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\begin{document}

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\section*{geometric theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{logical_definition}{Logical definition}\dotfill \pageref*{logical_definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{other_characterizations}{Other characterizations}\dotfill \pageref*{other_characterizations} \linebreak
\noindent\hyperlink{InTermsOfSheafTopoi}{In terms of sheaf topoi}\dotfill \pageref*{InTermsOfSheafTopoi} \linebreak
\noindent\hyperlink{FunctorialDefinition}{Functorial definition}\dotfill \pageref*{FunctorialDefinition} \linebreak
\noindent\hyperlink{hybrid_definition}{Hybrid definition}\dotfill \pageref*{hybrid_definition} \linebreak
\noindent\hyperlink{localization}{Localization}\dotfill \pageref*{localization} \linebreak
\noindent\hyperlink{morphisms_of_theories}{Morphisms of theories}\dotfill \pageref*{morphisms_of_theories} \linebreak
\noindent\hyperlink{gros_categories_of_models}{Gros categories of models}\dotfill \pageref*{gros_categories_of_models} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{geometric theory} has many different incarnations. A few are:

\begin{itemize}%
\item A geometric theory is a (possibly infinitary) [[first-order logic|first order]] [[theory]] whose [[model]]s are preserved and reflected by [[geometric morphisms]].


\item A geometric theory is a (possibly infinitary) [[first-order logic|first order]] [[theory]] whose axioms can be written as [[sequent|sequents]] in context of formulae constructed from the [[logical connective|connectives]] $\top$ ([[truth]]), $\wedge$ (finite [[logical conjunction|conjunction]]), $\bot$ ([[falsity]]), $\bigvee$ (possibly infinitary [[disjunction]]), and $\exists$ ([[existential quantification]]), as well as $=$ ([[equality]]).


\item A geometric theory is a [[syntax|syntactic]] description of a [[Grothendieck topos]].



\end{itemize}
The equivalence of these statements involves some serious proofs, including [[Giraud's theorem]] characterizing Grothendieck topoi.

\hypertarget{logical_definition}{}\subsection*{{Logical definition}}\label{logical_definition}

In logical terms, a geometric theory fits into a hierarchy of theories that can be interpreted in the [[internal logic]] of a hierarchy of types of [[categories]]. Since [[predicates]] in the internal logic are represented by [[subobjects]], in order to interpret any connective or quantifier in the internal logic, one needs a corresponding operation on subobjects to exist in the category in question, and be well-behaved. For instance:

\begin{itemize}%
\item Since $\wedge$ and $\top$ are represented by [[intersections]] and identities (top elements), and $=$ is represented by equalizer, these can be interpreted in any [[lex category]]. Theories involving only these are examples of [[cartesian theory|cartesian theories]].


\item Since $\exists$ is represented by the [[image]] of a subobject, it can be interpreted in any [[regular category]]. Theories involving only $\wedge$, $\top$, and $\exists$ are [[regular theory|regular theories]].


\item Since $\vee$ and $\bot$ are represented by [[union]] and [[bottom elements]], these can be interpreted in any [[coherent category]]. Theories which add these to regular logic are called [[coherent theory|coherent theories]].


\item Finally, theories which also involve infinitary $\bigvee$, which is again represented by an infinitary union, can be interpreted in [[infinitary coherent categories]], aka \emph{[[geometric categories]]}. These are \textbf{geometric theories}.



\end{itemize}
Note that the axioms of one of these theories are actually of the form

\begin{displaymath}
\varphi \vdash_{\vec{x}} \psi
\end{displaymath}
where $\varphi$ and $\psi$ are formulas involving only the specified connectives and quantifiers, $\vdash$ means entailment, and $\vec{x}$ is a [[context]]. Such an axiom can also be written as

\begin{displaymath}
\forall \vec{x}. (\varphi \Rightarrow \psi)
\end{displaymath}
so that although $\Rightarrow$ and $\forall$ are not strictly part of any of the above logics, they can be applied ``once at top level.'' In an axiom of this form for geometric logic, the formulas $\varphi$ and $\psi$ which must be built out of $\top$, $\wedge$, $\bot$, $\bigvee$, and $\exists$ are sometimes called \emph{positive} or \emph{geometric} formulas.

