The notion of geometric theory has many different incarnations. A few are:
A geometric theory is a first order theory whose models are preserved and reflected by geometric morphisms.
A geometric theory is a first order theory whose axioms can be written solely in terms of the connectives $\top$ (truth), $\wedge$ (finite conjunction), $\bot$ (falsity), $\bigvee$ (possibly infinitary disjunction), and $\exists$ (existential quantification).
A geometric theory is a syntactic description of a Grothendieck topos.
The equivalence of these statements involves some serious proofs, including Giraud's theorem characterizing Grothendieck topoi.
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:
Since $\wedge$ and $\top$ are represented by intersections and identities (top elements), these can be interpreted in any lex category. Theories involving only these are cartesian theories.
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 theories.
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 theories.
Finally, theories which also involve infinitary $\bigvee$, which is again represented by an infinitary union, can be interpreted in infinitary coherent categories, aka geometric categories. These are geometric theories.
Note that the axioms of one of these theories are actually of the form
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
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 positive 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 is automatically a Heyting category, but 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 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 $G'$.
Any finitary algebraic theory is, in particular, a cartesian theory, and hence geometric. This includes monoids, groups, abelian groups, rings, commutative rings, etc.
The theory of (strict) categories is not finitary-algebraic, but it is cartesian, and hence geometric; this generalises to (finitary) essentially algebraic theories.
The theory of torsion-free abelian groups is also cartesian, being obtained from the theory of abelian groups by the addition of the sequents $(n\cdot x = 0) \vdash (x = 0)$ for all $n\in \mathbb{N}$.
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
saying that if $x+y$ is invertible, then either $x$ or $y$ is so.
The theory of fields is also coherent, being obtained from the theory of commutative rings by adding $(0 = 1) \vdash \bot$ and also
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”) are not coherent.
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
asserting that for each $x$, either $x=0$ or $x+x=0$ or $x+x+x=0$ or …. Similarly, the theory of fields of finite characteristic is geometric but not coherent.
The theory of a real number is geometric. This is a 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:
The classifying topos of this theory is the topos of sheaves on the real numbers.
A geometric theory whose classifying topos is a presheaf topos is called a theory of presheaf type.
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 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 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 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 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 equivalent.
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 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 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 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 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 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 equifiers.)
From this point of view, a 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.
The following definition is sort of a “halfway house” between logic and geometry. Start with a first-order 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 geometric theory over $\Sigma$ is defined to consist of the following.
The equivalence of this definition with the others can be found in
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 $(E_i \to \coprod_i E_i)_{i\in I}$.
By framing this notion in the internal language of a topos $S$ we can talk of geometric theories over $S$, with models in bounded $S$-toposes (the 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
Since a Grothendieck topos is the “extensional essence” of a geometric theory, it makes sense to define a 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, forget along $h$, from $T'$-models to $T$-models in any $S$-topos.
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 $E$, 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 $S$-toposes.
Standard textbook accounts include
Peter Johnstone, section D1.1 of Sketches of an Elephant
Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic
Discussion in the context of computer science is in
Discussion with an eye towards geometric type theory is in