typical contexts
A local geometric morphism $f : E \to S$ between toposes $E,S$ is
such that a further right adjoint $f^! : S \to E$ exists
and such that one, hence all, of the following equivalent conditions hold:
When we regard $E$ as a topos over $S$, so that $f$ is regarded as its global section geometric morphism in the category of toposes over $S$, then we say that $E$ is a local $S$-topos. In this case we may label the functors involved as
to indicate that if we think of $\Gamma$ as sending a space to its underlying $S$-object of points by forgetting cohesion, then $Disc$ creates the discrete space/discrete object and $Codisc$ the codiscrete space/codiscrete object on an object in $S$.
This is especially common when $S=$ Set, in which case the final condition is automatic since all functors are $Set$-indexed. Hence in that case we have the following simpler definition.
A sheaf topos $\mathcal{T}$ is a local topos if the global section geometric morphism $\mathcal{T} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Set$ has a further right adjoint $CoDisc$, making an adjoint triple $(LConst \vdash \Gamma \vdash CoDisc)$
(As just stated, it is automatic in the case over $Set$ that this is furthermore a full and faithful functor.)
Another way of stating this is that a Grothendieck topos is local if and only if the terminal object $1$ is connected and projective (since this means precisely that $\Gamma = \hom(1, -)$ preserves colimits, and therefore has a right adjoint by virtue of an adjoint functor theorem). Another term for this: we say $1$ is tiny (atomic).
A geometric morphism $f : E \to S$ is local precisely if
there exists a geometric morphism $c : S \to E$ such that $f \circ c \simeq id$;
for every other geometric morphism $g : G \to S$ the composite $c\circ g$ is an initial object in the hom-category $Topos_{/S}(g,f)$ of the slice 2-category of Topos over $S$.
This is (Johnstone, theorem 3.6.1 vi)).
In particular this means that the category of topos points of a local topos has a contractible nerve.
The global section geometric morphism of any local $\mathcal{S}$-topos (over a base topos $\mathcal{S}$) is a Grothendieck fibration and a Grothendieck opfibration.
This appears in (Shulman).
The Freyd cover of a topos is a local topos, and in fact freely so. Every local topos is a retract of a Freyd cover.
This appears as (Johnstone, lemma C3.6.4).
In a local sheaf topos over Set, every inhabited object is globally inhabited:
every object $X$ for which the unique morphism $X \to *$ to the terminal object is an epimorphism has a global point $* \to X$.
Since in a local topos the global section functor $\Gamma$ is a left adjoint, it preserves epimorphisms. Since it is a right adjoint it preserves the terminal object. Therefore $\Gamma(X) \to \Gamma(*) \simeq *$ is an epimorphism in Set, hence a surjection, meaning that $\Gamma(X)$ is inhabited. Since $\Gamma(X) \simeq Hom(*,X)$ (see global section geometric morphism), the claim follows.
In a topos every epimorphism is an effective epimorphism. Therefore $X \to *$ being an epi means that $X$ is a (-1)-connected object. Therefore the above statement says in terms of (infinity,1)-category theory that a non-trivial local topos has homotopy dimension 0.
The same is true for any local (infinity,1)-topos.
Every local topos $\Gamma : E \to S$ comes with a notion of concrete sheaves, a reflective subcategory $Conc_\Gamma(E) \hookrightarrow E$ which factors the topos inclusion of $S$:
and is a quasitopos. See concrete sheaf for details.
Since a local geometric morphism has a left adjoint in the 2-category Topos, it is necessarily a homotopy equivalence of toposes.
For any local topos $\Gamma \colon \mathcal{E} \to \mathcal{S}$, the base topos $\mathcal{S}$ is equivalent to the category of sheaves for a Lawvere-Tierney topology $j$ on $\mathcal{E}$. A sound and complete elementary axiomatization of local maps of (bounded) toposes can be given in terms of properties of topos $E$ and topology $j$ (AwodeyBirkedal)
We discuss first
and then
themselves.
Let $\mathcal{E}$ be an elementary topos equipped with a Lawvere-Tierney topology $j : \Omega \to \Omega$.
Write $V \mapsto \sharp V$ for the $j$-closure operation on subobjects $V \hookrightarrow X$, the sharp modality
Write
for the reflective subcategory of j-sheaves.
We say that $j$ is an essential topology if for all objects $X$ the closure operation $\sharp : Sub(X) \to Sub(X)$ on posets of subobjects has a left adjoint $\flat \dashv \sharp$:
This appears under the term “principal” in (Awodey-Birkedal, def. 2.1).
We use the notation ”$\flat$” and ”$\sharp$” oppositely to the use on p.14 of Awodey-Birkedal. Our convention is such that it harmonizes with the terminology at cohesive topos and cohesive (infinity,1)-topos, where it makes interpretational sense to pronounce ”$\flat$” as “flat”.
The left adjoints $\flat : Sub(X) \to Sub(X)$ for all $X \in \mathcal{E}$ extend to a functor $\flat : \mathcal{E} \to \mathcal{E}$ on all of $\mathcal{E}$.