The interpretation of arbitrary uses of $\Rightarrow$ and $\forall$ requires a [[Heyting category]]. In fact, by the [[adjoint functor theorem]] for [[posets]], any geometric category which is [[well-powered category|well-powered]] is automatically a Heyting category, but \emph{geometric functors} are not necessarily Heyting functors. Likewise, a well-powered geometric category automatically has arbitrary intersections of subobjects as well, so we can interpret infinitary $\bigwedge$ in its internal logic, but again these are not preserved by geometric functors.

By the usual syntactic constructions (see [[internal logic]] and [[context]]), any geometric theory $T$ generates a ``free geometric category containing a model of that theory,'' also known as its \emph{[[syntactic category]]} $G_T$. This syntactic category $G_T$ has the universal property that for any other geometric category $G'$, geometric functors $G_T \to G'$ are equivalent to $T$-models in $G'$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Any finitary [[algebraic theory]] is, in particular, a cartesian theory, and hence geometric. This includes [[monoids]], [[groups]], [[abelian groups]], [[rings]], [[commutative rings]], etc.


\item The theory of ([[strict category|strict]]) [[categories]] is not finitary-algebraic, but it is cartesian, and hence geometric; this generalises to (finitary) [[essentially algebraic theory|essentially algebraic theories]].


\item The theory of [[torsion]]-free abelian groups is also cartesian, being obtained from the theory of abelian groups by the addition of the sequents $(n\cdot x = 0) \vdash_{x} (x = 0)$ for all $n\in \mathbb{N}$.


\item The theory of [[local rings]] is a coherent theory, being obtained from the theory of commutative rings by adding the sequent $(0 = 1) \vdash \bot$, for nontriviality, and

\begin{displaymath}
\exists z. ((x+y)z = 1) \vdash_{x y} (\exists z.(x z = 1) \vee \exists z.(y z = 1))
\end{displaymath}
saying that if $x+y$ is invertible, then either $x$ or $y$ is so.


\item The theory of [[fields]] is also coherent, being obtained from the theory of commutative rings by adding $(0 = 1) \vdash \bot$ and also

\begin{displaymath}
\top \vdash_{x} (x=0) \vee (\exists y.(x y = 1))
\end{displaymath}
asserting that every element is either zero or invertible. In the [[constructive logic]] that holds internal to the categories in question, this is the notion of a ``discrete field;'' other classically equivalent axiomatizations (called ``Heyting fields'' or ``residue fields'' -- see [[field]]) are not coherent.


\item The theory of [[torsion]] abelian groups is geometric but not coherent; it can be obtained from the theory of abelian groups by adding the sequent

\begin{displaymath}
\top \vdash_{x} \bigvee_{n\ge 1} (n \cdot x = 0)
\end{displaymath}
asserting that for each $x$, either $x=0$ or $x+x=0$ or $x+x+x=0$ or \ldots{}. Similarly, the theory of fields of finite [[characteristic]] is geometric but not coherent.


\item The theory of a [[real number]] is geometric. This is a [[propositional logic|propositional]] theory, having no sorts, and having one relation symbol ``$p\lt x\lt q$'' for each pair of rational numbers $p\lt q$. Its axioms are:

\begin{itemize}%
\item $(p_1\lt x\lt q_1) \wedge (p_2\lt x\lt q_2) \vdash (\max(p_1,p_2) \lt x \lt \min(q_1,q_2))$ (if $\max(p_1,p_2)\lt \min(q_1,q_2)$), and $(p_1\lt x\lt q_1) \wedge (p_2\lt x\lt q_2) \vdash \bot$ (otherwise).
\item $(p\lt x\lt q) \vdash (p\lt x\lt q') \vee (p' \lt x\lt q)$ (if $p\lt p' \lt q' \lt q$).
\item $(p\lt x\lt q) \vdash \bigvee_{p\lt p'\lt q'\lt q} (p' \lt x\lt q')$.
\item $\top \vdash \bigvee_{p\lt q} (p\lt x\lt q)$.

