention that since for any $k\in\mathcal{M}$ any arrow of the form $(h^{-1}(f))^{-1}(k)$ lies in $\mathcal{M}$, then, one can achieve the claim exactly by iterating the proof of Theorem 4.7. $\quad\quad\square$

The following proposition gives an answer to this question: When the weak ideal topology, defined as in (10), is an action preserving map?

Proposition 4.9**.**

Let $I$ be an ideal of $\widehat{C}$, $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega(C).$ If $I$ is stable under pullbacks (i.e. $\forall g,\ \forall f\in I_{C_{g}},\ g^{-1}(f)\in I_{D_{g}}$), then $j^{I}_{C}(S_{C})\cdot f\subseteq j^{I}_{C}(S_{C}\cdot f)$. The converse holds, i.e., $j^{I}$ is an $M$-action preserving map in $\widehat{\mathcal{C}}$, whenever in the category $\mathcal{C}$ any composable pair $(s,t)$ admits a pullback as follows:

[TABLE]

Moreover, the ideal $I$ satisfies in the converse of the stability property of pullbacks, i.e. for any arrow $g$ and any $f\in t(C_{g})$ if $g^{-1}(f)\in I_{D_{g}}$ then $f\in I_{C_{g}}$.

Proof. Take $C\in\mathcal{C},$ $f\in M(C)$ and $S_{C}\in\Omega(C).$ By (10) and (32), one has

[TABLE]

and

[TABLE]

To verify the first part of proposition, let $h\in j^{I}_{C}(S_{C})\cdot f$ and $k\in I_{D_{h}}$. Then one has $f\times(hk)=ff^{-1}(h)(h^{-1}(f))^{-1}(k)=(f\times h)\circ(h^{-1}(f))^{-1}(k)$. Put $g=(h^{-1}(f))^{-1}(k).$ Since $I$ is stable under pullbacks and $k\in I_{D_{h}}$ it yields that $g\in I_{D_{f\times h}}.$ Then, by (42), we get $f\times(hk)\in S_{C}.$

For establishing the inclusion $j^{I}_{C}(S_{C}\cdot f)\subseteq j^{I}_{C}(S_{C})\cdot f$, let $h\in j^{I}_{C}(S_{C}\cdot f)$ and $g\in I_{D_{f\times h}}$. Here, by assumption, the composable pair $(g,h^{-1}(f))$ form part of a pullback diagram as follows

[TABLE]

That the ideal $I$ satisfies in the converse property of the stability under pullbacks and that $g\in I_{D_{f\times h}}$ conclude that $k\in I_{D_{h}}.$ Finally, by (41), we have

[TABLE]

thus, $f\times(hk)\in S_{C}$ and the proposition is proved.$\quad\quad\square$

Also, we have

Proposition 4.10**.**

Consider a pullback square in $\widehat{\mathcal{C}}$ as follows,

[TABLE]

Then $W$ is a subact of $\Omega$ whenever $\phi$ is an $M$-action preserving map.

In particular, (weak) Grothendieck topologies on $\mathcal{C}$ associated to $M$-action preserving (weak) topologies on $\widehat{\mathcal{C}},$ are subacts of $\Omega.$

Proof. Let $C\in\mathcal{C},$ $m\in M(C)$ and $S_{C}\in W(C).$ We investigate $S_{C}\cdot m\in W(C).$ One has $\phi_{C}(S_{C}\cdot m)=\phi_{C}(S_{C})\cdot m$ since $\phi$ is an action preserving map in $\widehat{\mathcal{C}}$. Since $S_{C}\in W(C),$ by the diagram (43), we have $\phi_{C}(S_{C})=t(C)$ and thus, $\phi_{C}(S_{C}\cdot m)=t(C)\cdot m=t(C).$ Then, again by the diagram (43), $S_{C}\cdot m\in W(C)$ as required. The second assertion is clear. $\quad\quad\square$

