topos theory

# Connected topos

## Idea

If we view a (Grothendieck) topos as a generalized topological space, then a connected topos is a generalization of a connected topological space.

More generally, a connected geometric morphism $p:E\to F$ is a “relativized” notion of this, saying that $E$ is “connected as a topos over $F$.”

## Definition

### Connected geometric morphism

A geometric morphism $p:E\to F$ is connected if its inverse image part ${p}^{*}$ is full and faithful.

A Grothendieck topos $E$ is connected if the unique geometric morphism $E\to \mathrm{Set}=\mathrm{Sh}\left(*\right)$ is connected. If $E$ is the topos of sheaves on a topological space $X$ (or more generally a locale), then this is equivalent to the usual definition of connectedness for $X$ (see C1.5.7 in the Elephant).

Equivalently, a topos is connected if its global section geometric morphism exhibits discrete objects.

### Connected locally connected morphisms

For geometric morphisms which are also locally connected, connectedness can be phrased in an especially nice form.

###### Proposition

If $p:E\to F$ is locally connected, then it is connected if and only if the left adjoint ${p}_{!}$ of the inverse image functor (which exists, since $p$ is locally connected) preserves the terminal object.

###### Proof

On the one hand, if ${p}^{*}$ is fully faithful, then the counit ${p}_{!}{p}^{*}\to Id$ is an isomorphism, so we have ${p}_{!}\left(*\right)\cong {p}_{!}\left({p}^{*}\left(*\right)\right)\cong *$; hence ${p}_{!}$ preserves the terminal object.

On the other hand, suppose that ${p}_{!}$ preserves the terminal object. Suppose also for simplicity that $F=\mathrm{Set}$. Then any set $A$ is the coproduct ${\coprod }_{A}*$ of $A$ copies of the terminal object. But ${p}^{*}$ and ${p}_{!}$ both preserve coproducts (since they are left adjoints) and terminal objects (since ${p}^{*}$ is left exact, and by assumption for ${p}_{!}$), so we have

${p}_{!}\left({p}^{*}\left(A\right)\right)\cong {p}_{!}\left({p}^{*}\left(\coprod _{A}*\right)\right)\cong \coprod _{A}{p}_{!}\left({p}^{*}\left(*\right)\right)\cong \coprod _{A}*\cong A$p_!(p^*(A)) \cong p_!(p^*(\coprod_A *)) \cong \coprod_A p_!(p^*(*)) \cong \coprod_A * \cong A

Thus, the counit ${p}_{!}{p}^{*}\to Id$ is an isomorphism, so ${p}^{*}$ is fully faithful.

When $F$ is not $\mathrm{Set}$, we just have to replace ordinary coproducts with ”$F$-indexed coproducts,” regarding $E$ and $F$ as $F$-indexed categories.

(This is C3.3.3 in the Elephant.)

Strengthenings of this condition include

### Connected locally connected sites

###### Proposition

If $C$ is a locally connected site with a terminal object, then the topos of sheaves $\mathrm{Sh}\left(C\right)$ on $C$ is (not just locally connected) but connected.

###### Proof

As explained at locally connected site, when $C$ is locally connected, the left adjoint ${\Pi }_{0}:\mathrm{Sh}\left(C\right)\to \mathrm{Set}$ is simply obtained by taking colimits over ${C}^{\mathrm{op}}$. Now by the co-Yoneda lemma, the colimit over any representable presheaf is a singleton (i.e. a terminal object in Set):

$\underset{\to }{\mathrm{lim}}y\left(V\right)={\int }^{U\in C}C\left(U,V\right)={\int }^{U\in C}C\left(U,V\right)\cdot *=*\phantom{\rule{thinmathspace}{0ex}}.$\lim_\to y(V) = \int^{U \in C} C(U,V) = \int^{U \in C} C(U,V) \cdot * = * \,.

But if $C$ has a terminal object, then that terminal object represents the terminal presheaf, which is also the terminal presheaf. Hence under these conditions, ${\Pi }_{0}$ preserves the terminal object, so $\mathrm{Sh}\left(C\right)$ is connected.

## Properties

### Orthogonality

###### Proposition

Connected geometric morphisms are left orthogonal to etale geometric morphisms in the 2-category Topos.

