nLab
locally connected site

Idea

A locally connected site is a site satisfying sufficient conditions to make its topos of sheaves into a locally connected topos.

Definition

Let $C$ be a small site; we say it is a locally connected site if all covering sieves of any object $U\in C$ are connected, as full subcategories of the slice category $C/U$. (In particular, this means that all covering families are inhabited.)

Locally connected topoi from sites

Proposition

If $C$ is a locally connected site, then the sheaf topos $\mathrm{Sh}(C)$ is a locally connected topos.

This means that the inverse image functor $L\mathrm{Const}:\mathrm{Set}\to \mathrm{Sh}(C)$ has a left adjoint ${\Pi }_0$.

To see this, notice

Observation

If $C$ is locally connected, then every constant presheaf on $C$ is a sheaf.

Using this, we have

Proof

The constant presheaf functor $\mathrm{Const}:\mathrm{Set}\to \mathrm{Psh}(C)$ has a left adjoint given by taking colimits along $C^{\mathrm{op}}$ (this is one of the equivalent definitions of the colimit operatiion.) Since constant presheaves on $C$ are sheaves, $L\mathrm{Const}$ is just a factorization of $\mathrm{Const}$ through $\mathrm{Sh}(C)$, and thus it also has a left adjoint given by the colimit operation.

Observation

The colimit over a representable functor is always the singleton set.

So for $X\in \mathrm{Sh}(C)$ any sheaf, we may write it, using the co-Yoneda lemma as a coend over representables

$$X={\int }^{U\in C}X(U)\cdot U.$$X = \int^{U \in C} X(U) \cdot U \,.

The left adjoint functor ${\Pi }_0$ commutes with the coend and the tensoring in the integrand to produce

$${\Pi }_0(X)={\int }^{U\in C}X(U)\cdot *={\mathrm{colim}}_{U\to X}*.$$\Pi_0(X) = \int^{U \in C} X(U) \cdot {*} = colim_{U \to X} {*} \,.

We may think of this as computing the set of plot-connected components of $X$.

Observation

If $C$ furthermore has a terminal object, then colimits along $C^{\mathrm{op}}$ preserve the terminal object, so that $\mathrm{Sh}(C)$ is moreover a connected topos.

Note that a non-locally-connected site can still give rise to a locally connected topos of sheaves, but every locally connected topos can be defined by some locally connected site.

Examples

• any small subcategory of Top on connected topological spaces (with the standard open cover coverage).

• CartSp

• Any site whose topology is generated by a singleton pretopology, i.e. a Grothendieck pretopology in which all covering families consist of single arrows. For if a covering sieve on $U$ is generated by a single arrow $p:V\to U$, then $p$ is a weakly terminal object? of the sieve (qua full subcategory of $C/U$), so the sieve is connected.

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site / ∞-cohesive site