(…)
A Lawvere-Tierney topology $j$ is essential, $(\flat \dashv \sharp )$, precisely if for all objects $X$ there exists a least $\sharp$-dense subobject $U_X \hookrightarrow X$.
This appears as (AwodeyBirkedal, lemma 2.3).
By the discussion at category of sheaves we have that $\sharp$ is given by the composite
where the first morphism is sheafification and the second is full and faithful.
If now the left adjoint $\flat$ exists, it follows that this comes from a left adjoint $Disc$ to $\Gamma$ as
Therefore the $(Disc \dashv \Gamma)$-counit provides morphisms
whose image factorization $U_x \hookrightarrow X$ we claim provides the least dense subobjects.
To show that $U_X$ is dense it is sufficient to show that
is an isomorphism.
Composing this morphism with $CoDisc$ of the $(Disc \dashv\Gamma)$-unit on $\Gamma X$ (which is an isomorphism since $Disc$ is a full and faithful functor by the discussion at fully faithful adjoint triples) and using the $(Disc \dashv \Gamma)$ triangle identity we have
Using that also $coDisc$ is full and faithful and then 2-out-of-3 for isomorphisms it follows that $coDisc \Gamma Disc \Gamma X \stackrel{\simeq}{\to} coDisc \Gamma X$ hence
is indeed an isomorphism.
Moreover, by one of the equivalent characterizations of reflective subcategories we have (…)
An object $X \in \mathcal{E}$ is called discrete if for all $\Gamma$-local isomorphisms $f : A \to B$ the induced morphism
is an isomorphism (of sets, hence a bijection).
An object $X \in \mathcal{E}$ is called o-discrete if $\flat X \simeq X$.
Every discrete object is o-discrete.
On $\mathcal{E}$ a elementary topos with Lawvere-Tierney topology $j$ consider the following axioms.
Axiom 1. $j$ is essential.
Axiom 2 a. There is an object $G \in \mathcal{E}$ such that every object is a subquotient of the product of a discrete object with $G$.
Axiom 2 b. With $G$ as above, there is a discrete object $G'$ and an epimorphism $G' \to \flat G$.
Axiom 3. For all discrete objects $D$, if the internal hom $[X,D]$ is o-discrete, then $X$ is also discrete.
Axiom 4. Discrete objects are closed under binary products.
These axioms characterize local geometric morphisms $\mathcal{E} \to Sh_j(\mathcal{E}) \simeq D_j(\mathcal{E})$.
If $G$ is fixed to be the terminal object (in which case Axiom 2 b becomes empty), then they characterize local and localic geometric morphisms.
This is (Awodey-Birkedal, theorem 3.1) together with the discussion around remark 3.7.
If $C$ is a small category with a terminal object $* \in C$, then the presheaf topos $[C^{op},Set]$ is a local topos.
Notice that $Set \simeq [*,Set]$ is the presheaf topos over the point category, the category with a single object and a single morphism. Therefore the constant presheaf functor
can be thought of as sending a set $S \in Set$, hence a functor $S : * \to Set$ to the composite functor
Notice that in the presence of a terminal object in $C$, $Const$ is a full and faithful functor: a natural transformation $Const(S_1) \to Const(S_2)$ has components
where the vertical morphisms are $Const(U \to *)$, the point being that they exist for every $U \in C$ given the presence of the terminal object. It follows that such a natural transformation is given by any and one and the same function $f \colon S_1 \to S_2$.
The functor $Const$ has a left adjoint and a right adjoint, and these are – essentially by definition – the colimit and the limit operations
which send a presheaf/functor $F \colon C^{op} \to Set$ to its colimit $\underset{\rightarrow}{\lim} F \in Set$ or limit $\underset{\leftarrow}{\lim} F \in Set$, respectively.
Since adjoints are essentially unique, it follows that the global section functor $\Gamma \colon [C^{op}, Set]$ is given by taking the limit, $\Gamma \simeq \underset{\leftarrow}{\lim}$.
Observe that the terminal object $* \in C$ is the initial object in the opposite category $C^{op}$. But the limit over a diagram with initial object is given simply by evaluation at that object, and so we have for any $F \in [C^{op}, Set]$ that
hence that the global section functor is simply given by evaluating a presheaf on the terminal object of $C$.
Limits and colimit in a presheaf category $[C^{op}, Set]$ are computed objectwise over $C$ (see at limits and colimits by example). Therefore evaluation at any object in $C$ preserves in limits and colimits, and in particular evaluation at the terminal object does. Therefore $\Gamma$ preserves all colimits. Hence by the adjoint functor theorem it has a further right adjoint $CoDisc$.
We can compute it explicitly by the Yoneda lemma and using the defining Hom-isomorphism of adjoints to be the functor $CoDisc \colon Set \to [C^{op}, Set]$ such that for $S \in Set$ the presheaf $CoDisc(S)$ is given over $U \in C$ by
So in conclusion we have an adjoint triple $(Const \vdash \Gamma \vdash CoDisc)$ where $Const$ is a full and faithful functor. By the discussion at fully faithful adjoint triples it follows then that also $CoDisc$ is full and faithful.
The converse to prop. 1 is true if $C$ is Cauchy complete.