\end{itemize}
The [[classifying topos]] of this theory is the topos of [[sheaves]] on the [[real numbers]].


\item The theory of a set isomorphic to $\mathbb{N}$ is geometric, and makes essential use of the infinitary disjunctions. See [[geometric type theory]].


\item The theory of flat diagrams over a small category $\mathcal{C}$ is geometric. For each object $i$ of $\mathcal{C}$ there is a sort $\sigma_i$, and for each morphism $u\colon i \to j$ a function symbol $\alpha_u\colon \sigma_i \to \sigma_j$. The axioms are -

\begin{itemize}%
\item $\top \vdash_{x\colon\sigma_i} \alpha_{Id_i}(x) = x$


\item $\top \vdash_{x\colon\sigma_i} \alpha_{v}(\alpha_u(x)) = \alpha_{v u}(x)$ ($u\colon i \to j$ and $v\colon j \to k$)


\item $\top \vdash \bigvee_{i \in \mathcal{C}_0} (\exists x\colon \sigma_i) \top$


\item $\top \vdash_{x\colon\sigma_i, y\colon\sigma_j} \bigvee_{u\colon k\to i, v\colon k\to j} (\exists z\colon\sigma_k) (x = \alpha_u(z) \wedge y = \alpha_v(z))$


\item $\alpha_u(x) = \alpha_v(x) \vdash_{x\colon\sigma_i} \bigvee_{w\colon k\to i, u w = v w} (\exists z\colon \sigma_k) x = \alpha_w(z)$ ($u,v\colon i \to j$)



\end{itemize}


\end{itemize}
This theory is classified by the topos of presheaves over $\mathcal{C}$.

It is commonly more useful to consider the theory of flat \emph{presheaves} over $\mathcal{C}$, in other words the flat diagrams over $\mathcal{C}^{op}$. ([[Elephant]] calls these \emph{torsors} over $\mathcal{C}$, generalizing the established terminology for groups.) This is because the representable presheaves are flat, and so Yoneda's lemma transforms objects of $\mathcal{C}$ covariantly into models of the theory. In fact, the models of the theory are the filtered colimits of representables. For example, a finitary algebraic theory is classified by the topos of covariant functors from the category of finitely presented algebras to $Set$.

\begin{itemize}%
\item A geometric theory whose [[classifying topos]] is a [[presheaf topos]] is called a \emph{[[theory of presheaf type]]}.

\end{itemize}
\hypertarget{other_characterizations}{}\subsection*{{Other characterizations}}\label{other_characterizations}

\hypertarget{InTermsOfSheafTopoi}{}\subsubsection*{{In terms of sheaf topoi}}\label{InTermsOfSheafTopoi}

Categories of each ``logical'' type can also be ``completed'' with respect to a suitable ``exactness'' property, without changing their internal logic. Any regular category has an completion into an [[exact category]] (see [[regular and exact completion]]), any coherent category has a completion into a [[pretopos]], and any geometric category has a completion into an [[infinitary pretopos]].

However, [[Giraud's theorem]] says that an infinitary pretopos having a [[generating set]] is precisely a [[Grothendieck topos]]. Moreover, a functor between Grothendieck topoi is geometric (preserves all the structure of a geometric category) iff it preserves finite [[limits]] and small [[colimits]]. By the [[adjoint functor theorem]], this implies that it is the [[inverse image]] part of a [[geometric morphism]], i.e. an [[adjunction]] $f^* \dashv f_*$ in which $f^*$ (the ``inverse image'') preserves finite limits.

Thus:

Grothendieck topoi and inverse-image functors are in some sense the ``natural home'' for models of geometric theories.