Now let us consider an arbitrary element of $\Omega^{\Omega}(C)$ as an arrow ${\bf y}(C)\times\Omega\rightarrow\Omega$, for any $C\in\mathcal{C}.$ The arrow $\lambda:M\rightarrow\Omega^{\Omega},$ the cartesian transpose of $\widehat{\mu_{{}_{M}}}:\Omega\times M\rightarrow\Omega$, defined as in (32), is given by

[TABLE]

for any $C,D\in\mathcal{C},$ $m\in M(C),$ $g\in{\bf y}(C)(D)$ and $S_{D}\in\Omega(D).$ Indeed, any arrow of the form $\lambda_{C}(m):{\bf y}(C)\times\Omega\rightarrow\Omega$ may be called a (internal) translation on $\Omega.$ It is easy to see that

[TABLE]

for any $C\in\mathcal{C},$ $m\in M(C)$ and $S_{C}\in\Omega(C).$ We proceed to generalize these translations to form an action preserving weak topology on $\widehat{\mathcal{C}}.$

Let $\mathcal{F}=\{f_{C}:D_{f_{C}}\rightarrow C\}_{C\in\mathcal{C}}$ be a family of arrows of $\mathcal{C}$ such that for any $g:C\rightarrow D$ and any $S_{D}\in\Omega(D)$ one has

[TABLE]

(e.g., this holds if $g^{-1}(f_{D})=f_{C}$). Then, it is easy to see that the family $\mathcal{F}$ gives us a natural transformation $\alpha_{\mathcal{F}}:\Omega\rightarrow\Omega$ defined by

[TABLE]

for any $C\in\mathcal{C}$ and $S_{C}\in\Omega(C).$ Note that in (45) we have $(\alpha_{\mathcal{F}})_{C}(S_{C})=S_{C}\cdot f_{C}$ whenever $f_{C}\in M(C).$

In the following theorem we establish an action preserving weak topology on $\widehat{\mathcal{C}}.$ First, an arrow $f:A\rightarrow B$ in $\mathcal{C}$ is said to be idempotent in $\mathcal{C}/B$ if $f^{2}=f$ where $f^{2}=f\times f.$ We point out that if an arbitrary arrow $f:A\rightarrow B$ is monic, then it is idempotent since $f\times f=ff^{-1}(f)=f\circ{\rm id}_{A}=f.$

Theorem 4.11**.**

Let $\mathcal{F}=\{f_{C}:D_{f_{C}}\rightarrow C\}_{C\in\mathcal{C}}$ be a family of arrows in $\mathcal{C}$ which satisfies in the formula (44). Then, the arrow $\alpha_{\mathcal{F}}:\Omega\rightarrow\Omega$ given by (45) is an action preserving productive weak topology on $\widehat{\mathcal{C}}.$ Furthermore, $\alpha_{\mathcal{F}}$ is a topology on $\widehat{\mathcal{C}}$ if for any $C\in\mathcal{C},$ the arrow $f_{C}$ is idempotent.

Conversely, if $\alpha_{\mathcal{F}}:\Omega\rightarrow\Omega$ is a topology on $\widehat{\mathcal{C}},$ then one has

[TABLE]

for any $C\in\mathcal{C},$ $S_{C}\in\Omega(C)$ and $h\in(\alpha_{\mathcal{F}})_{C}(S_{C}).$

Proof. First of all, one can easily checked that $\alpha_{\mathcal{F}}:\Omega\rightarrow\Omega$ is a productive weak topology on $\widehat{\mathcal{C}}.$ To prove that $\alpha_{\mathcal{F}}:\Omega\rightarrow\Omega$ is an action preserving map, let $C\in\mathcal{C},m\in M(C)$ and $S_{C}\in\Omega(C).$ Then, we achieve

[TABLE]

Now, to prove the second part of theorem, let $f_{C}$ be an idempotent arrow in $\mathcal{C}/C$, for $C\in\mathcal{C}$. Then, for any $S_{C}\in\Omega(C),$ we have

[TABLE]

and so, $\alpha_{\mathcal{F}}$ is idempotent.