###### Proof

Since the functor ${\mathrm{Topos}}^{\mathrm{op}}\to \mathrm{Cat}$ sending a topos to itself and a geometric morphism to its inverse image functor is 2-fully-faithful (an equivalence on hom-categories), connected morphisms are representably co-fully-faithful in $\mathrm{Topos}$.

Therefore, for 2-categorical orthogonality it suffices to show that in any commutative (up to iso) square

$\begin{array}{ccc}A& \stackrel{f}{\to }& B\\ {}^{p}↓& & {↓}^{q}\\ C& \stackrel{g}{\to }& D\end{array}$\array{ A & \xrightarrow{f} & B \\ {}^{\mathllap{p}}\downarrow & & \downarrow^{\mathrlap{q}} \\ C & \xrightarrow{g} & D}

of geometric morphisms in which $p$ is connected and $q$ is etale, there exists a filler $h:C\to B$ such that $hp\cong f$ and $qh\cong g$.

However, if $X\in D$ is such that $B\cong D/X$ (such exists by definition of $q$ being etale), then for any topos $E$ equipped with a geometric morphism $k:E\to D$, lifts of $k$ along $q$ are equivalent to morphisms $*\to {k}^{*}\left(X\right)$ in $C$. In particular, $f$ is determined by a map $*\to {f}^{*}\left({q}^{*}\left(X\right)\right)\cong {p}^{*}\left({g}^{*}\left(X\right)\right)$, and since $*\cong {p}^{*}\left({p}_{*}\left(*\right)\right)$ and ${p}^{*}$ is fully faithful, this map comes from a map $*\to {g}^{*}\left(X\right)$ in $C$, which in turn determines a geometric morphism $h:C\to B$ which is the desired filler.

###### Proposition

Any locally connected geometric morphism factors as a connected and locally connected geometric morphism followed by an etale one.

###### Proof

Given $f:E\to S$ locally connected, we can factor it as $E\to S/{f}_{!}\left(*\right)\to S$. The second map is etale by definition, while the first is locally connected (the left adjoint is essentially ${f}_{!}$ again) and connected since it preserves the terminal object (by construction).

In particular:

• (Connected, Etale) is a factorization system on the 2-category $\mathrm{LCTopos}$ of toposes and locally connected geometric morphisms.

• The category of etale geometric morphisms over a base topos $S$, which is equivalent to $S$ itself, is a reflective subcategory of the slice 2-category $\mathrm{LCTopos}/S$. The reflector constructs ”${\Pi }_{0}$ of a locally connected topos.”

These results all have generalizations to ∞-connected (∞,1)-toposes.

## Examples

###### Proposition

The gros sheaf topos $\mathrm{Sh}\left(\mathrm{CartSp}\right)$ on the site CartSp – which contains the quasi-topos of diffeological space – is a connected topos, since the site CartSp is a locally connected site and contains a terminal object.

###### Proposition

Let $\Gamma :ℰ\to \mathrm{Set}$ be a connected and locally connected topos and $X\in ℰ$ a connected object, ${\Pi }_{0}\left(X\right)\simeq *$. Then the over-topos $ℰ/X$ is also connected and locally connected.

###### Proof

For every object $X$, we have that $ℰ/X$ sits over $ℰ$ by the etale geometric morphism.

$ℰ/X\stackrel{\stackrel{{X}_{!}}{\to }}{\stackrel{\stackrel{{X}^{*}}{←}}{\underset{{X}_{*}}{\to }}}ℰ\stackrel{\stackrel{{\Pi }_{0}}{\to }}{\stackrel{\stackrel{\Delta }{←}}{\underset{\Gamma }{\to }}}\mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$\mathcal{E}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}} \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} Set \,.

This makes $ℰ/X$ be a locally connected topos.

Notice that the terminal object of $ℰ/X$ is $\left(X\stackrel{\mathrm{Id}}{\to }X\right)$. If now $X$ is connected, then

${\Pi }_{0}{X}_{!}\left(X\stackrel{\mathrm{Id}}{\to }X\right)\simeq {\Pi }_{0}X\simeq *$\Pi_0 X_! (X \stackrel{Id}{\to} X) \simeq \Pi_0 X \simeq *

and so the extra left adjoint ${\Pi }_{0}\circ {X}_{!}$ preserves the terminal object. By the above proposition this means that $ℰ/X$ is also connected.

Revised on November 8, 2012 12:43:58 by Urs Schreiber (82.169.65.155)