Note, though, that geometric morphisms are generally considered as pointing in the direction of the \emph{direct} image $f_*$, which is the opposite direction to the geometric functor $f^*$. (This is because when $E$ and $F$ are the toposes of [[sheaves]] on [[sober space|sober topological spaces]] (or [[locales]]) $X$ and $Y$ respectively, then [[continuous maps]] $X \to Y$ correspond precisely to geometric morphisms $E \to F$.) Also, of course any Grothendieck topos is an [[elementary topos]] (at least, as long as one works in [[foundations]] for which [[Set]] is an elementary topos), and hence its internal logic also includes ``higher-order'' constructions such as function-objects and power-objects. However, these are not preserved by geometric functors, so they (like $\forall$ and $\Rightarrow$) are not part of geometric logic. (They are, however, preserved by [[logical functors]], a different sort of morphism between topoi.)

In particular, we can apply the ``exact completion'' operation to the syntactic category $G_T$ of a geometric theory to obtain an infinitary pretopos $Set[T]$. As long as the theory $T$ was itself small, $Set[T]$ will have a generating set, and therefore will be a Grothendieck topos. The universal property of the syntactic category, combined with that of the exact completion, implies that for any Grothendieck topos $E$, geometric morphisms $E\to Set[T]$ are equivalent to $T$-models in $E$. This topos $Set[T]$ is called the \textbf{[[classifying topos]]} of $T$.

In the other direction, if $C$ is any small [[site]], we can write down a ``geometric theory of cover-preserving [[flat functors]] $C^{op}\to Set$.'' By [[Diaconescu's theorem]] classifying geometric morphisms into sheaf topoi, it follows that $Sh(C)$ is a [[classifying topos]] for this theory. Therefore, not only does every geometric theory have a [[classifying topos]], every Grothendieck topos is the [[classifying topos]] of \emph{some} theory. Very different-looking theories can have equivalent classifying topoi; this of course implies that they have all the same models in all Grothendieck topoi (hence a Grothendieck topos is the ``extensional essence'' of a geometric theory). We say that two geometric theories with equivalent classifying topoi are [[Morita equivalence|Morita equivalent]].

\hypertarget{FunctorialDefinition}{}\subsubsection*{{Functorial definition}}\label{FunctorialDefinition}

We can approach the same idea by starting instead from the notions of [[Grothendieck topos]] and [[geometric morphism]]. The following approach is described in B4.2 of the [[Elephant]].

Suppose we consider what sorts of ``theories'' we can define in terms of Grothendieck topoi, that are preserved by inverse image functors. Any such theory should certainly define a [[2-functor]] $T\colon GrTop^{op}\to Cat$, where $GrTop$ is the 2-category of Grothendieck topoi and geometric morphisms, so for the moment let's call any such $T$ a ``theory''. The image $T(E)$ of a functor $E$ is supposed to be ``the category of $T$-models,'' and a \emph{classifying topos} for such a 2-functor will be just a [[representing object]] for it.

Of course, this notion of theory is far too general; we only want to consider theories that are constructed in some reasonable way. One theory that should certainly be geometric is the [[theory of objects]], $O$. This 2-functor sends a topos $E$ to itself, considered as a mere category, and an inverse image functor to itself, considered as a mere functor. The theory $O$ can be shown to have a classifying topos, the [[classifying topos for the theory of objects|object classifier]] $Set[O]$. Similarly, we have a theory $O_n$ of $n$-tuples of objects that should be geometric.

How can we construct more theories that ought to be geometric? We should start from some finite collection of objects (i.e. a model of $O_n$), ``construct some new objects and morphisms,'' and then ``impose some axioms on them.'' For any theory $T$, let's call a transformation $T\to O$ a \emph{geometric construct}. This is supposed to be ``an object constructed out of the axioms of $T$ in a natural way.'' More precisely, to every $T$-model in a topos $E$ it assigns an object of $E$, in a way that varies naturally with morphisms of $T$-models and inverse image functors.

Now define a \emph{simple functional extension} of $T$ to be the [[inserter]] of a pair of geometric constructs $T\;\rightrightarrows\; O$. A model of such a theory will consist of a model of $T$, together with an additional morphism between two objects constructed out of the given $T$-model. By iteratively applying such constructions, we can add in any number of new morphisms between ``constructed objects.''

