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

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 toposes from sites

We discuss that the sheaf toposes over locally connected sites are locally connected toposes.

Remark

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

The fact that all covering families are inhabited makes the constant presheaves be separated presheaves (see this example) and then the connectedness condition further makes them be sheaves.

Proposition

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

This means that the inverse image functor $L Const\colon Set \to Sh(C)$ has a left adjoint $\Pi_0$.

Proof

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

Proposition

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

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

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

The left adjoint functor $\Pi_0$ commutes with the coend and the tensoring in the integrand to produce

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

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

Proposition

If $C$ furthermore has a terminal object, then colimits along $C^{op}$ preserve the terminal object, so that $Sh(C)$ 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