nLab
connected topos

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:EFp\colon E\to F is a “relativized” notion of this, saying that EE is “connected as a topos over FF.”

Definition

Connected geometric morphism

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

A Grothendieck topos EE is connected if the unique geometric morphism ESet=Sh(*)E \to Set = Sh(*) is connected. If EE is the topos of sheaves on a topological space XX (or more generally a locale), then this is equivalent to the usual definition of connectedness for XX (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:EFp\colon E\to F is locally connected, then it is connected if and only if the left adjoint p !p_! of the inverse image functor (which exists, since pp is locally connected) preserves the terminal object.

Proof

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

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

p !(p *(A))p !(p *( A*)) Ap !(p *(*)) A*A p_!(p^*(A)) \cong p_!(p^*(\coprod_A *)) \cong \coprod_A p_!(p^*(*)) \cong \coprod_A * \cong A

Thus, the counit p !p *Idp_! p^* \to \Id is an isomorphism, so p *p^* is fully faithful.

When FF is not SetSet, we just have to replace ordinary coproducts with “FF-indexed coproducts,” regarding EE and FF as FF-indexed categories.

(This is C3.3.3 in the Elephant.)

Strengthenings of this condition include * a strongly connected geometric morphism, which is locally connected such that p !p_! preserves all finite products, and * a totally connected geometric morphism, which is locally connected such that p !p_! preserves all finite limits.

Connected locally connected sites

Proposition

If CC is a locally connected site with a terminal object, then the topos of sheaves Sh(C)Sh(C) on CC is (not just locally connected) but connected.

Proof

As explained at locally connected site, when CC is locally connected, the left adjoint Π 0:Sh(C)Set\Pi_0\colon Sh(C) \to Set is simply obtained by taking colimits over C opC^{op}. Now by the co-Yoneda lemma, the colimit over any representable presheaf is a singleton (i.e. a terminal object in Set):

lim y(V)= UCC(U,V)= UCC(U,V)*=*. \lim_\to y(V) = \int^{U \in C} C(U,V) = \int^{U \in C} C(U,V) \cdot * = * \,.

But if CC has a terminal object, then that terminal object represents the terminal presheaf, which is also the terminal presheaf. Hence under these conditions, Π 0\Pi_0 preserves the terminal object, so Sh(C)Sh(C) is connected.

Properties

Orthogonality

Proposition

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

Proof

Since the functor Topos opCatTopos^{op} \to 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 ToposTopos.

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

A f B p q C g D \array{ A & \xrightarrow{f} & B \\ {}^{\mathllap{p}}\downarrow & & \downarrow^{\mathrlap{q}} \\ C & \xrightarrow{g} & D}

of geometric morphisms in which pp is connected and qq is etale, there exists a filler h:CBh\colon C\to B such that hpfh p \cong f and qhgq h \cong g.

However, if XDX\in D is such that BD/XB \cong D/X (such exists by definition of qq being etale), then for any topos EE equipped with a geometric morphism k:EDk\colon E\to D, lifts of kk along qq are equivalent to morphisms *k *(X)* \to k^*(X) in CC. In particular, ff is determined by a map *f *(q *(X))p *(g *(X))*\to f^*(q^*(X)) \cong p^*(g^*(X)), and since *p *(p *(*))* \cong p^*(p_*(*)) and p *p^* is fully faithful, this map comes from a map *g *(X)*\to g^*(X) in CC, which in turn determines a geometric morphism h:CBh\colon 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:ESf\colon E\to S locally connected, we can factor it as ES/f !(*)SE \to S/f_!(*) \to S. The second map is etale by definition, while the first is locally connected (the left adjoint is essentially f !f_! again) and connected since it preserves the terminal object (by construction).

In particular:

  • (Connected, Etale) is a factorization system on the 2-category LCToposLCTopos of toposes and locally connected geometric morphisms.

  • The category of etale geometric morphisms over a base topos SS, which is equivalent to SS itself, is a reflective subcategory of the slice 2-category LCTopos/SLCTopos/S. The reflector constructs “Π 0\Pi_0 of a locally connected topos.”

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

Examples

Proposition

The gros sheaf topos Sh(CartSp)Sh(CartSp) 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 Γ:Set\Gamma : \mathcal{E} \to Set be a connected and locally connected topos and XX \in \mathcal{E} a connected object, Π 0(X)*\Pi_0(X) \simeq *. Then the over-topos /X\mathcal{E}/X is also connected and locally connected.

Proof

For every object XX, we have that /X\mathcal{E}/X sits over \mathcal{E} by the etale geometric morphism.

/XX *X *X !ΓΔΠ 0Set. \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 /X\mathcal{E}/X be a locally connected topos.

Notice that the terminal object of /X\mathcal{E}/X is (XIdX)(X \stackrel{Id}{\to} X). If now XX is connected, then

Π 0X !(XIdX)Π 0X* \Pi_0 X_! (X \stackrel{Id}{\to} X) \simeq \Pi_0 X \simeq *

and so the extra left adjoint Π 0X !\Pi_0 \circ X_! preserves the terminal object. By the above proposition this means that /X\mathcal{E}/X is also connected.

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