Finally, define a \emph{simple geometric quotient} of $T$ to be the [[inverter]] of a [[modification]] between a pair of geometric constructs $T\;\rightrightarrows\; O$. That is, we require that a certain naturally defined morphism between objects constructed out of $T$-models must be an isomorphism. (Applying equalizers, we see that this also includes the ability to set morphisms equal, i.e. to construct [[equifier]]s.)

From this point of view, a \textbf{geometric theory} is a theory $GrTop^{op}\to Cat$ obtained from some $O_n$ by a finite sequence of simple functional extensions and simple geometric quotients. Of course, once we know that each $O_n$ has a classifying topos, it follows immediately that any geometric theory has a classifying topos, since $GrTop$ has inserters and inverters.

\hypertarget{hybrid_definition}{}\subsubsection*{{Hybrid definition}}\label{hybrid_definition}

The following definition is sort of a ``halfway house'' between logic and geometry. Start with a first-order [[signature (in logic)|signature]] $\Sigma$ (this is the logical part). Then we have a 2-functor $\Sigma Str\colon GrTop^{op}\to Cat$ sending a topos $E$ to the category $\Sigma Str(E)$ of $\Sigma$-structures in $E$. A \textbf{geometric theory} over $\Sigma$ is defined to consist of the following.

\begin{itemize}%
\item For each Grothendieck topos $E$, we have a [[replete subcategory|replete]] [[full subcategory]] $T(E)$ of $\Sigma Str(E)$, such that
\item For each geometric morphism $f\colon E\to F$, if $M\in T(F)$ then $f^*M\in T(E)$ (i.e. $T$ is a subfunctor of $\Sigma Str$), and moreover
\item For any set-indexed jointly surjective family $(f_i \colon E_i \to E)_{i\in I}$ of geometric morphisms, and any $M \in \Sigma Str(E)$, if $f_i^* M \in T(E_i)$ for all $i$, then $M\in T(E)$.

\end{itemize}
The equivalence of this definition with the others can be found in

\begin{itemize}%
\item Olivia Caramello, ``A characterization theorem for geometric logic'', \href{http://arxiv.org/abs/0912.1404}{arXiv:0912.1404}.

\end{itemize}
It is not sufficient, in the third condition, to restrict to the case when $I$ is a singleton, but it is sufficient to consider the case when $I$ is a singleton together with all families of coproduct injections $(E_i \to \coprod_i E_i)_{i\in I}$.

\hypertarget{localization}{}\subsection*{{Localization}}\label{localization}

By framing this notion in the [[internal language]] of a [[topos]] $S$ we can talk of geometric theories over $S$, with models in [[bounded topos|bounded]] $S$-toposes (the [[relative point of view|relative version]] of ``Grothendieck topos''). As a simple example, if we have a sheaf $A$ of [[rings]] on a [[topological space]] $X$ we can describe left $A$-[[modules]] as models of a geometric theory over $Sh(X)$, the topos of sheaves on $X$, and this notion is definable in $Sh(X)$-toposes.

Similarly to the $Set$-based case, given a geometric theory $T$ over a topos $S$, we can form the $S$-topos $S[T] \to S$ that classifies $T$, for which the category of $T$-models in a bounded $S$-topos $E$ is naturally equivalent to the category of morphisms of $S$-toposes $E \to S[T]$. In other words

\begin{displaymath}
T Mod(E) \simeq Top_S(E,S[T])
\end{displaymath}
\hypertarget{morphisms_of_theories}{}\subsection*{{Morphisms of theories}}\label{morphisms_of_theories}

Since the classifying topos encodes the ``extensional essence'' of a geometric theory, it makes sense to define a \textbf{map of theories} $T \to T'$ to be a morphism of $S$-toposes $h: S[T'] \to S[T]$. Equivalently, of course, this is a $T$-model in $S[T']$. Composition with $h$ induces a functor, \emph{[[forgetful functor|forget]] along} $h$, from $T'$-models to $T$-models in any $S$-topos.