Conversely, if $\alpha_{\mathcal{F}}$ is idempotent, the third equality in (47)above shows that the duplication (46) holds. $\quad\quad\square$

It is easily to see that the modal closure operator associated to the weak topology $\alpha_{\mathcal{F}}$, defined as in (45), denoted by

[TABLE]

for any presheaf $F,$ corresponds to any subpresheaf $G$ of $F,$ the subpresheaf $\overline{G}$ of $F,$ given by

[TABLE]

for any $C\in\mathcal{C}.$ Moreover, the weak Grothendieck topology associated to $\alpha_{\mathcal{F}},$ is given by

[TABLE]

for any $C\in\mathcal{C}.$

The following demonstrates what stated in this manuscript, for some small categories.

Example 4.12**.**

Let $(L,\wedge,1)$ be a meet-semilattice with the greatest element $1$ which can be considered as a category with finite limits (see also [16]). Since any arrow in $L$ is monic, we deduce that $\neg\neg=j_{{}_{\rm Sub}}$ on $\widehat{L}.$ Now suppose that $\mathcal{M}$ is a class of arrows of $L$ which are closed under compositions and meets of arrows with common codomain, i.e., pullbacks. Also, it has identities. One can easily checked that the arrow $\widehat{\mu_{{}_{M}}}:\Omega\times M\rightarrow\Omega$ in $\widehat{L},$ is given by

[TABLE]

for any $a\in L,$ arrow $b\leq a$ in $\mathcal{M}$ and $S_{a}\subseteq\downarrow\!a.$ Meanwhile, by Theorem 4.7, the topology $\neg\neg$ on $\widehat{L}$ is $M$-action preserving since any composable pair $u\leq v\leq w$ of arrows in $L$ together with the chain $u\leq u\leq w$ of arrows in $L$ form a pullback in $L.$ For instance, let $L$ be the meet-semilattice $\{x,y,1\}$ in which $x\leq y\leq 1.$ Then, the class

[TABLE]

is an admissible class on $L.$ In this route, the class

[TABLE]

is a family of arrows in $L$ which satisfies in (44). 2. 2)

Let $\mathcal{C}$ be the category Dcpo (($\omega$-)Cpo, ContL) of directed complete posets (directed ($\omega$-)complete posets, continuous lattices) and continuous maps (see, [4]). The class of all Scott-open inclusions on any one of these categories Dcpo, ($\omega$-)Cpo and ContL constitutes an admissible class $\mathcal{M}$ (see, [23, 24]). Since these categories are cartesian closed, by Theorem 4.7, the topology $\neg\neg$ on any one of these is action preserving with respect to the class of all monics of these categories. 3. 3)

The class of all covering families on the categories of loci ${\mathbb{L}}$ (or ${\mathbb{G}}$ or ${\mathbb{F}}$) (up to isomorphism) constitutes an admissible class on ${\mathbb{L}}$ (or ${\mathbb{G}}$ or ${\mathbb{F}}$) since any element of these families is monic (for details, see [28]). One can easily checked that the topology $j_{{}_{M}}:\Omega\rightarrow\Omega$ defined as in (26), on $\widehat{{\mathbb{L}}}$ is given by, for each object $A\in{\mathbb{L}}^{\rm op}$ and any cosieve $S_{A}$ on $A$, the cosieve $(j_{{}_{M}})_{A}(S_{A})$ which consists of all $C^{\infty}$-homomorphisms $g:A\rightarrow B$ for which there exists an element $\eta_{i_{0}}$ of a cover (in dual form) $\{\eta_{i}:B\rightarrow B[s_{i}^{-1}]|~{}i=1,\ldots,n\}$ for which $\eta_{i_{0}}g\in S_{A}.$ Roughly, we deduce that $j_{{}_{\widehat{{\mathbb{L}}}}}\leq j_{{}_{M}}$ on $\widehat{{\mathbb{L}}}$. In asimilar way, one has $j_{{}_{\widehat{{\mathbb{G}}}}}\leq j_{{}_{M}}$ on $\widehat{{\mathbb{G}}}$ and $j_{{}_{\widehat{{\mathbb{F}}}}}\leq j_{{}_{M}}$ on $\widehat{{\mathbb{F}}}$.