nLab
locally connected site

Context

Topos Theory

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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 CC be a small site; we say it is a locally connected site if all covering sieves of any object UCU\in C are connected, as full subcategories of the slice category C /UC_{/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 CC is locally connected, then every constant presheaf on CC 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 CC is a locally connected site, then the sheaf topos Sh(C)Sh(C) is a locally connected topos.

This means that the inverse image functor LConst:SetSh(C)L Const\colon Set \to Sh(C) has a left adjoint Π 0\Pi_0.

Proof

By remark 1 it follows that the constant presheaf functor Const:SetPsh(C)Const \colon Set \to Psh(C) has a left adjoint given by taking colimits along C opC^{op} (this is one of the equivalent definitions of the colimit operatiion.) Since constant presheaves on CC are sheaves, LConstL Const is just a factorization of ConstConst through Sh(C)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 XSh(C)X \in Sh(C) any sheaf, we may write it, using the co-Yoneda lemma as a coend over representables

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

The left adjoint functor Π 0\Pi_0 commutes with the coend and the tensoring in the integrand to produce

Π 0(X)= UCX(U)*=colim UX*. \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 XX.

Proposition

If CC furthermore has a terminal object, then colimits along C opC^{op} preserve the terminal object, so that Sh(C)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 UU is generated by a single arrow p:VUp:V\to U, then pp is a weakly terminal object? of the sieve (qua full subcategory of C/UC/U), so the sieve is connected.

and

Revised on June 20, 2013 12:54:12 by Urs Schreiber (82.169.65.155)