\hypertarget{gros_categories_of_models}{}\subsection*{{Gros categories of models}}\label{gros_categories_of_models}

Define the [[gros category]] $T Mod$ of $T$-models to have as objects pairs $(E,A)$ where $E$ is an $S$-topos and $A$ is a $T$-model in $E$. A map $(f,g): (E,A) \to (F,B)$ is given by a morphism $f: F \to E$ of $S$-toposes and a homomorphism $g: f^*(A) \to B$ of $T$-models in $F$. The composition of maps should be evident. A map $h: T \to T'$ of geometric theories over $S$ induces a forgetful functor $T' Mod \to T Mod$ which leaves unchanged the $S$-topos of residence, which has a left adjoint $T Mod \to T' Mod$ which may change the topos. For if $a: E \to S[T]$ is a $T$-model in $E$, [[base change|pulling]] $a$ back along $h$ yields a $T'$-model, not in $E$ but in the [[pullback]]. This is a consequence of general facts about finite [[2-limits]] of the [[2-category]] of [[bounded topos|bounded]] $S$-toposes.

The 2-categorical version of $T Mod$ is useful in generalization of the [[spectrum]] construction: See at [[Cole's theory of spectrum]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item [[Barr's theorem]] says, that if a statement in [[geometric logic]] is deducible from a [[geometric theory]] using classical [[logic]] and the [[axiom of choice]], then it is also deducible from it in [[constructive mathematics]].

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[geometric type theory]]


\item [[geometric homotopy type theory]]


\item [[Cole's theory of spectrum]]


\item [[classifying topos for the theory of objects]]


\item [[theory of objects]]


\item [[theory of decidable objects]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Standard references are

\begin{itemize}%
\item [[Peter Johnstone]], sections B4.2, D1.1 of \emph{[[Sketches of an Elephant]]}


\item [[Michael Makkai]], [[Gonzalo E. Reyes]], \emph{First Order Categorical Logic} , LNM 611, Springer Berlin 1977.



\end{itemize}
A textbook account of (finitary) geometric logic can be found in

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
A systematic introduction to topos theory and geometric logic can be found in the following draft by O. Caramello:

\begin{itemize}%
\item [[Olivia Caramello]], \emph{Topos-theoretic background} , ms. 2014. (\href{http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf}{pdf})

\end{itemize}
For additional background on (finitary) geometric formulas consider:

\begin{itemize}%
\item Karel Stokkermans, [[Bill Lawvere]], and [[Steve Vickers]]: \emph{Catlist discussion March 1999} . (\href{http://comments.gmane.org/gmane.science.mathematics.categories/1058}{link})


\item H.J. Keisler: \emph{Theory of models with generalized atomic formulas} , JSL 25 (1960), pp.1-26.



\end{itemize}
Discussion in the context of [[computer science]] is in

\begin{itemize}%
\item [[Steve Vickers]], \emph{Geometric logic in computer science} (\href{http://www.cs.bham.ac.uk/~sjv/GLiCS.pdf}{pdf})

\end{itemize}
Discussion with an eye towards [[geometric type theory]] is in

\begin{itemize}%
\item [[Steve Vickers]], \emph{Locales and toposes as spaces} (\href{http://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdf}{pdf})

\end{itemize}
[[Stone duality]] for geometric theories is discussed in:

\begin{itemize}%
\item [[Henrik Forssell]], \emph{Topological representation of geometric theories}, \href{http://arxiv.org/abs/1109.0699}{arxiv/1109.0699}

\end{itemize}
A nice short exposition together with an unorthodox proposal to expand geometric logic with fix point operators can be found here:

\begin{itemize}%
\item [[Andreas Blass]], \emph{Topoi and Computation}, Bull. European Assoc.Theoret. Comp.Sci. \textbf{36} (1988) pp.57-65. (\href{http://www.math.lsa.umich.edu/~ablass/eatcs.pdf}{draft})

\end{itemize}
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