If $X$ is a topological space, or more generally a locale, then $Sh(X)$ is local (over Set) iff $X$ has a focal point $x$, i.e. a point whose only neighborhood is the whole space. In this case, the extra right adjoint $f^! : Set \to Sh(X)$ to the global sections functor $f_* : Sh(X) \to Set$ is given by the functor which computes the stalk at $x$. This can also be given without reference to $x$, by the formula
for sets $M$ and open subsets $U \subseteq X$.
For $C$ a local site, the category of sheaves $Sh(C)$ is a local topos over $Set$.
For instance CartSp is a local site. Objects in $Sh(C)$ are generalized smooth spaces such as diffeological spaces. The further right adjoint
is the functor that sends a set to the diffeological space on that set with codiscrete smooth structure (every map of sets is smooth).
Let $A$ be a partial combinatory algebra and let $A\sharp\subseteq A$ be a sub partial combinatory algebra of $A$. Then there is a (localic) local geometric morphism from the relative realizability topos? $\mathrm{RT}(A,A\sharp)$ to the standard realizability topos $\mathrm{RT}(A\sharp)$.
Let $LocTopos$ denote the 2-category of local Grothendieck toposes (over Set) with all geometric morphisms between them. Let $PTopos$ denote the 2-category whose objects are pointed toposes? (i.e. (Grothendieck) toposes $E$ equipped with a geometric morphism $s\colon Set\to E$), and whose morphisms are pairs $(f,\alpha)$ such that $f\colon E\to E'$ is a geometric morphism and $\alpha\colon s'\to f s$ is a (not necessarily invertible) geometric transformation.
Note that if $E$ is a local topos with global sections geometric morphism $e^*\dashv e_*$, then the adjunction $e_*\dashv e^!$ is also a geometric morphism $Set\to E$. In this way we have a functor $LocTopos \to PTopos$, which is a full embedding, and turns out to have a right adjoint: this right adjoint is called the localization of a pointed topos at its specified point. For example:
If $C$ is a small category and $U$ is an object of $C$, then the localization of the presheaf topos $[C^{op},Set]$ at the point induced by $U\colon 1\to C$ can be identified with the presheaf topos $[(C/U)^{op},Set]$ over the over category of $C$ over $U$. By the general properties of over toposes, this is equivalently the over-topos $PSh(C)/U$ (where $U$ is regarded in $PSh(C)$ by the Yoneda embedding).
If $X=Spec(A)$ is the Zariski spectrum of a commutative ring $A$, and $P\subset A$ is a prime ideal of $A$ (i.e. a point of $X$), then the localization of $Sh(X)$ at $P\colon 1\to X$ can be identified with $Sh(Spec(A_P))$, where $A_P$ denotes the localization of $A$ at $P$. Of course, this is the origin of the terminology.
A similar construction is possible for bounded toposes over any base (not just Set).
For $\mathcal{E}$ a Grothendieck topos and $X \in \mathcal{E}$ an object, the over topos $\mathcal{E}/X$ is local if $X$ is a tiny object (atomic object).
We check that the global section geometric morphism $\Gamma : \mathcal{E}/X \to Set$ preserves colimits. It is given by the hom-functor out of the terminal object of $\mathcal{E}/X$, which is $(X \stackrel{Id}{\to} X)$:
The hom-sets in the over category are fibers of the hom-sets in $\mathcal{E}$: we have a pullback diagram
Moreover, overserve that colimits in the over category are computed in $\mathcal{E}$.
If $X$ is a tiny object then by definition we have
Inserting all this into the above pullback gives the pullback
By universal colimits in the topos Set, this pullback of a colimit is the colimit of the separate pullbacks, so that
So $\Gamma$ does commute with colimits if $X$ is tiny. By the adjoint functor theorem then the right adjoint $\nabla : Set \to \mathcal{E}/X$ does exist and so $\mathcal{E}/X$ is a local topos.
As a special case this reproduces the above statement that slices $PSh(C)/j(U)$ of presheaf toposes over objects in the image of the Yoneda embedding are local: every representable functor is tiny (see there).
Let $A$ be a commutative ring (such as $\mathbb{Z}$ or a field), let $Spec A$ be the prime spectrum of $A$, and let $\mathcal{Z}_A$ be the big Zariski topos for $A$ (i.e. the classifying topos for local $A$-algebras). For each element $a$ of $A$, we have an open subset $D (a) = \{ \mathfrak{p} \in Spec A : a \notin \mathfrak{p} \}$, and these open subsets constitute a basis for the topology on $Spec A$. The full subcategory of the frame of open subsets of $Spec A$ spanned by these basic open subsets admits a contravariant full embedding in the category of finitely-presented $A$-algebras via the functor $D (a) \mapsto A [a^{-1}]$ (the well-definedness of this functor requires a non-trivial check!), and this functor moreover has the cover lifting property, so induces a local geometric morphism $\mathcal{Z}_A \to Sh (Spec A)$.
local topos / local (∞,1)-topos
and
Standard references include
and Chapter C3.6 of
A completely internal characterization of local toposes is discussed in
This is based on part 2 of
Free local constructions are considered in